The following selections are masterpieces in
differential equations
from a variety of sources. They are related to geometry or physics in some way.
Some of them are difficult to read. However, it is rewarding if you understand
them eventually.
Sturm's theorems on the Zeros
of Solutions of Second-order Equations [Arn1,
§27.7].
Differential calculus.
Preliminaries.
The vector product
Remark 1. Modern textbooks often ignore the definiteness of the vector product. [Wea1, p.3, l.18-p.4, l.13] provides a careful treatment.
Remark 2. The concept of moment originates from physics. Without the help of physical terms and a picture
[Kara, p.7, Fig. 1.6], it is difficult to understand the abstract definition
given in [Wea1, p.5, l.5-l.7].
Taylor's formula with integral remainder [Tay,
p.112, Theorem II].
Differentiating integrals
(Finite integrals
with respect to real t, real z): [Wid, p.350, Theorem 8]
® [Wid, p.352, Theorem 10] ®
[Sak, p.45, Theorem 1.5].
Remark. [Wid, p.353, Theorem 11; Example B] are corollaries of the above theorems.
(Infinite integrals with respect to real t, real z): [Wat1, §4.44, p.73, (I); p.74, Corollary].
(Finite integrals with respect to real t, complex z): [Wat1, p.92,
§5.31]. The proof of the complex case [Wat1, p.92,
l.13-l.19] is the same as the
proof of the real case [Wid, p.352, Theorem 10].
(Infinite integrals with
respect to real t, complex z): The case of infinite integrals [Gon, pp.526-527, Theorem
8.4] is derived from the case of finite integrals [Wat1, p.92,
l.13-l.19] by means of [Ru2, p.230,
Theorem 10.28]. By using [Ru2, p.27, Theorem 1.34], the statement given
in [Gon, pp.526-527, Theorem 8.4] allows González
to eliminate the superfluous hypotheses of the theorem
given in
[Wat1, p.92, §5.32]. The proof of [Gon,
pp.526-527, Theorem 8.4] is simpler than both the proof of [Lang1, p.270,
Lemma] and the proof of [Wat1, p.74, Corollary]. See also [Ru2, pp.246-247,
Exercise 16].
(Integrals on arcs in a complex plane, complex z): [Ahl, p.121, Lemma 3].
(Using a limiting procedure when the integrand is singular): [Jack, p.35, l.2-l.10].
Remark. In order to prove that the improper integral given in [Wat1, p.502,
l.10] is analytic in k, we may use the method of complex analysis [Con, p.177,
Lemma 7.16; p.178, p.178, Proposition 7.17; Ru2, p.230, Theorem 10.28] or the
method of real analysis [Ru2, p.246, Exercise 16]. The purpose of transforming
the left-hand side of the equality given in [Wat1, p.502, l.10] to the
right-hand side is to move the variable k from the path of integration to the
integrand.
The inverse function theorem [Spi1, p.35, Theorem.
2-11 [1]; functions in implicit form: Cou2, vol.
2, p263, l.-16-l.-8]; local linearlization of
differentiable mappings [Spi, vol.1, p.56, Theorem 9; p.59, Theorem 10]; the
implicit function theorem [Wid, p.56, Theorem 14; p.59, Theorem 16].
Remark 1. One may understand the essence of the implicit function theorem more
easily from the proofs of [Wid,
p.56, Theorem 14; p.59, Theorem 16]. These two theorems are clear and useful
versions of the implicit function theorem. Other versions are either too general to
highlight the key point, too abstract to help readers visualize the picture, or
too complicated to apply to practical problems.
Remark 2. The proof of [Kli,
p.6, Theorem 0.5.2] follows the model given in [Kli, p.6, l.11-l.20]. However.
in the proof of [Kli, p.6, Theorem 0.5.2], Klingenberg uses many identification
maps, but fails to define them explicitly. The proofs given in [Spi, vol.1, p.56,
Theorem 9; p.59, Theorem 10] are clearer and more accessible.
Remark 3. Before stating the implicit function theorem, Courant provides stimulating examples
to explain why he formulates the theorem as he does [Cou2, vol. II,
pp.219-221, §3.1.b]. After stating the
implicit function theorem, Courant shows us how we use this theorem to solve
practical problems [Cou2, vol. 2, pp.221-225, §3.1.c].
Motivation and application are integral parts of understanding a
mathematical theorem, but are often neglected in our learning process.
Remark 4. The key point of the inverse function [Cou2, vol. 2, p.262, l.2-l.5] is to solve x [x=X(u,y)] using the equation
[Cou2, vol. 2, p.261, (33a)(i)], and then solve y using the equation obtained by
substituting the expression x=X(u,y) into [Cou2, vol. 2, p.261, (33a)(ii)].
The purpose of mathematics is to teach students to observe, to think, and to solve problems,
not to engage in insignificant generalization of theorems.
Remark 5. To prove the theorem of the decomposition into primitive mappings,
Courant does nothing but make an insightful
side comment
[Cou2, vol. 2, p.264, l.-14-l.-9]
on his proof of the inverse function theorem.
In contrast, Rudin builds complicated machinery to prove this decomposition
theorem as a theorem separate from the inverse function theorem [Ru1, p.199, l.-9-p.201, l.6]. In my opinion,
c(x,h)
in [Cou2, vol. 2, p.264, (34q)] is more easily visualized than g_{m}^{-1}(y)
in [Ru1, p.200, (51)]. The purpose of mathematics is to help students
understand a theorem's
insight rather than bury it in complicated but insignificant structures.
Remark 6. The inverse function theorem is essentially a topological theorem. See
[Mun00, p.383, Theorem 62.3]. Remark 7. [Gon, p.543, Theorem 8.11] provides
the most effective method of finding inverse functions [1].
Area of a curved surface (in the form
of z = f(x,y) [Cou1, vol. 2, p.424, (29a); in the parametric form [Cou2, vol. 2,
p.428, (30c)]).
Curves in parametric form C(t) = (x(t), y(t)) [Cou2, vol. 1, p.339, l.16-p.346, l.-1].
Curves in implicit form F(x,y)=0 [Cou2, vol. 2, pp.230-235,
§3.2.a].
Curvature
(1). Definition [Cou2, vol. I, p.354, (14)].
(2). Curves in parametric form C(t) = (x(t), y(t)) [Cou2, vol. 1, p.355, (15)].
(3). Curves represented by functions y = f(x) [Cou2, vol. 1, p.357,(16)].
(4). Curves in implicit form F(x,y)=0 [Cou2, vol. 2, p.232, (14a)].
How we determine the sign of curvature
(1). Curves in parametric form C(t) = (x(t), y(t)) [Cou2, vol. 1, p.355, l.18-p.358, l.14].
(2). Curves in implicit form F(x,y) = 0 (a). The curve is given by F(x,y) = 0. (b).
The normal [Cou2, vol. 2, p.231, (12c)] points to the region F > 0, so
the normal determines the region F > 0. (c). If
the tangent to the curve near the point of contact lies in the region F
³ 0, then the curvature k is positive [Cou2, vol. 2, p.232, l.-3-l.-1].
(1). The oriented area within a simple closed oriented curve [Cou2, vol.
1, p.365, (20)].
Application in thermal physics [Cou2, vol. 1, p.419, l.19-l.18]. Example
[Hua, p.11, Fig. 1.4].
(2). Areas bounded by closed curves [Cou2, vol. 1, pp. 430-435,
§A.2].
Remark. According to [Wea1, vol. 1, p.12, Fig. 2], the direction of the
principal normal = the direction of the tangent + p/2.
If a curve is represented in parametric form, the normal direction is
defined as above [Cou2, vol. 1, p.346, l.-11-l.-5].
However, if a curve is represented in implicit form F(x,y) = 0, then the
direction of the principal normal = the direction of the tangent
- p/2 [Cou2, vol. 2,
p.231, (12c) & (12d)]. In this case, the normal direction coincides with the
direction of grad F. This direction enables us to determine which region satisfies F > 0
[Cou2, vol. 2, p.232, l.1-l.3].
Orientable surfaces. Orientation of an ordered basis [Cou2, vol.2,
p.196, (82b)]
® orientation of an ordered set of
independent vectors [Cou2, vol.2, p.200,
l.5] (the positive orientation agrees with the usual sense of counterclockwise
rotation when we view the ordered set from the positive side of an oriented
plane [Cou2, vol.2, p.200, l.-13-p.201, l.19])
® A surface oriented positively with
respect to the coordinate axes [Cou2, vol.2, p.578, l .-5]
(a region [Cou2, vol.2, p.580, l .10; a simple surface: p.634, l.-12] or a coordinate patch with parameters u, v [Cou2, vol.2,
pp.580-581, (40n, o, p, s)]) ® orientable surfaces ([O'N,
p.178, Theorem 7.5]; methods of realization [Cou2, vol.2, p.585, (41a) &
p.586, (41e); elementary surfaces in the n-dim Euclidean space: p.646, l.21]).
Gauss' theorem [Cou1, vol.2, pp.359-365]. Versions: Cartesian coordinates
[Cou1, vol.2, p.364, l.13], the vector form [Cou1, vol.2, p.364, l.17] and the
differential form [Cou2, vol.2, p.545, (2a)] (the differential form allows us
to use diffeomorphisms [Arn1, p.61, l.-6] to select
a coordinate system that is convenient for calculating the contour integral [Cou2, vol.2, p.550,
l.-1]) . Remark 1. In the 2-dim case, both the
divergence theorem and Stokes' theorem reduce to Gauss' theorem [Cou1, vol.2,
p.364, l.17; p.365, l.-14]. Remark 2. Written
in their differential forms, the divergence theorem and Stokes' theorem are
similar in the 3-dim case (Compare [Cou2, vol.2, p.601, (53) with [Cou2,
vol.2, p.612, (74)]).
The divergence theorem [Cou1,
vol.2, pp.385-387]. Versions: Cartesian coordinates [Cou1, vol.2, p.386,
l.10] (good for tensor analysis [Lan7, p.5, (2.2)]), the vector form [Cou1, vol.2,
p.387, l.-11] (good for geometric
interpretation).
Remark 1. The proof given in [Wangs, pp.21-24, §1-14] features an
unambiguous definition [Wangs, p.17, l.-8-l.-4]
of the vector element of area on a closed surface. [Wangs] uses the divergence theorem
to prove [Wangs, p.24, (1-66); p.35, (1-122) &
(1-123)].
Remark 2. The proof of the divergence theorem given in [Fan,
§2.6] is better than that given in [Wangs,
§1-14] because the former proof highlights
the key geometric idea and is independent of coordinate systems.
Stokes'
theorem [Cou1, vol.2, pp.392-396]. Remark 1. The idea of
its proof is based on the plane version of Gauss' theorem (A curved surface
can be approximated by inscribed polyhedrons (Warning [Cou2, vol.2, p.421, l.-10-p.422,
l.16]) or transformed by a coordinate patch [Cou2, vol.2, p.612,
l.-6-p.613, l.12].)
Remark 2. The proof given in [Wangs, pp.24-26, §1-15]
features an unambiguous definition [Wangs, p.17, l.17-l.23] of the vector
element of area on an open surface. [Wangs] uses Strokes' theorem to prove [Wangs,
p.27, (1-73); p.35, (1-124)].
Remark 3. In general, the geometric interpretation in [Kara, Chap. 4] is
clearer and more organized than that in [Fan, chap. 2]. However, the
definition given in [Fan, p.64, (2.101)] is simple and natural, while the
definition given in [Kara, p.68, (4.19)] is artificial. Furthermore, the proof
of [Kara, (4.19) Þ (4.25)] is much more complicated
than the proof of [Fan, (2.101) Þ (2.126)].
Consequently, Fano's approach makes it easier to deal with curl.
Generalization of a, b, c and d: [Spi1, p.124, l.-7-p.125, l.-3].
The integral solution of the Poisson equation [Kara, p.100, (5.75); Chou, p.31, l.-3-p.32,
l.9].
Remark. Jackson makes a serious mistake in [Jack, p.35, l.-11]
because he tampers with the given assumption. The
value of Ñr is given and Jackson does not
have the authority to make it equal to 0. The R that Jackson chooses in [Jack,
p.35, l.-17] can make r small, but cannot affect the
given value of
Ñr.
E = -ÑF Û Ñ´E = 0
in a simply connected domain.
Proof. Þ: Ñ´Ñu = 0 [Wangs, p.15, (1-48)].
Ü: Let
F (x) = ò_{C} E_{1}dx_{1}+E_{2}dx_{2}+E_{3}dx_{3},
where C begins at a fixed point x_{0} and ends at x. By Stokes'
theorem, is independent of the contour of integration.
Its physical meaning is given by [Wangs, pp.37-38, §1-20].
Other related comments [Kara, pp.94-95, §5.9].
Remark. Since [Chou, p.584, (I.78)] is similar to [Jack, p.30, (1.17)] and
[Chou, p.584, (I.79)] is similar to [Jack, p.180, (5.28)], the Helmholtz
theorem is essentially the combined statement of a and b. The construction
given in the proofs of the Helmholtz theorem uses Green's function [Jack, p.36, l.-7]
and has physical meanings, so the
proofs can be used to provide better proofs of a and b even though
slight restrictions are imposed on the
hypothesis of the Helmholtz theorem.
Formulas.
Laplacian.
spherical coordinates (various proofs: 1. direct
substitution using [Lev2, p.107, (5.62)-(5.64)]. 2. invariant character of the Euler equations
[Cou, vol.1, p.225, (63)]. 3. The divergence theorem [Cou2, vol.2, p.610,
(69)].)
Two remarks on [Wangs, p.34, (1-121)]
The complex expression of mechanical operations in [Wangs, p.34, (1-121)]
impedes intuitive understanding.
(A·Ñ) B
= (A_{x}
¶/¶x+ A_{y}
¶/¶y+ A_{z}
¶/¶z) (B_{x}, B_{y}, B_{z})
= ((A_{x}¶/¶x+ A_{y}¶/¶y+ A_{z}¶/¶z)B_{x},
(A_{x}¶/¶x+ A_{y}¶/¶y+ A_{z}¶/¶z)B_{y},
(A_{x}¶/¶x+ A_{y}¶/¶y+ A_{z}¶/¶z)B_{z})
(the operator applies to each component). This interpretation can be
summarized in a shorter form: [(A·Ñ)
B]_{i} = A·(ÑB_{i}).
If B = f v, where v is a constant vector, then
(A·Ñ)(f
v) = [A·Ñf
]v. This statement is equivalent to (i).
Remark 1. The proofs of most of the important formulas in differential and integral calculus can be found in [Wangs,
pp.1-38, chap. 1]. Since these proofs are based on physical
considerations, they are the most effective proofs.
Remark 2. For cylindrical coordinates, we use row two of [Wangs, p.29, (1-76)] to express
Ñ×A [Wangs, p. 31, (1-87)] and use row three of [Wangs,
p.29, (1-76)] to express Ñ´A [Wangs, p.31,
(1-88)]. We use the same idea to express
Ñ×A and Ñ´A for Cartesian coordinates and spherical coordinates.
Thus, we recognize a pattern that only through geometric considerations can
we find the most effective method to prove [Wangs, p.31, (1-87) & (1-88)].
Calculus of variations
Fermat's principle (the principle of least action) is a mathematical
theorem [Fomi, p.84,
Theorem]. Remark 1. The proof of [Fomi, p.84, Theorem] shows that Fermat's
principle is nothing but the equation of motion disguised in an artificial form.
Consequently, the
discussion given in [Fur, p.14, l.12-l.13] merely reveals Furtak's ignorance.
Philosophy serves to clarify a picture through a thorough study rather than
mystify the picture.
Remark 2. (Stationary points) The case of the global minimum and that of a
local minimum in a discussion of Fermat's principle are not distinguishable
unless the discussion includes both [Hec, p.106, Fig. 4.28] and [Born,
p.130, Fig. 3.12]. The case of local maximum and that of local minimum in a
discussion of Fermat's principle are not distinguishable unless the discussion includes [Hec, p.110, r.c.,
l.11-l.28]. Only after we distinguish among the above cases are we able to
find all possible cases for stationary points. Only after we study [Born,
p.730, (58); p.731, (60)] are we able to clarify
the relationship between Fermat's principle [Born, p.128, l.-16-l.-12]
and the weaker form of Fermat's principle [Born, p.129, l.1].
The modern formulation of Fermat's principle [Hec, p.109]:
y is an extremal of the functional J[y]
Þ
the variational derivative of J[y] vanishes [Fomi, p.28, l.-10; p.35, Theorem].
For physical applications, it is convenient to reduce the variational derivative to the ordinary derivative by using the
linking device
dy = eh(x) [Cou, vol. 1, p.184, l.20].
