Equalities in Differential Equations

  1. The identification of K'-l with Kl [Gon1, p.404, (5.11-1) and (5.11-2)]
        Both [Gon1, p.416, l.6] and [Gon1, p.418, (5.15-17)] require that K'-l be identified with Kl. Kl is constructed using [Gon1, p.405, (5.11-4)]; K'-l is constructed using [Gon1, p.407, (5.11-9)]. In order to identify these two numbers, we must identify the procedures through which they are constructed. The integral in [Gon1, p.405, (5.11-4)] and the integral in [Gon1, p.407, (5.11-9)] are the same except for the sign of l. The assignments of e in [Gon1, p.405, l.-2] and [Gon1, p.407, l.-4] are consistent with the transformation w = i w' [Gon1, p.407, l.12].

  2. (Taking square roots)
        In the theory of elliptic functions, we frequently encounter the problem of taking square roots on both sides of an identity. The sign determination can be troublesome. We may use the Taylor series expansion to solve this problem.
    Example 1. The proof of [Gon1, p.412, l.16].
    Example 2. We use [Gon1, (5.12-7) & (5.12-8)] to prove [Gon1, p.436, (5.22-9)].
        The sign of a square root can also be determined from a path of integration. For example, the minus sign in [Wat1, p.502, l.20] comes from the minus sign in [Wat1, p.502, l.5].
    Remark 1. As one reads [Gon1, p.399, l.-12-l.-8], one must have the following facts in mind. Suppose f is a polynomial. The Riemann surface of f1/2 has two sheets, the upper one and the lower one. The number of branch cuts is equal to the number of zeros of f. Whenever a curve crosses a branch cut, it must go from one sheet to the other sheet in order to be continuous.
    Remark 2. The path of the integral given in [Gon1, p.451, (5.28-5)] starts at e2 and ends at , while the path of the integral given in [Gon1, p.450, (5.27-13)] starts at and ends at e2. The two integrals seem to differ in their signs [Gon1, p.452, l.5]. How does one make these seemingly contradictory statements compatible? The answer is that -1(z) is a multiple-valued function [Gon1, p.450, l.-5]. The value of the integral given in [Gon1, p.450, (5.27-13)] depends on what choice one makes for the integral path [Guo, p.460, l.1-p.461, l.17]. (-w2) = e2 is allowed. See [Gon1, p.448, Theorem 5.47 (1) and (4)] or [Guo, p.461, l.-8]. In order to prove that the second part approaches zero [Guo, p.460, l.-3], we consider the typical case C z-1/2 dz, where C is a small circle centered at z = 0. Let z = reiq.
    Remark 3. In order to easily trace back to the definition of a square root and to avoid the trouble of determining the signs of a square root, we should not arbitrarily square a quantity.
        From the way Guo has presented his material in [Guo, p.508, l.11-p.509, l.7], it seems that the derived formulas [Guo, p.509, (6) & (7)] are correct simply because Guo has made lucky choices of signs when taking square roots. The key to correct Guo's problem is to never square a quantity arbitrarily [Guo, p.503, (4)]. Once we correct this problem, [Guo, p.508, (2)] can be derived directly from [Guo, p.484, (17)].
        Squaring a quantity may seem to facilitate calculations at the first glance. However, when one tries to reverse the effect by taking a square root in a later stage, one will discover that squaring a quantity only asks for the trouble of determining the signs.
        If we trace back to the roots, the values of the square roots involved in [Guo, p.509, l.2-l.7] are predetermined by and inherit from [Guo, p.508, (2)] which in turn is derived from [Guo, p.484, (17)]. [Gon1, p.464, l.-4-l.-3] explains how the square root given on the left-hand side of [Gon1, p.464, (5.33-14)] is defined.

  3. (Addition theorems for the elliptic functions of order two)
        The addition theorems for the Jacobian elliptic functions can essentially be derived from the addition theorem for w = Tan z [Gon1, p.413, (5.14-1)]. See [Gon1, p.444, Exercise 5.3(3); p.445, Exercise 5.3(8)]. The addition theorem for Tan z can be considered the solution of the total differential equation given in [Gon1, p. 414, (5.14-4)]. Similarly, the addition theorem for (z) [Gon1, p.460, Theorem 5.52] is the solution of the two equations given in [Gon1, (5.32-8) and (5.32-1)]. The latter equation is obtained by manipulating the poles. The proof of [Gon1, p.458, (5.32-1)] can also be proved by taking the logarithmic derivative of [Guo, p. 481, (1)] which is also obtained by manipulating the poles. By contrast, the proof of [Guo, p.482, (4)] is more organized than that of [Gon1, p.459, (5.32-5)].
        [Gon1, p.460, (5.32-6)] can also be derived from [Guo, p.482, (5)]. The constant term of the Laurent series of the left-hand side of [Guo, p.482, (5)] is 2(u) [Guo, p.482, l.-4] because (z) contributes 0 to the constant term [Gon1, p.453, (5.29-2)].
        [Guo, p.483, l.8-l.1] provides a third proof of [Gon1, p.460, (5.32-6)].
    Remark. We may use [Gon1, p.444, Exercise 5.3(3)] to prove [Gon1, p.435, (5.22-1)]. By the way, the first equality in [Gon1, p.435, (5.22-1)] is incorrect.

  4. Summation of a series
        If is sum of a series is not given, we must use one of the following two methods to find the sum:
    1. Using the Fourier series expansion [Edw, 9.2]
    2. Using the residue theorem [Gon, 9.12].

  5. Proofs and development for a large number of available equalities [Guo, p.505; pp.523-524, Exercises 9.2, 9.3 & 9.4]
    1. (The general method) In order to prove these equalities, we must find a general method that applies to the proofs of all the equalities. This general method is given in [Guo, p.504, l.-2-p.505, l.7].
    2. Find shortcuts for the proofs [Guo, p.504, l.8-l.11; l.18-l.19]
          For Exercise 9.2, it is enough to prove the formula given in [Guo, p.524, l.1] using the general method. The remaining formulas of this exercise can be derived from the formulas given in [Guo, p.505; p.524, l.1] using shortcuts.
    3. Why there is always a corresponding formula available no matter which way we form pairs [Guo, p.505, (6); p.524, Exercise 9.3]
          By choosing {1,4} and {2,3} to form two pairs, the corresponding formula given in [Guo, p.505, (6)] is available. This is because for both period 1 and period t, the pairs {J1, J4} and {J2, J3} have the same features [Guo, p.502, (7)]. In fact, no matter which way we form pairs, both the left column and the right column of [Guo, p.502, (7)] will have two similar pairs for the purpose of constructing the corresponding formula.

  6. The equality given in [Perr, p.280, l.5-l.6] can be derived from the theory of continued fractions.