Successive approximations for a second-order ODE
Goal: [Wat1, p.195, l.-7]
Design: [Wat1, p.195, l.6]
Principle based on: successive approximations
key idea: (d/dz)ò [b, z] (z-z) f(z)dz = -ò[b, z]
f(z)dz.
The reason for distinguishing regular singular points from irregular singular points
[Wat1, p.198, l.15-l.19]
If we want to appreciate the distinction between regular and irregular singular points, we must create an
example,
say 2x3y" + (1+x)y'+3xy = 0, whose indicial equation gives only
one root for a.
Integral transforms [Guo, §2.12]
Goal: Solve ODE's by integrals [Guo, p.79, (9)].
Design: [Guo, p.79, l.11-p.80, l.10]
Principle based on: [Guo, p.79, (6)]
Key idea: [Guo, p.80, l.14-l.19]
Examples: [Guo, p.80, l.-9-l.-1]
A contour design for the case given in [Guo, p.82, (ii)]
Goal: Solve ODE's using the Laplace transform. More specifically, the contour
must satisfy the condition given in [Guo, p.82, (7)].
Design: [Guo, p.83, Fig. 3]
Principle based on: a1 is a branch point of the Riemann surface for w = log (z-a1).
If a point P goes along the contour circling a1
counterclockwise once and then clockwise once, P will remain on the same sheet of
the Riemann surface. Consequently, the starting point and the ending point of the contour given in [Guo, p.83, Fig. 3] will have the same
value for log (z-a1).
Key ideas: It is unnecessary to consider the Riemann surface for w = (z-a1)l1
(z-a2)l2 [Guo, p.83, l.2-l.3]. All we have to do is paste the contour
given in [Guo, p.83, Fig. 3] on the Riemann surface for w = log (z-a1).
After considering (z-a1)l1, we paste
the contour on
the Riemann surface for w = log (z-a2). Remark. Instead of using the hint given in [Guo, p.83, l.5], I use [Ru2, p.27 Theorem 1.34] to justify the differentiation under the integral sign
given in [Guo, p.83, (12)].
Differentiability of angle functions [O'N, p.50, Exercise 12]
The possible radical change of a curve at the point whose curvature is zero [O'N, p.76, Exercise 19]