Remark. Due to the absence of the above crucial link between the variational
derivative and the ordinary derivative, the application given in [Fur, p.11,
l.13] fails to closely connect to its rigorous theoretical ground and the theoretical formula given in [Fomi, p.29, l.-10]
fails to appear in a form ready for useful applications.
The proof of [Fomi, p.15, Theorem 1] breaks the elegance of the proof given in [Cou, vol. 1, chap IV,
§3.1] into pieces and makes the essence
of the proof unrecognizable.
[Born, Appendix I], [Fomi] and [Sag]
The strong minimum in [Born, p.730, l.12]. the weak minimum in [Born,
p.731, l.8]. and the minimum in [Fomi, p.12, l.-5-p.13,
l.1; §24,
Theorem 1 & Theorem 2] all refer to a local minimum. In calculus of variations,
we are not interested in the global minimum. The precise definitions of a
strong extremum and a weak extremum are given by [Fomi, p.13,
l.4-l.12].
We can use [Born, p.719, (2)] and [Fomi, p.9, Lemma 1] to prove [Born,
p.720, (7a) & (7b)].
The statement given in [Born, p.135, footnote *] is proved in [Fomi,
p.72, (16)].
[Born, Appendix I, §§2-3],
[Fomi, §33] and [Sag,
§3.2; p.141, (3.3.2)] all discuss Hilbert's invariant integral.
The discussion given in [Born, Appendix I, §§2-3]
is the most elegant. The field given in [Born, Appendix I, §3]
refers to the field defining y' = v [Arn1, p.28, l.9-l.-10]
to which Euler's equations can be reduced.
The definition of fields given in [Fomi, p.132, Definition 1 & Definition 2]
is too complicated.
"Let P_{1}Q be any extremal" in [Born,
p.731, l.-8] should have been "Let P_{1}Q be any
curve".
Sturm-Liouville Systems [Bir, chap. 10 & chap. 11].
Regular systems:
Eigenfunctions that have different eigenvalues are orthogonal.
Singular systems: Square-integrable eigenfunctions
u and v
that belong to different eigenvalues are orthogonal [Bir, p.264, Theorem
2] if u and v satisfy the boundary condition of [Bir, 264, (12)]. Remark. In [Bir,
chap. 10,
§4, Example 5, Example 6 and
Example 7], Birkhoff only shows that certain values are eigenvalues.
Expanding on this idea, [Cou, vol.1, chap. V, §10]
explains why other values cannot be eigenvalues.
Sturm Comparison Theorem [Bir, p.268, Theorem 3].
The sequence of eigenfunctions [Bir, p.273, Theorem 5].
The continuous spectrum [Bir, p.291, Theorem 13; p.292, Corollary 2].
The discrete spectrum [Bir, p,292, Theorem 14].
Completeness of eigenfunctions [Bir, p.313, Theorem 11].
Applications to quantum mechanics.
[Lan3, p.60,
l.-7-l.-4, the oscillation theorem] can be easily derived from
[Bir, p.268, Theorem 3 or p.273, Theorem 5] (Although these two
theorems require that [a,b] be a finite interval, their
proofs are still valid if the endpoints of the interval are
±¥). However, using a general theorem to
prove a statement often blurs the essence of the statement.
The following more direct proof clearly reveals the essence of
the oscillation theorem.
Proof.
Let a and b are consecutive zeros of y_{n}. Say y'_{n}(a)>0
and y'_{n}(b)<0.
If y_{n+1}>0 in (a,b),
then RHS<0, but LHS>0.
If y_{n+1}<0 in (a,b),
then RHS>0, but LHS<0.
Hence y_{n+1} change sign in
(a,b).
To view a concept from a single perspective is often not enough. To gain a complete picture, we must
view the concept from a complementary perspective. Example (Conjugate points).
[Fomi, p.106, Definition] and [Bir, p.52, l.17-l.28].
Green's functions.
The physical origin: [Arf, p.511, l.8].
The general definition of Green's function ([Arf, p.512, (8.159)
® [Mor,
part 1, p.884, (7.5.37)] ® [Ru3, p.192,
l.-5]).
(Main usage) Expressing the solution of a nonhomogeneous self-adjoint differential equation in terms of Green's function [Arf, p.514, l.-11-l.-6].
(Existence) [Ru3, p.195, Theorem 8.5].
(Symmetry [Jack, p.40, l.9-l.12])
Proof. Let G_{0}(r)= (4pr)^{-1}, where r =
|x-x_{0}|^{-1}.
f(x) = lim _{d®0 }ò _{|x-x0|
= d }f (y)(¶G_{0}/¶n) dy.
Let u(x) = G(a,x), v(x) = G(b,x), and W' = W
- {x: |x-a|
= e or |x-b|
= e }.
0 = ò_{ W}' (uÑ^{2}v-vÑ^{2}u) dx
= ò_{ ¶W} (u(¶v/¶n-v(¶u/¶n)dx +A_{e}+B_{e},
where A_{e} = - ò_{ |x-a|
= e } (uÑ^{2}v-vÑ^{2}u) dx
and B_{e} = - ò_{ |x-b|
= e } (uÑ^{2}v-vÑ^{2}u) dx.
lim _{e®0 }A_{e} = ò_{ |x-a|
= e }(G_{0}(|x-a|)
+ H(x))(¶v/¶n)-v(¶(G_{0}(|x-a|)
+ H(x))/¶n)dx = -v(a)
because the contributions from harmonic H are 0. Similarly, lim _{e®0 }B_{e} = u(b).
Since ò_{ |x-a|
= e }G_{0}(|x-a|)(¶v/¶n)dx®0
as e®0, v(a) = u(b).
(Forms) Expressing Green's function in terms of
a complete set of eigenfunctions [Mor,
part 1, p.884, (7.5.39); the Fourier transform: Arf, (8.213) & (8.218)].
distributions [Ru3, p.378, l.-6].
(Effective methods of finding the Green function for a boundary-value problem):
The method of images (for planes and spheres) [Wangs, §11-2].
The Green function expansion in spherical coordinates using the
basis of spherical harmonics (for shells) [Jack, §3.9].
The Green function expansion in cylindrical coordinates using the
basis of modified Bessel functions [Jack, p.126, (3.145) & (3.149)].
The Green function expansion using the
basis of eigenfunctions [Jack, p.128, (3.160)].
(Addition theorem for spherical harmonics)
[Jack, pp.110-111, §3.6], [Coh, pp.688-689]
and [Mer, pp.250-251] all prove the addition theorem for spherical harmonics.
Among the three proofs, Jackson's is the most concise
and Cohen-Tannoudji's is the easiest to understand. Merzbacher's is
unnecessarily complicated. The simplicity of Jackson's proof is due
to the use of [Jack, p.109, (3.59)]. The key idea behind this
simplification is to eliminate the terms containing m¹0 when we only need to consider the case
where q=0. [Jack,
p.109, (3.59)] can be proved using [Mer, p.251, (11.98)] or [Coh, p.682, (30)].
[Jack, p.242, Fig. 11.3] helps explain the meanings of b
and g in [Mer, p.111, (3.67)]. The proof in [Guo,
pp.241-243, 5.14] is based on the same idea as Jackson's proof. For
the first part of the proof, the
explanation in [Jack, p.110, l.-17-l.-3]
is better than [Guo, p.242, l.-6-p.243,
l.8]. However, for the second part of the proof, the explanation in [Guo, p.243, l.9-l.-1]
is better than [Jack, p.111, l.1-l.7].
The Mittag-Leffler theorem. To emphasize the purpose
of the Mittag-Leffler theorem, [Guo, pp.17-20] shows that the theorem is an expansion theorem for meromorphic
functions. However, from [Ahl, p.185, Theorem 4] the reader cannot obtain the
required expansion except with luck. This is because g in [Ahl, p.185, l.-1]
is undetermined. In other words, Ahlfors leaves the problem unsolved.
Actually, the main part of Ahlfors' version of the Mittag-Leffler theorem turns
out to be trivial for all the applications given in [Ahl, pp.187-188]. The
significance of the Mittag-Leffler theorem lies in the specification of g in [Ahl,
p.185, l.-1]. Thus, the way Ahlfors formulates
the theorem shows that he only preserves its trivial part, leaving out
its essential part. As for other authors of texts on complex variables, their
formulations only keep the first part of [Ahl, p.185, Theorem 4] without
considering the origin of its second part. This shows that these authors do not even know the purpose of the Mittag-Leffer
theorem. For example, see [Ru2, p.291, Theorem 13.10]. In addition, the proof of
[Ru2, p.291, Theorem 13.10] uses Runge's theorem, which is quite complicated. In order to
facilitate understanding the essence of the Mittag-Leffer theorem, we should
avoid such an unnecessary complication.
The proof of [Ahl, p.185, Theorem 4]
[1]
is effective because we may easily specify n_{n }
[Ahl, p.186, l.16]. If b_{n}
and M_{n} are given, we may select n_{n}
such that
lim_{n®¥}
M_{n}^{1/nn
}/|b_{n}|=0
[Ahl, p.186, l.15]. For example, we may choose n_{n}
> log M_{n} [Ahl, p.186, l.16].
In contrast, it takes some effort to see that the proof of [Gon1, p.286, Theorem
4.3] is also effective. This is because González's formulation fails to provide
the tracking information necessary to find b_{n}
[Gon1, p.287, l.1]. Based on the definition of uniform convergence, it seems
that the existence of b_{n}
[Gon1, p.287, l.1] is hypothetical. However, if we investigate further and examine
the proof of [Con, p.31, Theorem 1.3(c)], we will find that b_{n}
can also be specified once the principal part at a_{n}
is given and the sequence {b_{i}^{(n)
}| i=0, 1, 2, …} in [Gon1, p.286, (4.2-2)] is
thereby completely determined.
Remark. Guo's proof
[Guo, p.18, l.5-p.20, l.9] of the Mittag-Leffler theorem is most effective, but
most of the time all we need to know for application is what the expansion is [Gon1, p.290,
l.13] rather than how we can effectively obtain the expansion. Given a series identity
[Gon1, p.288, l.-7], we may not know from where the
identity comes and we do not want to know how we can effectively construct the
identity from scratch. We simply want to use a theorem's statement to validate
the identity regardless of the effectiveness of the theorem's proof. If this is
the case, then the strategy and the method [Ahl, p.185, l.6-l.-12]
instead of effectiveness become our top priority. The statement given in [Gon1,
p.286, Theorem 4.3] would be sufficient for our purpose. [Gon1,
p.287, l.-3] gives a neat description about the uniform
convergence of [Gon1, p.286, (4.2-1)], while [Guo, p.21, l.-7-l.-6]
does not. Now let us compare the effectiveness of Guo's proof with that of
Gonzalez's proof. [Guo, p.17, (1)] is a strong hypothesis, so {j_{np}
| n =1, 2, …}
[Guo, p.18, (4)] have the same number of terms. In contrast, the number of terms
that p_{n} [Gon1, p.287, l.4] contains may vary
with n. In [Gon1, p.286, l.4], g_{n}
is assumed known. One may wonder how one acquires this information in reality.
The truth is that we deal with sophisticated cases primarily in theory, rarely in applications. The
information is often derived from inspections or other simple means.¬¬
The Weierstrass factorization theorem.
The Weierstrass factorization theorem serves to specify the
expansion of a given meromorphic function in the form of an infinite product [Guo,
pp.25-29, §1.7]. If the condition [Guo,
p.17, (1)] is satisfied, then the required expansion is given by [Guo, p.27,
(5); p.29, (7) & (8)]. In particular, if p=0 in [Guo, p.17, (1)] and we consider
only an entire function with simple zeros, then the expansion will be reduced to a simpler form, [Guo,
p.25, (1)]. The way that Ahlfors and Rudin formulate the Weierstrass factorization theorem
[Ahl, p.194, Theorem 7; Ru2, p.325, Theorem 15.10] indicates that they do not
know its origin.
Remark 1. If we want to expand a given entire function as an infinite product from
scratch, we use [Guo, p.25, Theorem 1]. If the zeros and their corresponding
orders are given and we want to find the function's general form or we want to
prove the validity of a given identity, we use [Gon1,
p.202, Theorem 3.16]. If we use [Guo, p.25, Theorem
1] instead, we have to do extra work: checking whether the given function
satisfies the hypothesis given in [Guo, p.25, l.-10].
For example, [Guo, p.26, l.-10-l.-8;
p.20, Example 1]. ¬¬ Remark 2. If a series is absolutely convergent, we may change the
order of its terms or regroup the terms in any manner without changing the sum [Ru1, p.69, Theorem 3.57]. In contrast, for an infinite product, if it
is absolutely convergent [Wat1, p.33, l.-10] and we
regroup its terms, it may become not absolutely convergent [Wat1, p.33, l.-4].
However, we may restore its absolute convergence by adding some exponential
factors [Wat1, p.34, l.9].
Euler's homogeneous differential equation [Col, pp. 109-111, chap. II,
§6, subsection. 20].
Existence of solutions about ordinary points
When a specific case is not equivalent to the general case, the conditions of
the former are stronger than those of the latter.
If a theorem's assumption changes from the general case to a specific case, we may
find a more effective method to prove the theorem based on newly available sources. If not, it is only because we have not
yet found an advantageous viewpoint. The proof given in [Guo,
§2.2] is designed for the general case: a
normal system of first-order differential equations. The proof given in [Jef,
§16.03] is designed for the specific case: a
second-order DE. Both proofs use the method of successive approximation, but the
latter proof is more effective. The integration by parts given in [Jef, p.475,
l.16] is a wise
move. Via Jeffreys' approach we see that it suffices to prove the convergence of
only one series [Jef, p.476, (13)] rather than two [Guo, p.50, (13)].
Solutions of the DE in the vicinity of a singular point [Guo, p.55, l.5-l.14].
The definition of analytic continuation has many versions, each of which serves
a different purpose [Ru2, chap.16]. When discussing
the solution near a singular point, the most appropriate definition of analytic
continuation is given in [Bir, p.227, l.25]. The definition is clearly
specified: there is no room for another interpretation. For other textbooks, one
often wonders to which definition of analytic continuation they refer.
When
discussing the solution near a singular point, we must study the standard
example first because the properties of the solution are quite similar
to those of the standard example. This important standard example is discussed
in [Bir, p.226, l.8-l.10; p.224, (2')].
The form of A given in [Bir, p.231, l.15] can be derived by direct
observation rather than by solving an algebraic equation [Guo, p.54, (13)].
Generalization often obscures the key idea [Guo, p.51, l.-8].
All we need for an analytic continuation is its simple simulation [Bir, p.227, l.-21]
rather than its formal definition. The failure to give a simple test [Bir, p.223, l.-5]
for a branch point will make the concept of single-valuedness and multi-valuedness very confusing [Cod,
p.108, l.-12 & l.-9].
These are reasons why the proof of [Bir, p.232, Theorem 4] clarifies the key
idea of the method, while [Cod, pp.108-111, chap. 4,
§1] does not.
The fundamental theorem for a regular singular point [Guo, p.56, l.13-l.16].
[Guo, p.58, (17)] produces a better estimate than [Bir, p.241, l.9] (Compare [Guo, p.59, (21)] with [Bir, p.241, l.20]).
The proof of [Bir, p.240, Theorem 6] provides a clearer explanation than [Guo, p.57, l.1-l.3].
[Col, pp.255-257, §15 &
§16] gives a meaningful generalization without losing the big picture.
[Bed, p.389, l.15-l.25] provides a more effective method of solving a
differential equation whose indicial equation has a multiple root.
The conclusions in [Guo, p.60, l.-7] is obtained
from the experience of the calculation in [Bed, pp.369-371,
§115]. On the one hand, a theory is synthesized
from the practical experience of solving various differential equations. On the
other hand, differential equations require a theory as a guide to classify them
into various cases [Bed, §111-§116].
In order to give the coefficients of a power series solution a closed form, we use the method in [Bed, pp.359-363,
§113]. However, if we just want a good
approximation for the solution, we prefer to find the recurrence relation
using the method in [Bed, pp.365-368, §114].
The Frobenius method [Guo, pp.61-63, §2.5]. It is better to use examples
[Bed, §113 & §116] to illustrate the key idea behind the method.
This because there are several problems with formulating the method into an advanced theorem:
A long passage [Guo, p.61, l.12-p.62, l.10] is required just to describe the setting of the theorem.
The argument becomes vague when it comes to a subtle point, see [Guo, p.63, l.6].
Some precious experiences can only be obtained from practical examples, but cannot be stressed in
the form of a theorem
[Bed, p.361, l.-6-l.-5; p.377, l.8-l.10].
Fuchsian equations with three singularities [Bir, pp.251-254, chap. 9,
§13].
[Bir, p.251, l.-14-l.-4] explains why we need to study the
second order Fuchsian DE with three singularities.
If we were to treat ¥ the same as the
finite points in the extended complex plane, the argument and the calculations
involved in proving [Guo, p.69, (3)] would be much simpler. See the proof of [Bir,
p.253, (42)]. Guo first proves [Guo, p.68, (1)] and then lets c
® ¥. Guo's proof is unnecessarily complicated
because he treats ¥ as an exceptional point in the
extended complex plane.
Prove [Wat1, p.207, (II)], where z = (Az_{1}+B)(Cz_{1}+D)^{-1}.
Proof. By [Ru2, p.298, l.10-l.13], it suffices to prove that the equality
given in [Wat1, p.206, l.22-l.24] remains the same form under the linear
fractional transformation z_{1} = z^{-1}.
By the equality given in [Wat1, p.206, l.17],
2z^{-1} + (1-a-a')[az_{1}(z_{1}-a^{-1})]^{-1} + (1-b-b')[bz_{1}(z_{1}-b^{-1})]^{-1} + (1-g-g')[cz_{1}(z_{1}-c^{-1})]^{-1}
= (1-a-a')(z_{1}-a^{-1})^{-1} + (1-b-b')(z_{1}-b^{-1})^{-1} + (1-g-g')(z_{1}-c^{-1})^{-1}.
Remark. The proof given in [Bir, p.251, l.16-l.22] is akaward.
Irregular singularities.
The origin of the form of normal solutions [Guo, p.74, l.4-l.-6].
The Hermite equation [Guo, p.76, (16)] comes from searching for the normal solutions of the Weber equation [Guo, p.75, (11)].
Remark. [Coh, p.536, (15); p.537, (22); p.540, (43); p.541, (51)] provide the physical origin of the Hermite equation, but
[Coh, chap. V] fails to point out the important role that the Hermite equation plays in the normal solutions for
an irregular singular point.
The gamma function [Guo, chap. 3].
[Gon, p.615, Corollary 8.23] has the following two applications:
[Guo, p.95, (10)].
The extension of
G(z) [Gon, p.615, l.-7].
Remark. Guo uses [Guo, p.97, (1)] to extend the domain of the gamma function [Guo,
p.98, l.2], while Jeffreys uses [Jef, p.462, (6)] to extend the domain of the gamma
function. It is easier to calculate the function value of the latter extension
than that of the former one.
Gauss' formula for y(z)
[Gon1, p.318, (4.6-26)].
Remarks. [Gon1, p.318, l.3-l.5] provides a simpler proof of [Gon1, p.318,
(4.6-21)] than does [Wat1, pp.116-117, Example 6]. The proof of the first equality given in [Wat1, p.246, l.-6]
can be found in [Guo, p.98, (4)]. [Ru2, p.33, Exercise 7] shows that Guo's
approach given in [Guo, p.108, l.-9] is incorrect.
The beta function [Gon, pp.615-616, Theorem 8.47].
Gauss' multiplication formula [Gon1, p.304, l.-11-p.307, l.1].
Hankel's formula [Wat1, p.245, l.-11]
The argument given in [Wat1, p.244, l.-8-l.-2]
is consistent with the hypothesis given in [Guo, p.102, l.7] (
| arg (-t)
| < p should have
been | arg (-t)
| £
p), but the argument given in [Guo, p.101, l.-7-l.-1]
is not.¬
Stirling's
formula [Wat1, p.251, l.-6; p.252, l.2 & l.8]
The complete theorem concerning Stirling's
formula is given in [Wat1, p.251, l.-8-p.252, l.10].
The estimate O( |z|^{-2n})
given in [Guo, p.111, l.9] should have been O( |z|^{
-(2n+1)}).
The constant O really depends on n, but the way Guo uses the notation O in [Guo, p.111, (2)]
does not show this dependence. Therefore, [Guo, p.111, (2)] loses significant information contained in
[Wat1, p.252, l.8]. As for [Lang1, p.277, G7] or
[Gon1, p.320, (4.6-32)], they count only the first three terms in the asymptotic
expansion of log G(z). Consequently, these two
versions of Stirling's formula do not live up to the standard of high precision.¬ Remarks. Binet's second formula [Wat1,
p.251, l.-6-l.-5] is
proved in [Guo, p.122, (4)]. The series expansion given in [Wat1, p.251, l.-3]
is proved in [Edw1, p.655, l.-12-l.-6]. [Wat1, p.251, l.-1] is
proved in [Guo, p.120, (6)].
The Hermite formula
[Guo, p.119, (3)] is obtained by changing the integrand in [Guo, p.117, l.-7] to
j(z)[exp(2zi)-1]^{-1} and then changing the contour to
that of its image reflected below the x-axis.
Changing the contour alone cannot produce the desired result. The procedure
stated in [Guo, p.119, l.5] is incorrect.
The Riemann z-function.
The convergence involved in [Guo, p.123, (1)] can be proved by [Sak, p.433, chap. IX, Theorem 8.6(b)].
Remark.
The Bohr-Mollerup Theorem [Con, p.175, Theorem 7.13] and [Gon1, p.311, Theorem
4.14] essentially say that a function's Laurent expansion [Con, p.105, l.1-l.2] uniquely determines the
function.
Infinite Processes.
The Bolzano-Weierstrass Theorem. In contrast with the proof given in [Sma, p.4, the Bolzano-Weierstrass
Theorem], the proof of [Ru1, p.35, Theorem 2.42] appears very awkward. This is
because [Ru1, p. 35, Theorem 2.41(b)] gives a more effective method than [Ru1,
p.32, Definition 2.32] for checking whether a
set is compact.
Dedekind cuts.
Dedekind cuts can be viewed in a rigorous way or in an intuitive way. The former view is good
only for proving the completeness of the real line. Smail adopts the later view to prove the theorem in [Sma, p.5,
§6]. In contrast, Rudin adopts the former
view to prove [Ru1, p.11, Theorem 1.36]. All kinds of devices [Ru1, p.4,
Definition 1.9; p.9, Theorem 1.32 (the generalization of the Dedekind cut of
rational numbers to that of real numbers)] designed for proving completeness are
used in this proof. Thus, by opening a Pandora's box, Rudin unnecessarily complicates
his proof.
The convergence of a series.
The tests of convergence:
Simple tests [Sma, pp.73-77, §91-§100].
Ratio test [Sma, p.78, §101].
Root test [Sma, p.83, §103].
Kummer's test [Sma, pp.86-87, §107].
Raabe's test [Sma, p.88, §108].
Dini's theorem [Sma, pp.90-91, Theorem 109].
Cauchy's Condensation test [Sma, pp.91-92, §110].
Maclaurin's integral test [Sma, pp.93-95, §111].
Bertrand's tests [Sma, §113,
§114, §116].
Cahen's test [Sma, p.99, l.-5-l.-3].
Gauss' test [Sma, p.101, §118].
Remark. Generalization makes a theory shallow, while effectiveness increases a
theory's depth. However, modern textbooks in mathematical analysis often ignore many
effective tests mentioned above. To classify the series
into categories according to which test can be used to prove their convergence or to create effective computer programs to determine the convergence of a series,
it is necessary to have the knowledge of the above tests.
The ratio test should be used before the root test because the former test is more
efficient in computation and the latter test is more general [Sma,
p.84, l.-10]. The ratio test and Rabbe's test are special cases of Kummer's
test [Sma, p.88, l.1-l.3]. Rabbe's test is used when the ratio test fails [Sma, p.88, l.-9].
Using Dini's theorem to prove that S(n ln n)^{-1}
diverges is simpler than using Cauchy's Condensation test (Compare [Sma, p.91,
l.9] with [Ru1, p.54, Theorem 3.29]. The proof of [Sma, pp.95-96, Example] is
simpler than that of [Ru1, p.54, Theorem 3.28]. This shows that following the
more important feature [e.g., integration is more closely related to the concept
of summation than condensation] of summation will lead to a more effective proof for
convergence. The solution of a deeper problem requires a more delicate device. It can be said that all the above tests are brought out one by one
in the process of determining the convergence of the hypergeometric series [Sma,
p.102, (3)].
Errors [Sma, p.82, §102].
Rapidity of convergence and divergence [Sma, pp.103-105,
§120].
Analytic continuation.
The method for expanding the domain of an analytic function [Wat1, p.96, l.-17-p.97,
l.24].
Remark 1. The statement given in [Wat1, p.96, l.-12-l.-10]
can be proved using [Ru2 p.224, Theorem 10.16].
Remark 2. It is more difficult to see the big picture of analytic continuation
from [Ru2, p.347, l.-5-p.349, l.-10]
than from [Wat1, p.96, l.-17-p.97, l.24]. Thus, in
order to see the big picture, we should adopt a concrete setting to illustrate
a method.
Using geometric series to expand a circular domain [Wat1, p.98, l.1-l.10].
Using
Borel's integral to expand a circular domain [Wat1,
§7.81].
How we construct an analytic function f in |z| <1 such that
|z| =1 is the natural boundary
[Ru2, p.344, Definition 16.3] of f [Wat1, p.98, §5.501; Ru2, p.347, Example 16.7].
Uniqueness theorems
Fix a chain. If an analytic function is given for the first circular disc,
then the function for the last circular disc is uniquely determined
[Ru2, p.348, l.8].
Fix a curve g. If
g starts at the center of the circular disk D, then (f, D) admits at
most one analytic continuation along g no matter
what chains cover g [Ru2, p.348, l.-3-p.349,
l.1].
The monodromy theorem [Ru2, p.351, Theorem 16.15]
Fix a simply connected domain W.
Fix the start point and the end point of a family of curves in
W. If an analytic function is given in the
neighborhood of the start point, then the function in the neighborhood of the
end point is uniquely determined no matter what what curve in the
family is chosen.
Remark. [Wat1, p.97, l.-20-l.-8]
is more confusing than [Ru2, p.351, Theorem 16.15]. Thus, in order to gain clarity, we should adopt an abstract setting
(without using the Taylor series) to eliminate unrelated factors.
Application Based on intuition, we think that the analytic continuation is
only useful for nominal expansions, but fails to provide an effective method for calculation. For example, [Guo,
p.102, (2)] is an analytic continuation of [Guo, p.93, (1)] and [Guo, p.151,
(6)] is an analytic continuation of [Guo, p.136, (5)]. However, by combining the
concept of analytic continuation with the knowledge of the form in [Guo, p.161,
l.10], we can obtain [Guo, p.161, (8)] which greatly facilitates calculating the
value of the hypergeometric function when |z| > 1.
Integral solutions versus series solutions.
Why we need to represent the solution in an integral form [Guo, p.150, l.4-l.8;
p.162, l.4-l.12].
The basic principle of solving differential equations by integrals [Guo, p.78, l.9-p.80, l.10].
Another advantage of integral solutions [Guo, p.84, l.-9-l.-8].
Hypergeometric functions
Classification of the solutions of the hypergeometric equation [Guo,
§4.3]
Barnes' contour integrals for the hypergeometric function [Guo,
§4.6]
Branch point at z = 1: Read [Wat1, p.291, l.3-l.5] and the section entitled "The Hypergeometric Function and Branch Points" of the following webpage: http://mysite.du.edu/~jcalvert/math/complex.htm
Remark 1. Instead of using the hints given in [Wat1, p.290, l.-2-l.-1], I justify the interchange of the order of integration given in [Wat1, p.290, l.-8] as follows:
Use [Wat1, p.287, l.4; p.289, l.-9].
Let t = k+iv.
If k¹0, $d>0: p/2-d<|arg t|<p/2+d
as v®±¥.
t^{s }exp (-p|I(s)|) = O (exp (-(p/2)+d)|I(s)|)).
Remark 2. The equality given in [Wat1, p.291, l.3-l.5] can be proved as follows:
Proof. Let C^{-} = [-i(N+1/2), i(N+1/2)] + S^{-}, with S^{- }:
t = (N+1/2)e^{iq}, p/2 ³ q³ -p/2, be the path of integration.
Then let N®+¥. The poles of G(c -
a - b - t)G(- t) lie on the right of the
integration path and the poles of G(a+t)G(b+t)
lie on the left. The right-hand side of the equality given in [Wat1, p.291,
l.3-l.5] is the sum of the resides of the integrand
at the poles of G(c -
a - b - t)G(- t).
Branch point at z =
¥: Read [Wat1, p.289, l.3-l.5]
.
[Guo, p.171, (5) & (6)] show that the generating function of the Jacobi
polynomials is closely related to the Euler transform [Guo, p.88,
(23)].
The Chebyshev polynomial is a special case of the Jacobi polynomial [Guo, p.175, l.-6].
[Guo, p.157, (5)] can be used to remove the restriction Re
g> Re b>0. This removal is not only good for evaluating F(a,b,g,1) but also for
extending the domain of F(a,b,g,z)
[Leb, p.240, l.15-p.241, l.6]. In other words, it is good for dealing with the
integral representation of a function closely related to the beta function [Guo, p.245, (9.4.3)].
The formulation in [Leb, p.245, l.6-l.7] is more precise and concise than that in [Guo,
p.153, l.-9-l.-7].
Furthermore, the former provides a proof, while the latter does not.
Suppose we discuss a linear transform of the variable of the hypergeometric function.
The use of the integral representation of the solution of the hypergeometric
equation will cause a problem that requires justification [Lev, p.247, l.13]. If
we directly relate the solution to the hypergeometric equation [Guo, p.70,
(13)], then we can avoid this problem.
Generalized hypergeometric series [Guo, pp.189-190,
§4.15; Sma, p.102, Example (3); Luk, p.136,
l.-7-p.137, l.2].
The generating function for Legendre polynomials [Guo, §5.3]
Integral representations of P_{n}^{m} : [Hob, p.188, (10)] if m+n is not a negative integer;
factor (m^{2}-1)^{m/2} should have added to the right-hand side of the equality given in [Hob, p.192, l.2-l.3]
if m+n is a negative integer.
Expressing P_{n}^{m} in term of hypergeometric functions: [Guo, p.249, (8)] if m is not a positive integer; [Guo,
p.250, (12) & (13)] if m is a positive integer.
Definition of Q_{n}^{m} [Wat1, p.316, l.-25-l.-5;
Hob, chap. V, §125]
For
|m|>1, Q_{n}^{m} is given by [Hob, p.195, (19) or p.196, l.7-l.13]. If n is not a positive integer, by analytic continuation, we can extend the domain
of Q_{n}^{m} to the entire plane with the cut from +1 to -
¥ [Hob, p.195, (18); p.196, l.1-l.6]. If n is an integer, Q_{n}^{m} is given by [Hob,p.195, (20)].
Remark 1. The requirement R(n+1)>0 given in [Wat1, p.316, l.-9] is used to prove that the integrals along
two small circles centered at -1 and 1 are small. For example, at t =1, the integral behaves like (re^{iq}-1)^{n+1}®0
as r®0.
Remark 2. Deriving the formula given in [Wat1, p.316, l.-7] from that given in [Wat1, p.316, l.-11]:
Proof. Q_{n }= (4i sin np)^{-1}(ò_{[-1, 1] }(t^{2}-1)^{n}2^{-n}(z-t)^{-n-1}dt + ò_{[1, -1]} e^{-2pin}(t^{2}-1)^{n}2^{-n}(z-t)^{-n-1}dt).
Remark 3. [Guo, §5.8] introduces Q_{n } by using the general formula given in [Guo, p.61, (27)].
Guo's method is tedious and fails to show the essential difference between the
Legendre functions of the first kind and the Legendre functions of the second
kind. In contrast, [Wat1, § 15.2; p.316, l.-
25-l.- 5] points out that their difference is determined by how the contour
of integration surrounds the branch points of the integrand.
In addition, Watson's method is much simpler.
Recurrence relations for Q_{n} [Sne1,
pp.81-82, §19]. Remark. Giving a complete list
[Guo, p.230, (1)-(5)] of recurrence relations for Q_{n}
is not as important as providing all the tools needed to prove these relations.
Without proper tools, the readers may have problems overcoming the obstacles.
If y(x) is a solution of Legendre's equation, then (x^{2}-1)^{m/2}y^{(m)} is
a solution of Legendre's associated equation [Sne1, p.86, l.10-l.18].
Remark. Mathematical discussion should speak of essence and should avoid empty words. For example,
the direct substitution stated in [Guo, p.234, l.13-l.15] can easily lead to a
mess in calculation. In contrast, [Sne1, p.86, l.10-l.18] provides a useful
guide for avoiding such a mess.
Completeness: {P_{n}(x)} is a basis in
L^{2}(|x|£
1) [Cou, vol. 1, p.82, l.-1]; {Y_{lm}(q,
j)} is a basis in L^{2}(x^{2}+y^{2}+z^{2}=1)
[Guo, p.245, l.-5]. Remark. Completeness can be
expression by the closure relation [Coh, pp.99-100, §d;
p.665, (D-26)]. [Boh. p.324, l.11-l.14] treats the completeness of the surface
harmonics as a postulate. It should be proved as a theorem.
Derivation of the confluent hypergeometric equation
the hypergeometric equation [Guo, p.297, l.2]
® the equation given on [Guo, p.297, l.4] [replace z by z/b]
® [Guo, p.297, (1)] [let b=b®¥]
Derivation of the Whittaker equation
the confluent hypergeometric equation given on [Guo, p.300, l.-13]
® [Guo, p.300, (1)] [let y = e^{z/2}z^{-g/2}w(z)]
® [Guo, p.300, (1)] [let m = (g-1)/2, k = (g/2) - a]
In [Guo, p.304, l.-4-l.-1],
Guo gives two proofs of [Guo, p.304, (8)]: a formal proof and a rigorous proof.
Judging by the way he speaks, it seems that the formal proof is not a proof
because it is not rigorous. In
fact, a formal proof is the most direct, intuitive and effective proof because it avoids
repetition (see [Guo, p.304, l.-2-l.-1]), but it requires justification. We can use [Ru2, p.27, Theorem 1.34]
to justify the formal proofs mentioned both in [Guo, p.304, l.-4]
and in [Guo, p.298, l.1-l.3] (see the series' uniform and absolute convergence
in [Guo, p.136, (4)] and the integral's uniform and absolute convergence in [Guo,
p.155, l.7-l.8]). Furthermore, by following the proof of [Guo, p.155, (9)], I
find that [Guo, p.304, (8)] is valid in |arg(-z)|<p.
The restriction |arg(-z)|<p/2
mentioned in [Guo, p.304, l.-5] is unnecessary. The
proof of [Sla, p.37, (3.1.15)] does require the condition
|arg(-z)|<p/2,
but its contour of integration is different. The Cambridge publication provided
this proof of Slater's in 1960. Slater's proof is only one point of view, not an authoritative
one though, and it is outdated now. Studying science requires an independent
judgment, not just copying someone else's result.
The asymptotic expansion of W_{k, m}(z), when |z| is large [Wat1, §16.3].
Remark. [Guo, §6.6] provides two other proofs of the formula given in [Wat1, p.343, l.12].
The second solution of the equation for W_{k, m}(z) [Wat1, §16.31].
Mellin-Barnes contour integrals for W_{k, m}(z) [Wat1, §16.4]
If k±m+1/2 is 0 or a positive integer, then the poles of G(-s-k±m+1/2) and those of G(s) may intersect [Wat1, p.343, l.-1].
Prove the formula given in [Wat1, p.344, l.4].
Proof. Read [Guo, p.311, l.7-l.9].
Prove the statement given in Wat1, p.344, l.5].
Proof. On the line s = it, |z^{s}| = e^{(-arg z)t}.
Combining it with the estimate given in [Wat1, p.344, l.4],
we have (the integrand given in [Wat1, p.343, (C)]) = O(e^{(|arg z|-(3/2)p)|t|}|t|^{-2k-1/2}).
Consequently, the integral given in [Wat1, p.343, (C)] converges uniformly in {z| |arg z| £ (3/2)p - a}.
Prove the estimate given in [Wat1, p.344, l.-9-l.-8].
On the line s = -N-1/2 + it, |z^{s}| = |z|^{-N-1/2 }e^{(arg z)t}.
By the method given in [Guo, p.311, l.7-l.9], we have G(s)G(-s-k-m+1/2) G(-s-k+m+1/2) = O(|t|^{N-2k }e^{(arg s -2p)|t|}, where |arg s| £ (p/2) + d.
|z^{s}| = |z|^{-N-1/2 }e^{-(arg z)t}, where |arg z| £ (3/2)p - a.
Prove that A=1 and that B=0 [Wat1, p.345, l.-12].
Proof. Let T_{N}= 1+S_{n=0}^{N} {m^{2}-(k-1/2)^{2}}…{m^{2}-(k-n+1/2)^{2}}/[n!z^{n}].
I(z)/[e^{-z/2}z^{k}] = T_{N} + O(z^{-N-1/2}) [Wat1, p.344, l.-5].
W_{k,m}(z)/[e^{-z/2}z^{k}] = T_{N} + O(z^{-N-1/2}) [Wat1, p.343, l.8].
Hence (z^{N}/[e^{-z/2}z^{k}])[I(z) - W_{k,m}(z)]®0 as z®¥ (*).
If B¹0, then the largest term of (z^{N}/[e^{-z/2}z^{k}])[I(z) - W_{k,m}(z)] is Be^{z}z^{N-2k }which approaches
¥ instead of 0 as z®¥. This contradicts (*).
Therefore, B=0.
If A¹1, then the largest term of (z^{N}/[e^{-z/2}z^{k}])[I(z) - W_{k,m}(z)] is (A-1)(z^{N}/[e^{-z/2}z^{k}]) which approaches
¥ instead of 0 as z®¥. This also contradicts
(*). Therefore, A=1.
Prove the statement given in [Wat1, p.346, l.11-l.14].
proof. Let C_{R} be s = Re^{iq}, where -p/2£q£p/2.
On C_{R}, Re (-s log s) is less than or equal to -2^{-1/2}R log R +(p/[4´2^{1/2}])R if |arg s|£p/4 and
is less than or equal to (p/2)|Im s| if p/4£|arg s|£p/2.
[sec ps]^{2} = O(e^{-2p|Im s|}).
Although there is no discrepancy between [Wat1, p.346, Example 2] and [Guo,
p.314, (5)] when k is an integer, in terms of methodology the latter provides a better treatment.
Remark. By [Wat1, p.345, l.13], the integral given in [Wat1, p.343, (C)] contributes the value pi at the point s=0 [Gon, p.683, Lemma 9.4]. This is because s=0 is a pole of G(z) and
is degenerated by the integration path s = re^{iq }(-p/2£q£p/2), where r®0+. Note that this is different from the integration because a point can only contribute
the value 0 to a Lebesgue integral.
Using the method of separation of variables to solve the Laplace equation in parabolic cylinder coordinates will lead to the Weber equation
[Guo, p.320, (1); Leb, pp.281-283, §10.1].
Remark. Notice the distinction between Lebedev's [Leb, p.281, (10.1.1); the vertical
cylinder generated by a parabolic curve] and Landau's [Lan1, p.151, l.-6-l.-3;
the surface of revolution generated by a parabolic curve] parabolic coordinates.
Prove that the integral given in [Wat1, p.349, l.12] is well-defined.
Proof. It suffices to prove the statement given in [Guo, p.83, l.-4].
-(p/2) £ arg (m+z) £ (p/2) - d.
Let p - d/2 £ arg t £ p. Then
(p/2)+(d/2) £ arg [(m+z)t] £ (3p/2)-d.
cos (arg [(m+z)t]) £ -sin (d/2).
|e^{(m+z)t}| £e^{-|(m+z)t|sin (d/2)}.
Differentiations of vectors and tensors [Haw, chap. 11].
Remark. For the proof of [Haw, p.162, l.-1], see [Lov,
p.80, l.11-l.13].
Reduction of product representation [Hei, pp.67-71,
§9].
Remark. [Hei, p.69, l.-4] can be derived from [Coh,
p.1028, (8)-(11)]. [Hei, p.69, l.-1] can be derived from [Coh, p.1029,
(15)-(19)].
Differential geometry.
Prerequisite: [Fin].
Remark. This book provides a solid understanding of curves and surfaces of the
second degree. After we finish reading this book, at least we will not call a hyperbolic paraboloid a hyperboloid
[O'N, p.204, l.-15]. Their definitions and shapes are
quite different.
Structures.
Definitions of a topological manifold:
Metrizable [Spi, p.624, Theorem 1].
Second countable [Mat, p.97, Theorem 1].
Essential contents.
Local theory of space curves [Wea1, chap. 1; Kre, chap. 2].
The explanation given in [O'N, p.56, l.-3-l.-2;
p.57, l.1-l.3] is not as clear as that given in [Wea1, vol. 1, p.11, l.-14-l.-7].
The definition of the osculating plane in [Wea1, p.12, l.-19]
is not as rigorous as that in [O'N, p.61, l.15-l.16]. In addition, Kreyszig's
three-point approximation [Kre, p.34, l.-7] is not
as good as O'Neill's Taylor series approximation [O'N, p.61, l.12-l.13]. The
latter method is not only simpler, but also leads to a stronger result. The
three points described in [Kre, p.34, l.-7] are not
only on a specific plane, but also on a specific parabola [O'N, p.61, l.12].
It is important to relate the circular curvature to its circle [Wea1,
vol. 1, p.13, l.8] as closely as possible [Wea1, vol. 1, p.13, l.8-l.21]. [O'N, p.62, Lemma
3.6] has done some related work, but not sufficient.
Why does a minus sign appear in the formula B' = - tN?
O'Neill says, "The minus sign is traditional [O'N, p.58, l.4-l.5]." His
statement explains nothing. In fact, there are two things needing
clarification. First, how do we interpret the sign of
the torsion geometrically? The answer can be found in [Kre, p.38, l.-18-p.39,
l.12]. Once we have this intepretation, we can ask the second question: Why do we have to put a
minus sign in front of tN? The answer can be found in [Kre,
p.37, Fig.11.1]. The interpretation given in [Wea1, vol. 1, p.14, l.-1-p.15,
l.4] is very confusing.
[Wea1, pp.21-23, §5] discusses the
concept of contact only for a special case. In contrast, [Kre, pp.47-51,
§14] symmetrically discusses the concept
for the general case. Furthermore, the geometric meaning of contact of order m
[Kre, p.47, Definition 14.1] is that the two curves intersect on m+1
consecutive points [Wea1, l.12, l.-20].
Existence of curves with prescribed curvature functions [Kli, p.14,
Theorem 1.3.6].
Remark. The proof of [Kre, pp.42-45, Theorem 13.1] is much easier to
understand than that of [Kli, p.14,
Theorem 1.3.6] because the former is more concrete and includes more details.
Although Klingenberg can see the big picture, he omits too many details. Omitting details
indicates rashness, which is the root
of mistakes and disorders. For example, [Kli, p.43, Corollary] is incorrect
(see [Lau, p.53, Theorem 5.1.3]); the order of the two definitions in [Kli,
p.44, Definitions 3.4.4] should have been reversed.
Suppose the parametric curves are orthogonal.
(a). The curves v=const. are geodesics Û E is a function of u only.
(b). The curves u=const. are geodesics Û G is a function of
v only.
Remark. Although the statement given in [Wea1, vol. 1, p.101, l.14-l.17] is correct,
the related statement that we often use is its converse.
Remark. [Kre, p.51, Theorem 14.4] says that the osculating sphere has contact of third order with a curve at P. Contact of a surface with a curve
is an algebraic concept defined by the Taylor series. [Wea1, vol. 1,
p.21, l.-4-l.-1] says
that the osculating sphere passes through four consecutive points on the curve
at P. The geometrical concept of consecutive points is intuitive, but
it is difficult to make its definition rigorous. We also have to establish the
link between the geometric interpretation and the above algebraic
interpretation. This important link can be found in [Kre, p.34, l.-8-p.35,
l.11]. [Kre, p.48, l.1-l.2] says that contact of the second order is also
known as osculation. However, Kreyszig fails to explain why this algebraic
concept has a geometrical meaning. A good explanation can be found in [Cou2,
vol. I, p.359, l.-2-p.360, l.20]. As for a rigorous
treatment of an envelop constructed by consecutive members of a family
of curves, see [Cou2, vol. 2, p.294, l.6-p.295, l.19]. [Wea1, vol. 1, p.30,
l.16] gives another rigorous definition of an evolute, but Weatherburn's
definition makes it difficult to see the most effective method for constructing an evolute.
Local theory of surfaces.
The normal curvature.
Remark. The geometric meaning of [Wea1, vol. 1, p.62, l.12] is similar [Wea1,
vol. 1, p.59,
l.10-l.17] to that of [O'N, p.61, l.4].
The first (or metric) fundamental form of a surface [Lau, p.25, (3.17)].
Remark. The geometric proof, on the positive definiteness of the first fundamental form,
given in [Wea1, p.54, l.4-l.6] is simple and intuitive. In contrast, the proof given in [Kre, p.70, (20.11)] is abstract because it is based on [Kre, p.17, (5.13)].
Our knowledge of the shape of a surface comes from examining the normal curvatures of the curves on the surface [Lau, p.28, l.-12-l.-7;
p.53, l.4-l.7].
Fundamental theorem of surface theory (For the statement, see [Kli, p.64, l.-9-p.65,
l.3; Wea1, vol. 1, p.95, l.-2-p.96, l.2]; for the
proof, see [Lau, p.129, l.14-p.130, l.-7]).
Remark 1. (Integrability theory for systems of first order partial differential
equations) In [Spi, vol.1, p.252,
l.4-p.253, l.11] Spivak shows that the integrability conditions for systems of first order partial differential equations
have the same origin (the equality of mixed partial derivatives) as the integrability conditions
for 1-form (2-dim: [Spi, vol.1, p.250, Proposition 0]; 3-dim: [Cou2, vol.2,
p.104, the fundamental theorem]; n-dim: [Spi1, p.94, Poincaré
Lemma]). The notations used in [Spi, vol.1, p.254, Theorem 1] and [Lau, Appendix II,
§2] are quite similar. The similarity makes it easy for us
to compare these two proofs. Note that in [Spi, vol.1, p.254, Theorem 1] f_{i}
is of class C^{¥}, while in [Lau,
Appendix II,
§2] f_{a}
is of class C^{2}. Spivak claims that there exist
e_{1},
e_{2} > 0 such
that ("t^{1}Î[-e_{1},e_{1}])
("tÎ[-e_{2},e_{2}])
([Spi, vol.1, p.256, (3)] holds). Treat t^{1} as a
parameter (Spivak did not prove his claim correctly. We must use [Pon, p.170, Theorem 13]
instead. This is the only way to prove [Spi, vol.1, p.254, Theorem 1].
Unfortunately, this important tool, [Pon, p.170, Theorem 13], is left out in
[Arn1], [Bir], [Har] and many other popular textbooks in ordinary differential
equations). Thus, Spivak solves a system of differential equations for each
component of (t^{1},…,t^{m}).
In contrast, Laugwitz solves only one system of differential equations [Lau,
p.226, (4)]. However, we must prove that y^{i}(x^{b})
is well-defined, more specifically that it is independent of the paths of integration
[Lau, p.226, l.-14-l.-9].
Then the existence of a solution of [Lau, p.225, (1)] follows from the existence
of a solution of [Lau, p.226, (4)]. For the proof of uniqueness, Laugwitz's
consideration in [Lau, p.226, l.10-l.14] is insufficient, we need also take
[Lau, p.226, l.-14-l.-9]
into account. See [Spi, vol.1, p.254, l.-7].
Remark 2. (g^{~}_{ij})_{;k} = 0 [Lau, p.130, l.14] can be proved using [Lau, p.118, Theorem 11.2.1].
Remark 3. The explanation given in [Kli, p.65, l.-10-l.-9]
is clearer and more specific than that given in [Lau, p.130, step 3].
Studying [Wea1, vol. 1, §62] requires
the background given in [Fin, §§166-169;
§§340-343].
[Cou2, vol. 2, p.307, l.-7] links differential forms
to an integrals, so we can understand the origin and purpose of differential forms.
Note that ò_{G }L
depends on the G and not on the particular parametric
representation [Cou2, vol. 2, p.307, (55c)].
[Cou2, vol. 2, p.307, l.-7; p.323, (64c)] enable us
to distinguish the dummy variables of integrals from the coordinates on a
surface.
[Cou2, vol. 2, p.317, l.2] links a differential form to a manifold. [Cou2, vol.
2, p.314, l.4-l.7] links differential forms to tensors. Actually, in [Spi,
vol.1. p.282, l.1], a k-form on a manifold M is defined as a section of
W^{k}(TM), i.e., an
alternating covariant tensor field of order k.
Spivak's approach to differential forms
Define Á^{m}(V)
[Spi, vol. 1, p.159, l.1] ®
Define the tensor product TÄS
[Spi, vol. 1, p.159, l.6] ®
[Spi, vol. 1, p.275, Proposition 1 (1)] ®
Define wÙh [Spi,
vol. 1, p.275, l.-1]
® [Spi,
vol. 1, p.279, Theorem 3] ®
Use [Spi, vol. 1, p.282, l.4] to add a C^{¥}-manifold
structure on W^{k}(TM)
® Define a k-form
on a manifold M as a section of
W^{k}(TM)
[Spi, vol. 1, p.282, l.1].
Remark 1. The inner product motivates us to define a symmetric 2-tensor [Spi1, p.77, l.17]; determinants motivate us to define alternating k-tensors [Spi1,
p.78, l.9].
Remark 2. Á^{k}(TM)
is defined in [Spi, vol. 1, p.160, l.9].
Poincaré Lemma
(1). Proofs. An open set star-shaped with respect to 0 [Spi1, pp.94-95]
® A
manifold M smoothly contractible to a point [Spi, vol. 1, p.306, Corollary 18].
Remark. [Spi, vol. 1, p.321, Problem 23] links the above two proofs and shows
that for a star-shaped domain we can solve w=dh explicitly in terms of integrals.
(2). Examples.
1-forms [1]; 2-forms [Spi, vol. 1, p.297, l.-10-l.-1].
(3). [Spi, vol. 1, p.298, l.11-p.300, l.3] shows that we can find a
counterexample of [Spi,
vol. 1, p.306, Corollary 18] if M is not smoothly
contractible to a point.
Integration of forms [O'N, chap. IV, Sec. 6]
In R^{n}, we integrate a function on a volume. In a manifold M, we integrate a k-form on a k-chain [Spi, vol. 1, p.337, l.4-l.6].
The fundamental theorem of calculus ® Stokes' theorem
on the integral of dw over a k-chain in an n-dim
manifold M, where
w is a (k-1)-form on M [Spi1, p.102,
l.13-l.20; Spi, vol. 1, p.345, l.5-l.6].
(1).
Spivak relates the minus sign in [Spi, vol. 1, p.339, l.6] to the signed arrows
in the third figure on [Spi, vol. 1, p.338]. This relationship leads nowhere.
Only after we relate the minus sign to orientations [O'N, p.173, l.5] can we
find a method to generalize the meaning of the minus sign to higher dimensions.
Stokes' theorem on the integral of dw over an oriented n-dim
manifold-with-boundary M, where
w is an (n-1)-form on M
with compact support [Spi1, p.124, Theorem 5-5; Spi, vol. 1, p.354, Theorem 6].
(1). The sum in [Spi, vol. 1, p.348, l.8] is actually finite [Spi,
vol. 1, p.348, l.10].
Proof. Assume {p: f(p)¹0}
Ç support w = Æ. ò_{M}
f×w = ò_{c}
f×w, where support w Ì
c([0,1]^{n})
= ò_{[0,1]}^{n}
c*(f×w)
= 0 [Spi1, p.90, l.2].
(2). In order to define ò_{M}
w, we must stipulate that support w
Ì the interior of c([0,1]^{k}) [Spi,
vol. 1, p.353, l.8-l.9].
This interior with respect to M contains points of in the image of c _{
(n, 0)} [Spi, vol. 1, p.353, l.9-l.10]. Consequently, ò_{¶M}
w in [Spi, vol. 1, p.353, l.11] is well-defined.
(3). The induced orientation on R^{n-1}´{0} =
¶H^{n} is (-1)^{n} times the usual
orientation
[Spi, vol. 1, p.353, l.4].
When Spivak wrote [Spi1], he did not know how to prove this statement, so he treated
it as a rule. The only justification he could give at that time was that Stokes'
theorem will not have a neat appearance unless we adopt this rule [Spi1, p.119,
l.-4-l.-3; p.124,
l.11-l.22]. When he wrote [Spi], he finally be able to prove the above statement
[Spi, vol. 1, p.353, l.1-l.4] using the properties of the determinant.
(4). In order to prove the the first equality in the line marked by (*) in [Spi, vol. 1, p.353, l.-3],
we must note two things: First, support w = the
closure of {p: w(p)º0 does
not hold}. If w = f dx^{1}Ù…Ùdx^{n},
f(p)=0 may not make w(p)º0.
w(p)º0 only if fº0
in a small neighborhood of p. Second, support w (the
shaded area of the second figure on [Spi, vol. 1, p.353]) lies in a small
neighborhood of a point on c_{(n, 0)} and does not
touch of the three other sides of the quadrilateral. In vector analysis, it is
easy to visualize the divergence theorem [Wangs, §1-14].
In contrast, in differential geometry, it is difficult to visualize the proof of
[Spi, vol. 1, p.354, Theorem 6] because the proof is divided into three
segments: (a). Smoothly (using partition of unity) break the integral over a
manifold into the sum of integrals over small parts. (b). For each small part, [Spi,
vol. 1, p.354, Theorem 6] is the same as [Spi, vol. 1, p.343, Theorem 4]. (c). Smoothly
reassemble the manifold with its small parts.
(5). Classical "Strokes' type" of theorems:
[Spi1, p.134, Theorem 5-7; p. 135, Theorem 5-8 & Theorem 5-9].
In order to facilitate applications, we must translate Spivak's manifold
versions
into the fluid mechanics versions adopted by physics textbooks [Wangs, p.21, (1-59); p.24, (1-67)]. [O'N,
p.170, Theorem 6.5] is Green's theorem [Spi1, p.134, Theorem 5-7]. [O'N, p.154,
Definition 4.4] is essentially the equality given in [Spi1, p.134, l.-1].
If nÎN is even, then there does not exist a nowhere zero vector field on S^{n} [Spi,
vol. 1, p.377, Corollary 15]. {1}.
Proof. Assume there is a nowhere zero vector field on S^{n},
then we can construct a homotopy between the antipodal map A and the identity
map i. By [Spi, vol. 1, p.376, Corollary 14], deg A= deg i. However, by [Spi,
vol. 1, p.373, Theorem 12] {1}, deg A = (-1)^{n+1}
and deg i = 1.
The indices of the singular points of a vector field [Arn1,
§36]
(1). How we define the index of a curve [Arn1, p.309, l.15-l.18] Remark. The index of
a curve is a variant of the degree of f : S^{1}
® S^{1 }[Dug, p.342, l.-13; 1]. (a). The index of a curve vs. the oriented basis of a vector space [Arn1,
p.309, l.17-l.18].
(b). The index of a curve vs. the relationship between a
domain and its boundary [Arn1, p.313, l.15-18].
(2). How the index of a singular point is related to the index of a curve [Arn1, p.312, l.-3-l.-1].
How phase assignments to the logarithmic function affect the value of a contour integral.
For a confusing point, a good book will provide explanations
with greater detail for clarity. In contrast, a bad book will omit the
explanation and bog its readers down.
Example 1. Watson makes a vague
explanation [Wat, p.166, l.15-l.17] of how he derives [Wat, p.166, (3)] from [Wat, p.166, (2)]. [Guo,
p.369, (4)] is equivalent to [Wat, p.166, (3)]. However, Guo writes down [Guo,
p.369, (4)] directly without any explanation. The following explanation may
clarify the subtle point:
Watson's approach (compare [Wat, p.165, Fig. 5] with [Wat, p.165, Fig,4]): If we replace the second integral in [Wat, p.166, (2)] by the second integral in
[Wat, p.166, (1)], what adjustment should we make? Originally, arg (t^{2}-1)
at B is -p. After the replacement, arg (t^{2}-1)
at B becomes +p. Therefore, we
must add a factor exp (-2pi)
for each factor (t^{2}-1)
to compensate for our replacement. In total, we must multiply the integrand by exp [-2pi(u-½)]
for the adjustment because we have (t^{2}-1)^{u-½}.
Finally, we have to multiply the integrand
by -1 because the
contours of the two integrals have opposite directions.
Guo's approach (compare the contour in [Guo, p.370, Fig.29; arg (t^{2}-1)
at Q is -2p]
with that of the integral in [Guo, p.369, l.-3; arg
(t^{2}-1) at Q is
assigned to be 0]):
For the contour in [Guo, p.370, Fig.29], arg (t+1)
is 0 at P and -p at Q; arg (t-1)
is 0 at P and -p at Q. To understand this clearly,
one must be
able to evaluate arg (t+1) and arg (t-1)
at every point A along the contour in [Guo, p.370, Fig.29] by counting how many
turns the contour makes around -1 and +1 up to Point
A respectively.
Example 2. Watson directly writes down the formula for H_{u}^{(2)}
in [Wat, p.170, l.12] without detailed explanation. In contrast, [Guo, p.371,
l.4-l.12] gives more detailed explanation. Still, Guo fails to explain how he
obtains the first term in the bracket in [Guo, p.371, l.8]. The reader might
think his phase assignment is arbitrary and depends on his whim. Here I supply the
following detailed reasoning to clarify the point:
Consider the starting point t=1+i¥. Originally, arg
(t^{2}-1)=-p.
After replacing t^{2}-1
by 1-t^{2}, arg (1-t^{2})
becomes 0. Therefore, we must multiply the integrand by exp (-pi)
for each factor 1-t^{2}
to compensate for the replacement.
Example 3 [Guo, p.250, l.-1]. A is the integral on
the small counterclockwise circle around s=1; B is integral on the small
counterclockwise circle around s=0. Let the contour in [Guo, p.249, Fig.13]
intersect with the real axis to the right of s=1 at V_{1}
and V_{2}, where V_{1}
is closer to s=1 than V_{2}. Let the contour in [Guo,
p.249, Fig.13] intersect with the real axis to the left of s=0 at U_{1}
and U_{2}, where U_{1}
is closer to s=0 than U_{2}. Now consider the
small clockwise circle around s=1. Since arg V_{1}
= arg V_{2} +2p (phases
inherited from the contour in [Guo, p.249, Fig.13]), where V_{1}(V_{2})
is on the small clockwise (counterclockwise) circle around s=1), the value of s^{u}
on the clockwise circle = e^{2upi}
´ the value of s^{u}
on the counterclockwise circle.
Example 4. If Im(m)>0, the value of the integral in [Hob, p.200, l.-8] equals Le^{-2npi}-Me^{-2npi}.
Proof. By the definition of M, the initial phase of t-1 at
the starting point C is
p [Hob, p.200, l.-15].
However, at the end point C of the contour (-1+, 1-),
the phase of t-1 is -p.
Remark 1. To evaluate the phase of the integrand in a contour integral, we should not mix the assumptions with the consequences. Otherwise, we have to
worry about the problem of consistency. For example, [Hob, p.184, l.-19-p.185,
l.9] distinguishes the assumptions from the consequences, while the stipulation
in [Guo, p.251, l.-1-p.252, l.5] fails to do so.
Remark 2. The rules of phase assignment:
To evaluate a contour
integral, we must assign the phase of the starting point of the contour for each factor of
the integrand.
[Hob, p.184, l.-19-p.185, l.9] assigns the
phases of m+1, m-1, t+1, t-1,
t-m. All the assignments are based on one simple
rule: ordinary assignment of a polar angle.
[Hob, 192, Fig. (a) & (b)] shows
that when we deform a contour continuously into another one and the deformation
cannot avoid crossing a branch
point, then the value of the contour integral may change. The two figures in
[Hob, p.199] show a method of handling this type of problem. In the first
figure the phase of t-1 at A is assigned to be +p
because the entire contour is on the upper piece of the Riemann surface with
the branch point +1. In the second figure the phase of t-1
at A is assigned to be -p because the entire
contour is on the lower piece of the Riemann surface with the branch point +1.
In [Hob, p.193, the right-most
figure], we find a convenient point A and assign its phase relative to a branch
point so that the phase of
the starting point C relative to that branch point can be determined [Hob, p.193, l.-15].
Binomial distributions.
To introduce the Gaussian [Reic, §4.E.2] and Poisson
[Reic, §4.E.3] distributions, Reichl emphasizes
the conditions under which a binomial distribution becomes a Gaussian or Poisson distribution.
In this way he provides an effective method of constructing the Gaussian and
Poisson distributions and explains their origins. In
contrast, [Lin, §3.2 &
§3.3] just directly give the definitions of
these two distributions without explaining their origins. Simply characterizing
the properties of the Poisson distribution does not provide an effective
construction of the limiting process. Lindgren's approach preserves the
final answer, but leaves out the important constructing process. Thereby, his
definitions of the Gaussian and Poisson distribution obscure their relationships
to the binomial distribution.
When Reichl defines [Reic, p.185, (4.34)], he
may have a specific example [Reic, p.189,
(4.40) & Lin, p.163, l.-6] in mind. However, because he
fails to introduce that
example before this definition, the definition becomes very hard to understand.
Quality checklist for a theory of tensors
Does the theory distinguish a bound vector from a free vector? Good: [Kre,
p.103, l.12-l.16]. Poor: [Pee, §8].
Does the theory mention that the allowable coordinate transformations form a
group? Good:
[Kre, p.101, l.20-l.21]. Poor: [Pee, §8].
Does the theory have a clear definition of a tensor field? Good: [Kre,
p.111, l.12]. Poor: [Pee, §8].
Does the theory have a consistent scheme for development? [Kre, (31.1), (31.2), (31.3) & (32.1)] are proved using the same scheme, while [Pee,
p.230, (8.14)] is given by stipulation.
Does the theory have a geometric interpretation for the contravariant
or covariant components of a vector? Good: [Kre, p.116, Fig. 35.1 & Fig. 35.2].
Poor: [Pee, §8].
When we use the elements of a vector space as contravariant vectors [Kre,
p.121, l.-11-l.-10] and
the elements of its dual as covariant vectors [Kre, p.123, l.10] to define tensors, do we
relate it to the classical definition with a proper justification? Good: [Kre,
p.122, l.9; p.123, l.-7-l.-6].
Poor [Spi, vol. 1, chap. 4].
Intrinsic geometry.
To call Riemannian geometry intrinsic is an oversimplification. We must specify
its details:
(Using intrinsic expressions) Using tensors to express physical quantities.
(Using intrinsic generalizations) Generalizing intrinsic concepts.
Example. From the Gaussian curvature of a sphere [Ken, p.28, l.-3-l.-2] to the Gaussian curvature of a surface
[Ken, p.30, (3.1)].
(Using intrinsic calculations)
Example. From [Ken, p.28, l.-6] to [Ken, p.30, l.-10].
Establishing theorems in intrinsic (tensor) forms.
Parallel transport, in the sense of the connection of Levi-Civita, along a
path [Kre, pp.228-231, §78].
Remark. [Ken, p.215, A.6.1] gives a more rigorous proof of [Kre, p.231, (78.6)].
Special surfaces.
Envelopes; developable surfaces [Wea1, pp.38-65].
Remark. The examples given in [Kre, pp.254-255] further distinguish between
the characteristic [Kre, p.253, (86.5)] and the intersection of nearby members of a family of
surfaces [Wea1, p.40, l.-11].
Minimal surfaces
Descriptions of the local shape of a minimal surface
Using the concept of the Gaussian curvature [Kre, p.236, l.-19-l.-18].
Using the concept of minimum area [Kre, pp.236-237, Theorem 81.1].
Using the concept of spherical mapping [Kre, p.238, Theorem 81.2].
The global shape of a minimal surface [Kre, pp.244-245, Theorem 84.1].
Ways to visualize a family of real minimal surfaces [Kre, pp.245-246, Theorem 84.2 & Theorem 84.3]].
Surfaces of constant Gaussian curvature and non-Euclidean geometry.
The geodesic mapping of a pseudospherical surface into a plane [Kre, p.271,
l.-14-l.-2].
The geodesic mapping of a spherical surface [Kre, p.153, l.2] into a plane [Kre,
p.271, l.-2-p.272, l.6].
The surfaces with constant Gaussian curvature are the only surfaces that can be mapped geodesically into a plane
[Kre, p.272, Theorem 91.1].
The curvature tensor
Geometric interpretation of the curvature tensor [Kre, pp.292-295,
§100].
Remark. [Lau, p.107, Fig. 30] provides more information than [Kre, p.292, Fig. 100.1]. However,
the argument given in [Lau, §10.3] is incorrect.
How a space with a vanishing curvature tensor simplifies the representation of an affine connection and the concept of parallel transport
[Lau, p.109, Theorem 10.3.1].
Bessel functions
[Jack, p.426, (9.85); Guo, p.377, (4)] defines n_{l}(z) to be (p/2z)^{1/2} Y_{l+½}(z),
while [Coh, p.947, (62-a) and (62-b)] use the definition n_{l}(z)
= - (p/2z)^{1/2} Y_{l+½}(z).
Expressing a plane wave as a linear superposition of free spherical waves [Coh,
p.948, (65)].
In order to introduce a special function, we should focus on its general
properties. Consequently, we would like to use its integral representation [Guo,
p.353, (5)] as its definition. The strategy for explaining the gamma function can be used as
our blue print. The construction via the integral representation focuses on the locations of
the function's singular points, its behavior at the origin
and its asymptotic behavior [Coh, p.945, l.-5-p.947,
l.15]. Then we endeavor to evaluate the function using its
series representation [Guo, p.348, (7)].
If n is not an integer, J_{n} and J_{-n} are linear independent solutions of Bessel's equation [Wat1, p.359, l.7-l.9; Guo, p.348,
l.8-l.10; p.365, (2)].
Asymptotic series for J_{n}(z) when |z| is large
[Guo, p.378, (3), (4); p.379, (5), (6)] The statement given in [Wat1, p.368, l.13-l.16] can be proved as follows: The asymptotic expansion of W_{k, m} given in [Guo, p.307, (2)] converges when |z|>1 [Wat1, p.24, l.18-l.19].
Therefore, the expansion is analytic in {z| |arg z|<p,
|z|>1} and the ~ sign can be replaced by equal sign in {z| |arg z|<p,
|z|>1}. This identity can be extended to the domain {z| |arg z|<3p/2,
|z|>1}.¬
If we fix n and consider J_{n}(z) as a function of z, then z =
¥ is a removable singularity [Guo, p.349, l.5-l.6; Wat1, p.368, l.-7-l.-5]. If we fix z and consider J_{n}(z)
as a function of n, then n =
¥ is an essential singularity [Guo, p.348, l.9; p.349, l.7].
If n is an integer, Y_{n} is a solution of Bessel's equation [Wat,
p.58, l.-17-p.59, l.9].
The expansion of Y_{n} in an asending series [Wat, p.62, (3)]
Relations between Hankel functions and Whittaker functions [Guo, p.373, (16)
& (17)]
The zeros of Bessel functions whose order n is real [Wat1, p.361, l.10-l.11; l.14-l.15; Guo, p.422, l.7-l.8]
Representations by continued fractions
[Perr, p.300, l.3; p.299, (20)]
[Inc1, p.181, l.-12-l.-9; p.180, l.-1]
Fourier analysis of random processes [Matv, §14]
Remark. For detailed and rigorous definitions of a stochastic process, the power spectrum, the
autocorrelation function, and stationarity, read [Fri91,
§8.1, §8.2,
§8.3].
Definition of an asymptotic expansion [Wat1, p.151, l.5-l.11; Guo, p.30, l.3-l.15].
Remark.
The side remark given in [Wat1, p.1151, l.11] should be removed from the definition. Otherwise, readers may mistake it for another condition to be satisfied.¬
Uniqueness of an asymptotic expansion [Wat1, §8.32; Wat, §7.22]
The idea used to prove the statement given in [Wat1, p.154, l.1-l.2] can also be used to prove that the discrepancy between [Wat1, p.347, l.-7] and [Wat1, p.348, l.-3-l.-1] as well as the discrepancy between [Wat, p.201, (1)] and [Wat,
p.201, (2)] is only apparent.
Even though the series given in [Wat1, p.252, l.-10] is the
asymptotic expansion of
f(x), the series diverges [Wat1, p.252, l.-3-l.-1].
However, the series can be used to estimate G(x)
[Wat1, p.253, l.3-l.8], solutions of ODEs with singularities of the
second kind [Cod, p.148, l.14-l.16] or other integrals [Inc1, §7.323]. Furthermore, there are cases that the use of
asymptotic series of J_{0}(x) gives a more precise estimate than the use of its
convergent ascending series [Inc1, p.173, §7.321]. There are two important methods of
obtaining asymptotic expansions
Method of steepest descents [Jef, §17.04]
Features (Debye's observations): Saddle points [Jef, p.504, l.10-l.20;
Born, p.749, l.10-l.26]; paths of steepest descent [Guo, p.381, l.10-p.383, l.-3]. Remark. The
fact that the real part of an analytic function has no maximum [Guo, p.381, l.22] can be
proved by the statement given in [Ru2, p.259, l.-8-l.-7].
Main result: [Jef, p.505, (18)].
Method of stationary phase [Jef, §17.05]
Main result: [Jef, p.507, (10)].
Remark. For the first term approximation, we do not need Watson's lemma. For the
higher term approximation we do need Watson's lemma. Jeffreys'
proof of Watson's lemma given in [Jef, §17.03] is not devoid of shortcomings. For example, Jeffreys fails to specify the
angular section of a in which [Jef, p.502, (9)] is valid (see [Guo, p.34, l.-2]).
For a good formulation and proof of Watson's lemma, read Theorem 2 of the
following webpage:
http://homepage.tudelft.nl/11r49/documents/wi4006/watson.pdf
Waston's lemma essentially says that ò_{[0,¥]} e^{-zt}
f(t)dt is dominated by the values of f(t) in a neighborhood of t = 0 for large z and that we may estimate the integral by replacing f(t) with its local expansion for t = 0.
Asymptotic development of solutions [Inc1, §7.31] Prove the statement given in [Inc1, p.171, l.6-l.7].
Proof. In [Inc1, §7.31], we need not worry about the convergence of series if we assume that p and q are analytic at x = ¥.
Fix n. u_{1}+(u_{2}-u_{1})+…+(u_{n+1}-u_{n})
= h+(C_{1}/x)+…+(C_{n-1}/x^{n-1})+(C_{n}+e/x^{n}) [Inc1, p.170, l.-6].
|(u_{n+2}-u_{n+1})+(u_{n+3}-u_{n+2})+…|<M_{n+2}[(K/x)+(K^{2}/x^{2})+…] [Inc1, p.170, l.13]
<(K/x)^{n+1}(2M_{2}) = o(x^{-n}).
Remark. "m>n" given in [Inc1, p.170, l.-11] should be replaced by
"m ³ n-1".¬
Hamilton-Jacobi theory [Ches, §8-11]
Hamilton-Jacobi theory is the theory that discusses
Jacobi's method [Sne, chap. 2, §§13-14] of
solving first-order PDEs. The only way to understand the Hamilton-Jacobi
theory is to consider it from the perspective of partial differential equations [Ches,
chap. 8]. If you just read [Go2, chap. 10], [Lan1,
§47] or
any physicist's textbook on mechanics, you will never understand the theory's
insight
because these books fail to provide the important geometrical meanings of solutions of
first-order PDEs. One often wonders how one can write equations like [Ches p.176,
(8-34)] in mechanics given that x and p are related to each other somehow. This is
because [Ches, p.176, (8-26)] is obtained by substituting u=p and w=q into
F(x,y,u,v,w)=0. One also wonders how integration constants are related to
parameters . This is also explained in [Ches, chap.
8] [1]. However, studying [Ches, chap. 8]
requires some background in differential geometry [Wea1, chap. I-chap. IV]. The Hamilton-Jacobi
equation [Lan1, p.147, (47.1)], derived from the consideration of mechanics, is
a special case of the first-order differential equation [Ches, p.184, (8-61)].
Furthermore, [Ches, p.184, l.5-p.186, l.-4] shows
that every first-order PDE can be reduced to a Hamilton-Jacobi equation [Ches,
p.186, (8-72)]. Here I just highlight the geometrical meanings of Hamilton's
equations and the Hamilton-Jacobi equation: the solutions of Hamilton's equations
are the characteristics of the Hamilton-Jacobi equation [Ches, p.188,
l.13-l.15]. For detailed construction, see [Ches, p.194, Theorem 8-2 & Theorem
8-3].
Remark 1. Jacobi's method is better than Charpit's method because it can be directly generalized to the case of n independent variables
[Sne, p.80, l.1-l.11].
Remark 2. For the reason why one of the n+1 parameters is additive, [Lan1, p.148,
l.13] gives a better explanation than [Ches, p.193, l.-14].
How we formulate differential equations on a C^{r}-manifold M.
Endow the tangent bundle TM with the structure of a smooth manifold [Arn1,
§34.2].
If f : M ® N is C^{r},
define a corresponding C^{r}-bundle map f_{*} : TM
® TN [Spi, vol.1, p.104, (c)].
Define vector fields on M [Arn1, §34.5].
[Arn1, p.304, Theorem, (1)]. Remark. In proving [Spi, vol.1, p.203, Theorem 5], Spivak fails to provide a concrete example
for the proof. This makes it difficult for readers to see what is going on. In
contrast, before Arnold proves [Arn1, p.304, Theorem] he gives a concrete
example in [Arn1, p.303, l.-5-l.-2]
to help readers see the big picture.
The geometrical meaning of [X,Y] [Spi, vol.1, p.218, l.-4-p.225,
l.-1].
Atlases of manifolds
The sphere S^{2} [Arn1, p.291, Fig. 235].
The projective space RP^{n} [Arn1, p.291, Fig.237].
The imbedding theorem in dimension theory [Mun, p.310, Theorem 9.6]
Note that the existence of the imbedding given in the proof of [Mun, p.310, Theorem 9.6]
is logical [Wan3, p.109] because the proof uses Baire's category theorem. Only
in special cases [Mun, p.305, Fig. 13] can the existence of imbedding be
constructive.
The Jordan curve theorems: [Mun00, p.390, Theorem 63.4; p.392, Theorem 63.5].
Remark. [Mun00, p.390, Theorem 63.4] provides a big picture of [Dug, p.362,
Theorem 5.4].
Imbeding graphs in the plane: [Mun00, p.395, Lemma 64.1; p.396, Theorem 64.2; p.397, Theorem 64.4].
The winding number of a simple closed curve
[Mun00, p.399, Fig. 65.2]. Proof. Use [Mun00, p.377, Lemma 61.2] and [Mun00,
p.401, Theorem 65.2].
[Mun00, p.404, Theorem 66.2].
[Mun00, p.405, Lemma 66.3].
Remark. Although Munkres is perfect in proving the theorems given in [Mun00,
§65, §66],
somehow he fails to convey the important message in these two sections to the
readers. A reader wants to see the forest rather than trees. The above three theorems characterize the concept of the winding number from various perspectives
(an induced homomorphism from one fundamental group to another [Mun00, p.398,
l.16], a lifting of a homotopy, a complex line integral) and prove that all the characterizations are
equivalent.
The orientation of a manifold
Nonsingular linear maps are divided into two groups: orientation preserving &
orientation reversing [Spi, vol. 1, p.114, l.-5]
®
An orientation of a vector space [Spi, vol. 1, p.116, l.3]
®
An orientation preserving isomorphism from (V, m) to (W,
n) [Spi, vol. 1, p.116, l.10]
®
Assign the standard orientation on a trivial bundle e^{n}(X).
If X is connected, an equivalence f : e^{n}(X)
®
e^{n}(X) is either orientation preserving
or orientation reversing [Spi, vol. 1, p.116, l.-7].
® The above property (compatibility condition) of the trivial bundle can be used to define an orientation
on a non-trivial n-plane bundle [Spi, vol. 1, l.-3].
Apply this definition to the tangent bundle of a C^{¥}-manifold.
The degree of of a continuous map f: S^{n}
® S^{n} is written D(f)
D(f) is independent of triangulations of S^{n}
[Dug, §XVI.1, Lemma 2 & Lemma 4].
When n=1, D(f) = the winding number of f(S^{1})
[Dug, p.335, l.-6-p.336, l.13]. [1].
D(f) is a homotopy class invariant [Dug, p.339, Theorem 1.1].
The degree of f: (V^{n+1}; S^{n})
® (V^{n+1}; S^{n}) [Dug, p.340, l.4-l.-8]
The degree of a proper map between two connected oriented n-manifolds [Spi, vol. 1, p.373, l.-1]. Remark. If one
directly uses the advanced definition of the degree of a map given in [Spi, vol. 1, p.373, l.-1]
without tracing back to its origins, then one will miss not only the concept's
insightful meanings but also its step-by-step development.
Maps into spheres
Brouwer's theorem [Dug, p.340, Theorem 2.1]
Remark. [Dug, p.341, Corollary 2.2(2)] shows that the condition of smoothness
given in the hypothesis of [Arn1, p.311, Theorem] is irrelevant to the
conclusion.
[Spi, vol. 1, p.377, Corollary 15] applies to C^{¥} vector fields,
while [Dug, p.343, Theorem 3.3] applies to continuous vector fields. The extra
differential structure enables Spivak to obtain a stronger result. Note that [Spi,
vol. 1, p.376, Corollary 14] and [Spi, vol. 1, p.377, l.3-l.5] are results of
homotopy theory and can be proved using topological methods. However, [Spi, vol.
1, l.6-l.9] involves the solutions of differential equations. It cannot be
replaced by a topological argument.
Remark. The proof of [Dug, p.342, Proposition 3.2] provides a neat flow
chart and essentially does not use reduction to absurdity, while the proof of
[Mun00, p.350, Theorem 55.5] uses nontrivial reduction to absurdity in
[Mun00, p.350, l.-13-l.-3]. Thus,
the former proof is more constructive.
([Dug, p.347, Theorem 6.1] vs. [Mun00, p.356, Theorem 57.1]) The proof of [Mun00, p.356, Theorem 57.1]
is insightful simply because it avoids using triangulations. The use of
triangulations will complicate the proof [Dug, p.347, Theorem 6.1]. The concept
of fundamental groups helps organize the proof of [Mun00, p.356, Theorem 57.1].
The lemma given in [Dug, p.347] is the key to the induction
step for the proof of [Dug, p.347, Theorem 6.1]. In order to prove that the map
is non-nullhomotopic, [Dug, p.347, Theorem 6.1] uses degrees, while [Mun00, p.356, Theorem 57.1]
uses covering spaces, a concept equivalent to that of degrees.
In order to prove [Dug, p.349, Corollary 6.2 (1)], we may apply [Dug, p.347, Theorem 6.1] to
either f [Dug, p.349, l.-9-l.-5]
or f|S^{n-1} [Mun00, p.357, Theorem 57.2].
Hopf's theorem [Dug, p.352, Theorem 7.4] [1]
Remark. The map[f] ® D(f) of [S^{n}, S^{n}] into the set Z of positive integers is bijective [Dug, p.352, l.-4-l.-1].
The Jordan curve theorem
The Jordan separation theorem [Mun00, p.379, Theorem 61.3; Dug, p.358, Theorem 2.4]
Remark. [Mun00, p.377, Lemma 61.1] shows that the separation theorems for
subsets of S^{2} are equivalent to the separation
theorems for the corresponding subsets of R^{2}.
Invariance of Domain [Mun00, p.383, Theorem 62.3; Dug, p.359, Corollary 3.2]
Remark. [Mun00, p.381, l.-17-l.-11]
provides the insightful meaning of [Mun00, p.383, Theorem 62.3].
The Jordan curve theorem [Dug, p.362, Theorem 5.4; Mun00, p.390, Theorem 63.4]
Remark. The assumption "let x,y be two points of E^{2}"
given in [Dug, p.362, l.2] should have been changed to "let x,y be two points of
E^{2}-(AÈB)".
Remark. The proof of [Mun00, p.379, Theorem 61.3] uses [Mun00, p.368, Theorem
59.1], while the proof of [Mun00, p.390, Theorem 63.4] uses [Mun00, p.385,
Theorem 63.1(c)]. The assumptions of [Mun00, p.368, Theorem 59.1] and those of
[Mun00, p.385, Theorem 63.1(c)] are the same except that UÇV
is path connected in the former theorem and is not path connected in the latter
theorem [Mun00, p.385, l.12-l.16].
Elliptic functions
The origins of elliptic integrals
Finding the arc length of an ellipse [Guo, p.459, l.-9-l.-1].
Finding the arc length of a lemniscate [Gon1, pp.358-359].
If P(x) is cubic, the elliptic integral given in [Guo, p.456, (1)] can be
reduced to the following three fundamental types: I_{0},
I_{1} and J_{1} [Guo,
p.457, (7)]. See [Guo, p.457, l.-9-p.459, l.2].
Properties of elliptic functions [Gon2, §5.2 &
§5.3].
Periods of a meromorphic function The definition of doubly periodic functions [Gon1, p.363, l.4-l.8] should have been
justified by
[Ahl, p.257, Theorem 1] and [Ahl, p.257, Theorem 1] should have been
supplemented by [Gon1, p.367, Theorem 5.4]. Otherwise, the discussions are
not complete.
The classification of elliptic functions of order two [Guo, p.466, l.4-l.18]
Remark 1. The explanation given in [Pon1, pp.291-292, Example 4] is better than that given in [Guo, p.466,
l.-4-p.467, l.10].
Remark 2. In order to unify the theories of elliptic function and to make
the structure of elliptic functions transparent, González
uses Tan z to define Ã(z)
[Gon1, p.446, Definition 5.14] and Jacobian elliptic functions [Gon1, p.421,
(5.17-1)-(5.17-3)]. However, this approach neglects the important fact that
Ã(z) and Jacobian
elliptic functions are the typical elliptic functions that motivated Jacob and Weierstrass to study the theory of the elliptic functions.
The proof of [Guo, p.469, (1)] given in [Guo, p.468, l.-2-p.469,
l.7] is simpler than that given in [Gon1, p.452, l.-2-p.454,
l.3].
Representations of an elliptic function
In terms of s(z) [Guo, pp.476-477,
§8.9.1]
In terms of V(z) [Gon1, p.480, Theorem 5.62; Guo. p.478,
§8.9.2]
In terms of Ã(z) [Gon1, pp.468-469,
§5.34; Guo, p.479, l.1-p.480, l.2]
The coordinates of a cubic curve [Guo, p.486, (7)] can be expressed in terms of elliptic functions
[Guo, pp.485-486, §8.11].
The coordinates of a quartic curve can be expressed in terms of elliptic functions
[Guo, pp.486-489, §8.12].
Proof. A general quartic curve may be written in the following form: j_{4}(x, y) +
j_{3}(x, y) +
j_{2}(x, y) +
j_{1}(x, y) = 0,
where j_{k}(x, y) is a
homogeneous equation of degree k (k = 1,2,3,4) [Guo, p.486, (7)].
Find the roots of x^{3}j_{4}(1,
t) + x^{2}j_{3}(1,
t) + xj_{2}(1, t) +
j_{1}(1,
t) = 0. Note that the coordinates of the curve h^{2}
= R [Guo, p.655, (7)] can be expressed in terms of elliptic functions.
In order to clarify the meaning of [Guo, p.502, (6)], one must remember that
h
= h_{1},
w = w_{1},
w' = w_{3}, and
w_{2} =
- w_{1}
- w_{3} [Guo, p.498, l.-9; p.471, (14);
p.473, (10)]. Remark. In [Guo, p.499, (7)] we choose C = i, so
q'(0) = 2w_{1}
[Guo, p.500, l.9].
An elliptic function's integral forms
In [Guo, p.512, l.-3-p.514, l.3], Guo shows
that [Guo, p.512, (14)] can be reduced to Legendre's standard form [Guo, p.510,
(3)]. He divides his proof into two cases. Though his first case is simple, his
second case is tricky and complicated. The following view may make the
complicated transformations in Guo's proof appear more natural.
The standard integral form of an elliptic function is determined by its
differential equation. For Tan, its standard integral form is given by [Gon1,
p.395, (5.7-2)]. For sn^{-1},
its standard integral form is given by [Gon1, p.426, (5.17-10)]. For
Ã^{-}^{1},
its standard integral form is given by [Gon1, p.450, (5.27-13)]. [Gon1, p.395,
(5.7-2)] can be transformed to the standard integral form of sn^{-1}
[Gon1, p.438, l.14] because sn and Tan are related by [Gon1, p.421,
(5.17-1)]. Because Ã and Tan are related by [Gon1,
p.446, (5.27-1)], we may reduce [Gon1, p.450, (5.27-13)] to the standard
integral form of sn^{-1}
[Gon1, p.426, (5.17-10)].
Remark. In the formula given in [Guo, p.527, l.11], s
[Guo, p.513, l.-5] is fixed by w_{1} and
w_{3}. By contrast, in
[Gon1, p.447, (5.27-10)], w_{1} and
w_{2} are determined by
g. g is given by
[Gon1, p.447, (5.27-6)] and is more flexible than s
given in
[Guo, p.513, l.-5]. In fact, e_{2} in [Guo, p.513, (16)]
= e_{1} in [Gon1, p.447, (5.27-6)];
w_{2} in [Guo, p.527,
Exercise 9.11] = w_{1} in [Gon1, p.450, (5.27-10)];
g^{4} = [Gon1,p.447, (5.27-6)] = 9a^{2
}+ b^{2} [Guo,
p.513, (16)] = s^{4} [Guo,
p.513, l.-5].
Geometric representations of Jacobian elliptic functions [Guo,
§10.2] Remark 1. The proof of [Guo, p.541,
(1)] is simpler and more direct than that of [Gon1, p.429, (5.17-20)]. Remark
2. For intuitiveness
[Wat1, p.479, l.7] and simplicity [Wat1, p.481, l.1], sn should be defined as [Gon1,
p.421, (5.17-1)] instead of [Wat1, p.492, (A)].
Reducing a quartic to Legendre's standard form [Guo, p.546, (3)]
If the quartic is expressed as the product of linear factors, the method is
given in [Guo, p.550, l.-2-p.551, l.-6].
If the quartic is expressed as the product of sums of squares [Wat1, p.513, l.-11-p.514, l.11],
the method is given in [Guo, p.549, l.1-p.550, l.3].
Remark. If the quartic has four real roots, we may use the method a, other we
must use the more complicated method b [Guo, p.550, l.-5-l.-4].
Abel's method of proving the addition-theorem for Ã(z)
[Wat1, §20.312] can be considered a geometric interpretation of
the addition-theorem for Ã(z) [Wat1,
§20.3].
The inversion problem for the Ã-function [Sak1,
chap. VIII, §11,
§12 & §13;
Gon1, §5.41; Wat1, p.480, l.11-p.485, l.-11]. Remark.
There are some gaps and loopholes in [Wat1, p.455, l.22-l.32; p.484, l.1-l.29].
In contrast, [Sak1, chap. VIII, §11,
§12 & §13]
gives an excellent presentation on the modular function J(t)
and amends the above shortcomings: [Sak1, chap. VIII,
§11] provides clear
definitions of automorphic functions and modular functions; [Sak1, p.397, 2°]
fills the gap in [Wat1, p.484, §21.712];
[Sak1, p.398, l.1-p.399, l.4] closes the loopholes of the argument given in
[Wat1, p.484, §21.712]. [Gon1,
§5.41] makes some additional improvements:
[Gon1, p.491, l.13-l.23] is better than the proof of [Sak1, p.395, (12.7)];
[Gon1, p.494, l.8-p.495, l.6] is better than the argument given in [Sak1, p.398, l.15-p.399, l.7].
However, [Gon1, §5.41] fails to point out
the most important point: an elliptic function is a special case of an automorphic function [Sak1, p.388, l.20-l.26].
Evaluation of the elliptic integrals
The general method [Guo, p.457, l.-9-p.459,
l.5]
In terms of Ã:
[Guo, p.480, l.3-p.481,
l.10] (a special case of the general method) or
[Sak1, p.408, Theorem 14.9]
(using [Sak1, p.382, Theorem 8.6])
Remark 1. The first step is to change y^{ 2} = P(x),
where P(x) is a quartic polynomial, into the form given by [Sak1, p.405, (14.4)].
This form is related to Ã [Gon1, p.446, (5.27-3)].
Remark 2. In order to prove [Sak1, p.408, Theorem 14.9], we must use [Sak1,
p.406, Lemma 14.5] whose proof in turn uses [Sak1, p.403, Theorem 13.1].
In terms of Tan: [Gon1, §5.40]
(a special case of the general method)
Remark 1. The first step is to change w^{ 2} = P(x),
where P(x) is a cubic polynomial, into the form given in [Gon1, p.482, l.-9]. This form is related to Tan [Gon1, p.395,
(5.7-2)].
Remark 1. The definition of sn u given in [Guo, p.530, (1)]
fails to explain what motivates us to study this function.
The discussion given in [Wat1, §21.61] amends this drawback
and helps link [Guo, chap. 9] with [Guo, chap. 10].
Integration using the residue theorem
[Gon, p.683, Lemma 9.4] is an analytic statement. Its geometric meaning is described in [Lev,
p.206, l.12-l.19]. The proof of Jordan's lemma is divided into two cases: (1). The
subtended angle of the circular arc is less than or equal to p [Gon,
p.680, Lemma 9.2]; (2). The subtended angle of the circular arc is greater than
p [Sil, p.253, Lemma].
ò_{(-¥,+¥)} f, where f satisfies the conditions given in [Gon,
p.689, l.8-l.12]: [Gon, p.689, l.7-p.692, l.-11].
ò_{(-¥,+¥)} f, where f satisfies the conditions given in [Gon, p.692, l.-10-l.-9]:
[Gon, p.692, l.-10-p.693, l.5].
Remark. In [Pen, p.327, Example 7.9.2], we choose a sector as the contour for integration because
q = 2p/n is the primitive
period of w = z^{n} = (Re^{iq})^{n}.
ò_{(-¥,+¥)} f(x)cos
ax dx or ò_{(-¥,+¥)} f(x)sin
ax dx, where
a > 0 and f satisfies the conditions given in [Gon, p.692, l.8-l.11]:
[Gon, p.693, l.6-p.695, l.13].
(PV) ò_{(-¥,+¥)} f, where f satisfies the
same conditions as those in B except that f has simple poles on the real axis:
[Gon, p.697, l.9-p.700, l.-9].
(PV) ò_{(-¥,+¥)} f(x)cos
ax dx or ò_{(-¥,+¥)} f(x)sin
ax dx, where
a > 0 and f satisfies the same conditions
as those in C except that f has simple poles on the real axis:
[Gon, p.700, l.-8-p.704, l.-1].
(Integrands with a single branch point) ò_{(0,+¥)}
x^{a
}f(x)dx [Gon, p.706, l.3-p.708, l.13].
(Integrands with a single branch point)
(PV) ò_{(0,+¥)}
x^{a
}f(x)dx [Gon, p.708, l.14-p.712, l.10].
ò_{(0,+¥)} f(x)(ln
x)^{n}
dx, where n is a positive integer and f satisfies the same conditions as those
in A:
[Gon, p.713, l.4-p.717, l.2].
ò_{(0,+¥)} f, where f satisfies the conditions
1, 2, and 4 given in [Gon, p.717, l.4-l.8]:
For [Gon, p.717, Example 1], González made a
mistake in [Gon, p.718, l.7]. Therefore, read [Pen, p.331, Exercise 10] instead.
The discussion about the inverse Laplace transform given in [Gon, p.719,
l.5-p.721, l.12] is incomplete. For a concise account of the subject, read [Lev,
chap. 4, Sec. 7]. In [Lev, chap.4, Sec. 7], Levinson's real analysis proof of
[Lev. p.229, (7.9)] requires that f satisfies [Lev, p.228. (7.6)]. [Gon, p.721,
l.3-l.4] shows that this condition can be weakened by using the theory of
residues [1].
Remark. In [Pen, p.326, Example 7.9.1], we choose a rectangle as the contour for integration because z = 2ip
is the primitive period of w = e^{z}.
(Integrands with two algebraic branch points a and b) ò_{(a,
b)} f(x)dx or (PV)ò_{(a,
b)} f(x)dx, where f has an algebraic multiple-valued extension to
the complex plane [Gon, p.721, l.-2-p.730, l.9].
Remark. Suppose we circle around a branch point for w = [(z-a)(z-b)]^{1/2}.
The sign of the square root w on the cut when we encounter the branch cut for
the second time will be opposite to the sign of w on the cut when we encounter the branch
cut for the first time. Consequently, the sign of f_{0}(x)
on the lower boundary is opposite to the sign of f_{0}(x)
on the upper boundary [Gon, p.726, l.13-l.16]. This is how a branch cut for a
square root is designed. The computation given in [Gon, p.726, l.16] is
confusing.
[Ahl, p.154, Example 1]
[Ahl, p.159, Example 5]
Remark. When evaluating an integral, we should not only calculate its value,
but also specify the strongest convergence under which the integral
converges to that value.
The absolute convergence is stronger the ordinary convergence.
The absolute convergence of the integral given in [Ahl, p.154, Example 2] can be proved using
the condition given in [Gon, p.689, l.12].
The asymmetric (ordinary) convergence is stronger than the symmetric convergence
[Gon, p.686, l.1].
The use of a semicircle can only prove the symmetric
convergence of the integral given in [Gon, p.693, (9.11-11)], while the
use of a rectangle proves the asymmetric convergence of the integral
given in [Gon, p.695, (9.11-15)].
[Ahl, p.156, Fig. 25] is used to prove the asymmetric convergence of the limit given in [Ahl, p.157, l.4], while [Gon, p.698, Fig. 9.16] is used to prove the symmetric convergence of the integral given in [Gon, p.698, (9.11-21)].
The one-sided convergence (lim _{R®¥}
ò _{
[0, R]}) is stronger than the two-sided convergence (lim _{
e®
0+, R®¥}
ò _{[e, R]}).
Ahlfors finds the value of lim _{R®¥}
ò _{
[0, R]} (sin x)x^{-1} dx in [Ahl,
p.157, l.-11] using the concept of removable
singularity. In contrast, González
only finds the value of lim _{
e®
0+, R®¥}
ò _{[e, R]}
(sin x)x^{-1} dx in [Gon, p.703,
(9.11-28)].
The non-cut convergence is stronger than the cut convergence [Gon,
p.686, 2.(c)]. Ahlfors finds the value of lim
_{M® -¥, N®¥}
ò _{
[M, N]} (sin x)x^{-1} dx in [Ahl,
p.157, l.15], while González
only finds the value of lim _{
e®
0+, R®¥}
(ò _{[-R,
-e] }
+
ò _{[e, R]}) (sin x)x^{-1} dx
in [Gon, p.703, (9.11-27)].¬
Summation
Cesàro's method of summation [Zyg, vol.
1, chap. III, §1].
Asymptotic expansions (content: [Guo, §1.8];
application: calculating a given function's value within the prescribed accuracy [Wat1, p.150, l.-5-l.-1]).
Connected sets in the plane [Sak, Introduction
― theory of sets, §9].
Absolute continuity
f is absolutely continuous [Ru2, p.175, Definition 8.15]
Û
f is an integral [Ru2, p.178, l.13-l.15].
Remark 1. It is more important to recognize the natural links of absolute
continuity than to know everything about absolute continuity [Ru2, pp.128-133,
Absolute Continuity].
Remark 2. By [Ru2, p.132, Theorem 6.11], the concept of absolute continuity can apply to a more abstract setting: a pair of measures.
Tests for convergence
Tests for a series
Root test [Ru1, p.57, Theorem 3.33 (a)]
Ratio test [Ru1, p.57, Theorem 3.34 (a)]
Remark. The hypothesis of the ratio test satisfies the hypothesis of the root
test, so the hypothesis of the root test is weaker than that of the ratio test.
Consequently, the root test is a more refined test.
The convergence test for a power series
is based on the root test for a series [Ru1,
p.60, Theorem 3.39]
Tests for a trigonometric series [Zyg, vol.1, p.4, Theorem 2.6 & Theorem 2.7]
Tests for a Fourier series
The Dini test [Zyg, vol.1, p.52, Theorem 6.1 & l.17]
The Dirichet-Jordan test [Zyg, vol.1, p.57, Theorem 8.1; p. 60, Theorem
8.14]
The Dini-Lipschitz test [Zyg, vol.1, p.63, Theorem 10.3]
Lebesgue's test [Zyg, vol.1, p.65, Theorem 11.5; p.66, l.1-l.4]
fÎl^{p}_{1/p}, where p>1 [Zyg, vol.1, p.66, Theorem 11.10]
Remark 1. If f satisfies the Dini-Lipschitz condition in an interval containing x, then f satisfies
[Zyg, vol.1, p.65, (11.6); p.66, (11.7)]. See [Zyg, vol.1, p.66, l.9-l.10].
Remark 2. The hypothesis of the Dini test satisfies the hypothesis of Lebesgue's
test. More specifically, the fact that the first term of [Zyg, vol.1, p.52, (6.2)]
is finite implies [Zyg, vol.1, p.65, (11.6)]; the fact of the second term of [Zyg, vol.1, p.52,
(6.2)] is finite implies [Zyg, vol.1, p.66, (11.7)] [1].
Tests for a Fourier integral
The Dini test [Tit, p.14, Theorem 4]
The Dirichet-Jordan test [Tit, p.13, Theorem 3]
Uniform convergence
in angular regions
Power series: [Sak, chap. III, §2].
Dirichlet series: [Sak, chap. IX, §8].
Remark. [Zyg, vol.1, p.100, Theorem 7.6] and the statement given in [Zyg, vol.1, p.98, l.16-l.17] can be viewed as special cases of [Sak, p.433, Theorem 8.7].
Remark 1. As for estimation of orders of error for various summations of Fourier series, we are not interested in any particular function. Instead, we are interested in a group of functions.
Remark 2. The one-to-one correspondence between the classes and the orders of
error estimates such as A and B reminds us of the one-to-one correspondence
between subgroups and subfields in the Galois theory [Jaco, vol.3, p.41,
Fundamental Theorem of the Galois Theory].
From Fourier series for periodic functions to Fourier integrals for
non-periodic functions
The function conjugate to f [Zyg, vol.1, p.51, l.7] ® the Hilbert transform
of f
[Zyg, vol.2, p.243, l.-17].
[Zyg, vol.2, p.242, Theorem 1.3] [1].
Remark 1. This theorem shows that the partial sums of S[f]
and the integral given in [Zyg, vol.2, p.242, (1.1)] are essentially the same. [Zyg, vol.2, p.243,
(1.7)] shows S_{w}^{~}(x_{0})
- f^{ ~}(x_{0}) and s_{n}^{~}(x_{0})
- f_{a}^{~}(x_{0}) are essentially the
same. If a new theory (Fourier integrals) is essentially the same as an old
theory (Fourier series), we should not allow them to go separate ways [Wat1,
§9.7; Ru3, chapter 7]. Instead, we should constantly link the new theory back to the old one
during the new theory's development by proving that the corresponding concepts
and theorems are essentially the same. Otherwise, we cannot take advantage of the theorems
of the old theory as we develop the new one. In addition, only by
comparing the old theory to the new one may we understand the limitations of the
new theory [Zyg, vol.2, p.244, l.-4-p.245, l.4].
Remark 2. This theorem provides a convenient tool for us to transform theorems [Zyg,
vol.1, p.52, Theorem 6.1; p.57, Theorem 8.1] in the theory of Fourier series
directly to corresponding theorems in the theory of integrals. This tool also
allows us to immediately prove the latter theorems by the former theorems. We
need not spend time formulating the corresponding theorems and then proving them
from scratch as Titchmarsh did in [Tit, p.14,
Theorem 4; p.13, Theorem 3]. Since the convergence established in [Zyg, vol.2,
p.242, Theorem 1.3] is a strong convergence, this strong convergence will automatically establish
the
weak convergence in a theorem for Fourier integrals if the same weak convergence
in the corresponding theorem for Fourier series is already established. ¬
Just as S_{n} is the n-th partial sum of the Fourier series S[f], S_{w}(x)
given in [Zyg, vol.2, p.244, (1.10)] is a partial integral of the Fourier
integral given in [Zyg, vol.2, p.244, (1.12)].
Just as the series conjugate to the series given in [Zyg, vol.1, p.1, (1.5)] is
given in [Zyg, vol.1, p.3, (1.7)], the integral conjugate to the last integral in [Zyg,
vol.2, p.244, (1.12)] is defined to be the last term of [Zyg, vol.2, p.244,
(1.16)].
[Zyg, vol.1, p.89, Theorem 3.4]
® [Zyg, vol.2, p.246, the first statement of Theorem 1.21 (l.4-l.5)].
[Zyg, vol.1, p.90, Theorem 3.9] ® [Zyg, vol.2, p.246, the second statement of Theorem 1.21 (l.5-l.7)].
Conformal mappings [Sak, chap. I, §15;
Ahl, pp.73-76, §2.3]
The Arzela-Ascoli theorem (Compare [Sak, p.54, Theorem 4.4] with
[Ahl,p.214, Theorem 11])
A theorem is not merely a pile of information. The formulation of a
theorem should stress its essence. If AÞB
is essential and BÞA is trivial, then we
should say AÞB rather than AÛB.
Otherwise, the theorem's statement may obscure its essence.
[Ahl, p.214, Theorem 11] generalizes [Sak, p.54, Theorem 4.4] from S^{2}
to a complete metric space. The generalization not only sacrifices the
theorem's natural setting [Sak, p.53, Theorem 4.1] to accommodate the
metric space, but also complicates the formulation of the theorem's
hypothesis. [Sak, p.54, Theorem 4.4] has one hypothesis. In contrast,
[Ahl, p.214, Theorem 11] has two hypotheses [Ahl, p.214, l.7-l.8].
[Ahl, p.216, Theorem 12] should have stated that a locally bounded
family is equicontinuous on compact sets. [Ahl, p.216, l.12-l.19] repeats
the argument given in [Ahl, p.214, l.-6-p.215,
l.22]. See the proof of [Sak, p.51, Theorem 3.3].
The Cauchy-Riemann equations [Ru2, p.250, l.1-p.251, l.15]
Remark. [Sak, p.58, l.12-p.59, l.17; p.99, l.8-l.24; p.113, l.-3-l.-2] indicate that Saks fails to grasp the essence of
the Cauchy-Riemann equations. He made a mess because he worried about
what he should not.
Rouché's theorem
The proof given in [Lang1, p.158, Theorem 1.4] is clear and organized.
The formulation of Rouché's theorem given in
[Lang1, p.158, Theorem 1.4] is better than that given in [Ru2, p.242,
Theorem 10.43]. The version of Rouché's
theorem given in [Sak, p.157, Theorem 10.2] is convenient for
applications. In addition, [Sak, p.157, l.-10]
highlights the key point of this theorem: it is derived from homotopy
theory.
In the proof of [Sak, p.157, Theorem 10.2], the boundary of the closed
set F is
different from S L_{j} [Sak, p.157,
l.-4]. The proof given in [Ru2, p.242,
Theorem 10.43] also involves two curves: G
and G_{0}. In contrast, the proof
given in [Lang1, p.158, Theorem 1.4] involves only one curve
g.
[Sak, p.155, Theorem 10.1] should have been divided
into two parts: The first part should have stated that Cauchy's theorem,
Cauchy's formula, and [Sak, p.154, Theorem 9.2] are not only valid for
rectangles but also for any cycle homologous to 0 in a region [Ahl,
p.145, Theorem 18].
The second part should have proved that the
constructed curve given in [Sak, p.155, l.-7-p.156,
l.2] is a finite sum of closed curves homologous to 0. In this way one
trivial
proof can do all that the three proofs do. See [Sak, p.156, l.2-p.157,
l.6]. Note that the proof pattern of III [Sak, p.156, l.-4-p.157,
l.6] is different from that of I and II [Sak, p.156, l.4-l.-5].
¥
In complex variables,
is not different from any other point on S^{2}.
If we exclude ¥ from the domain and range of
a meromorphic function, the function will become holomorphic. Thus, the
discussion of open mapping theorem for holomorphic functions [Ru2,
p.231, l.1-p.233, l.-5] is incomplete, while
the discussion of open mapping theorem for meromorphic functions [Sak,
chap. III, §12] is complete.
In the open plane [Sak, p.17, l.2], the convergence
to a complex number is different from the divergence to
¥. However, on S^{2} the
conventional divergence to ¥ and the
convergence to ¥ are the same. The
definition of a normal family given in [Sak, p.50, l.16-l.21] uses the
former convention rather than the latter one.
Weierstrass' preparation theorem [Sak, chap. III,
§14] Remark. We may prove the statement given in [Sak, p.168, l.19-l.20]
using a method similar to the one given in [Jaco, vol. 1, p.108,
l.12-p.109, l.7]. This approach requires that we embed the entire
reduction procedure in the induction step. Whenever we use the method of
mathematical induction in a proof, we should limit the scope of the
induction step as narrowly
as possible even if the induction is finite. In terms of the computer
language, when we use a subroutine recursively, we must restrict the
content of the subroutine to a minimum. Otherwise, it may consume a lot
of time and memory. Thus, Saks provides a better proof in [Sak, p.169, l.-12-p.170,
l.10] because he uses the mathematical induction to establish only two
general formulas in [Sak, p.169, l.-4-l.-3;
p.170, l.5-l.6].
Jacobson's approach given in [Jaco, vol. 1, p.108, l.-14-p.109,
l.7] has another drawback. The exhaustive search for the next highest
term [Jaco, vol. 1, p.109, l.2-l.4] of the homogeneous symmetric
polynomial is ineffective. The general formulas given in [Sak, p.169, l.-4-l.-3;
p.170, l.5-l.6] can easily be inferred by considering simple cases k=1
and k=2.¬
Runge's theorem [Sak, p.176, Theorem 2.1]
The construction given in [Sak, p.177, l.3-l.11] is effective.
In contrast, the proof of [Ru2, p.288, Theorem 13.6] is ineffective.
First, the Hahn-Banach theorem used in [Ru2, p.288, l.-5] introduces the
axiom of choice. We should avoid using this dubious axiom. Second, Rudin
unnecessarily uses the complicated Riesz representation theorem in [Ru2,
p.288, l.-3-l.-2].
One may easily translate Saks' proof into a computer program, but not
Rudin's.
An annulus vs. the Laurent series [Ahl, chap. 5,
§1.3]
Saks should have omitted [Sak, chap. 3, §5]
even though he uses [Sak, p.142, theorem 5.7] in [Sak, p.144, l.-11].
Saks should have replaced [Sak, chap. 3, §5]
with the passage given in [Sak,
p.196, l.-11-p.199, l.12] because the two
passages use the same method and the theorems in the latter passage are
more general. [Sak, p.142, Theorem 5.7] should have been
regarded as a trivial corollary of [Sak, p.198, Theorem 9.6].
The degree of connectivity of a region [Sak, chap. IV,
§12]
The Riemann-zeta function
Domain expansion of the
generalized Riemann-zeta function: s > 1 [Wat1, p.265,
l.18]; s < 0 [Wat1, p.269, l.2]; s¹1
[Wat1, p.270, l.7].
Estimates: [Wat1, p.276, l.7-l.14]
Remark 1. Corrections:
a. s £ d given in [Wat1, p.276, l.7] should have been corrected as s £ -d.
b. The statement given in [Wat1, p.275, l.11] is incorrect. It should
have been replaced by the following two lines:
(1-d£s£1) Þ z(s, a) = O{|t|^{1-s} log |t|}.
(1£s£1+d) Þ z(s, a) = O(log |t|).
c. The statement given in [Wat1, p.275, l.-14] is incorrect. It should be corrected as follows:
z(s) = O[exp{2^{-1} p
|t| + (2^{-1}-s-it)(log
|1-s| + i arc tan [(-t)/(1-s)])}]z(1-s).
d. In [Wat1, p.275, l.-11], we must specify the range of s as follows:
($K>0)(-K<s<1-d).
The constant implied in the symbol O given in [Wat1, p.275, l.-11]
depends on K.
Remark 2.
The constants implied in the symbol O's given in [Wat1, p.276, l.7-l.9]
depend on d.
Remark 3. The constant implied in the symbol O given in [Wat1, p.276, l.11] changes if we replace the divider 1/2 of the s-axis with another number
b between 0 and 1. This is because we have to let d
equal b rather than 1/2.
The asymptotic expansion of log G(s+a) [Wat1, §13.6]
Remark 1. Corrections: a. Kz^{-n-1/2}e^{-d|t|t(-n-1/2)}|t|
given in [Wat1, p.277, l.-4] should be replaced by
Kz^{-n-1/2}e^{-d|t|}|t|^{t(-n-1/2)}.
b. The absolute convergence of the double series can be proved as follows:
S_{n=1}^{¥}S_{m=2}^{¥}m^{-1}(|z|/a)^{m}(1+n/a)^{-m }£ (S_{n=1}^{¥}(1+n/a)^{-2})(S_{m=2}^{¥}(|z|/a)^{m}).
The hint given in [Wat1, p.276, l.-11] is
useless.
Remark 2. The constant implied in the symbol O given in [Wat1, p.278, l.2]
depends on d, where |arg z| £ p-d, so
the constant implied in the
symbol O given in [Wat1, p.278, l.-11] depends on d.
Remark 3. The constant implied in the symbol O given in [Wat1, p.276, l.2-l.9] depends on d, where d is related to the range of s.
In contrast, the constant implied in the symbol O given in [Wat1, p.277, l.-11] does not depend on d, where d is related to the arg z. However, the quantity inside the parenthesis behind O
given in [Wat1, p.277, l.-11] depends on d.
By the integral given in [Wat1, p.278, l.1], the constant implied in the symbol O given in [Wat1, p.278, l.-11] depends on d.
Remark 4. d does not affect the equality
given in [Wat1, p.277, l.12]. However, if we want to choose an T_{0} so that for every
T³T_{0},
(2pi)^{-1}ò_{[(3/2)-Ti, (3/2)+Ti]} pz^{s}(s sin ps)^{-1}z(s, a)ds approach log
[G(a)(G(z+a)^{-1}]
+ z
G'(a)G(a)^{-1} within the admissible error, then
T_{0} depends on d.
A similar discussion applies to the equalities given in [Wat1, p.278, l.-2;
p.279, l.4]. Remark
5. If one can understand all the constants implied in the symbol O's in
[Wat1, §13.6], one would truly understand the symbol O. The problem with
[Guo, §3.21] is that the authors fail to truly understand the symbol O.
The parameters on which a constant implied in symbol O's in [Guo, §3.21]
depends are not traceable. For example, both T_{0} (see Remark
4) in [Guo, p.126,
(2)] and the constant implied in the symbol O given in [Guo, p.127, (5)] depend on d, but the authors
specify the range of z by |arg z| < p
[Guo, p.126, l.12; p.127, l.-3] rather
than |arg z| £ p-d. Back to [Wat1]. If many
readers have a difficult time understanding a topic in a textbook, it
means that the topic in that book is not well written. In my opinion, if
the symbol O depends on the parameter d, we should write O_{d} instead of O
to clarify its meaning.¬
Zero-free regions [Ell, p.131, Theorem 4.5]
The number of zeros in {s+it | 0<s<1,
0 < t £ T} [Ell, p.159, Theorem 5.7]
Zeros on the critical line Re s = 1/2 [Ell, p.173, Theorem 5.11].
Integral equations
Fredholm's integral equation [Boc, §6-§8; Wat1, §11.2, §11.21, and §11.22]
The statement given in [Wat1, p.217, Example 1] should have been inserted between [Wat1, p.216, l.-10] and
[Wat1, p.216, l.-9]. See [Boc,
p.35, l.13-15]. The argument given in [Wat1, p.218, l.-5-p.219, l.10]
meaninglessly repeats that given in [Wat1, p.216, l.-9-p.217,
l.9]. Without using [Wat1, p.217, Example 1], both the argument in [Wat1, p.217,
l.10] and that in [Wat1, p.219, l.11] would not work.
Just as the eigenfunctions of a Sturm-Liouville system can be considered the eigenfunctions of a self-adjoint differential operator,
the eigenfunctions of the homogeneous integral equation of the form given in [Wat1, p.227, l.14] can be considered eigenfunctions of a self-joint integral operator
[Wat1, §11.5, §11.51, §11.6, & §11.61].
Remark 1. The inequality U_{2n+2}/U_{2n }³ n^{n} given in [Wat1, p.223, l.-4] should be corrected as U_{2n+2}/U_{2 }³ n^{n} .
Remark 2. The dimension of the eigenspace corresponding to a characteristic number l
is finite [Wat1, p.228, l.9]. Otherwise, there would be an infinite number of l which appear on the diagonal of the integral operator and we
would not be able to define the norm of the operator.
Abel's integral equation [Boc, §3]
Schlömilch's integral equation [Wat1, §11.81]
Continued fractions
Contractions for faster convergence [Perr, §43]
Equivalence of continued fractions and series [Perr, §45]
Remark. [Perr, p.210, (19)] can be proved by [Ru2, p.321, Lemma 15.3].
Conditional versus unconditional convergence [Perr, §48]
Broman and Stern's criteria for divergence [Perr, §49]
Convergence for positive elements [Perr, §50]
The Pringsheim criteria for convergence [Perr, §53]
The van Vleck-Jensen criteria for convergence [Perr, §54]
Periodic continued fractions [Perr, §55]
limit-periodic continued fractions [Perr, §56]
Viewing a continued fraction as a sequence of recursion formulas [Perr, §57]
(The Sturm-Picone Comparison Theorem) [Inc1, §10.31; Nag, p.704, Theorem 17; p.713, group project C]
Let f_{1} be a nontrivial solution to the Sturm-Liouville equation
(d/dx)[p_{1}(dy/dx)]+Q_{1}y=0, a<x<b,
and let f_{2} be a nontrivial solution to
(d/dx)[p_{2}(dy/dx)]+Q_{2}y=0, a<x<b.
Assume that p_{1}≥p_{2}>0 and Q_{1}≤Q_{2} for x in [a,b]. Then between any two consecutive zeros x_{1} and x_{2} of f_{1} in [a,b], there is a zero of f_{2}, unless f_{1} and f_{2} are linearly dependent on [x_{1},x_{2}], in which case Q_{1}(x)ºQ_{2}(x) on [x_{1},x_{2}].
Proof.
(The Picone formula)
Let f_{2}≠0 in [x_{1},x_{2}]. By [Inc1, p.226, l.9-l.10],
ò_{[x1,x2]} (Q_{2}-Q_{1})f_{1}^{2}dx + ò_{[x1,x2]} (p_{1}-p_{2})(f_{1}')^{2}dx + ò_{[x1,x2]} p_{2} [f_{1}'-([f_{1}f_{2}']/f_{2})]^{2}dx =0.
If f_{2}≠0 in (x_{1},x_{2}), but f_{2} is 0 at x = x_{1} or x = x_{2},
then the Picone formula still holds [Inc1, p.226, l.-21-l.-20].
If Q_{1}(x)ºQ_{2}(x) on [x_{1},x_{2}]
is not true, then f_{2} must have a zero in (x_{1},x_{2}). Otherwise the left-hand side of the Picone formula is greater than 0.
If Q_{1}(x)ºQ_{2}(x) on [x_{1},x_{2}], then f_{1} and f_{2} are linearly dependent on [x_{1},x_{2}].
Proof. In an interval where p_{1}(x)ºp_{2}(x), f_{1} and f_{2} satisfy the same differential equation and their Wronskian is 0.
In an interval where p_{1}(x)¹p_{2}(x), f_{1}' = f_{2}' = 0. Consequently, their Wronskian is also 0.