[整理版]习题解答：数学物理方法（吴崇试编著第二版）

1、Methods of Mathematical PhysicsSolution of Exercise ProblemsYan Zeng March 9, 20085Contents1 Complex Numbers and Complex Functions52 Analytic Functions52.1 Exercises in the text52.2 Exercises at the end of chapter73 Complex Integration104 Inftnite Series134.1 Exercises in the text134.2 Exercises at

2、the end of chapter145 Local Expansion of Analytic Functions185.1 Exercises in the text185.2 Exercises at the end of chapter196 Power Series Solution of Second Order Linear ODE246.1 Exercises in the text246.2 Exercises at the end of chapter247 Residue Theorem and Its Applications337.1 Exercises in th

3、e text337.2 Exercises at the end of chapter388 Function529 Laplace Transform559.1 Exercise in the text559.2 Exercise at the end of chapter5610 Function6211 Complex Functions in Mathematica6812 Equations of Mathematical Physics6813 General Solutions of Linear PDE6913.1 Exercise in the text6913.2 Exer

4、cise at the end of chapter6914 Separation of Variables7214.1 Exercise in the text7214.2 Exercise at the end of chapter7615 Orthogonal Curvilinear Coordinates8315.1 Exercise in the text8315.2 Exercise at the end of chapter8416 Spherical Functions8516.1 Exercise in the text8516.2 Exercise at the end o

5、f chapter8717 Cylinder Functions8717.1 Exercise in the text8717.2 Exercise at the end of chapter8818 Summary of Separation of Variables8818.1 Exercise in the text8818.2 Exercise at the end of chapter8819 Applications of Integral Transforms9020 Method of Greens Function9421 Introduction to Calculus o

6、f Variation9421.1 Exercise in the text9421.2 The RayleighRitz method and its application to the SturmLiouville problem9621.3 Solutions of the exercise problems from Gelfand and Fomin 4, Chapter 89721.4 Exercise at the end of chapter10022 Overview of Equations of Mathematical Physics10222.1 Summary o

7、n the classification of second order linear PDE10222.2 Exercise at the end of chapter103A The Black-Scholes partial differential equation106A.1 Derivation of the Black-Scholes PDE and its boundary conditions106A.2 Simplification of the Black-Scholes PDE via change-of-variable107A.3 Solution of the s

8、implified PDE and the Black-Scholes call option pricing formula108This is a solution manual of selected exercise problems from Methods of Mathematical Physics, 2nd Edition (in Chinese), by Wu Chong-Shi (Peking University Press, Beijing, 2003).1 Complex Numbers and Complex FunctionsExercises are omit

9、ted since they are straightforward.2 Analytic Functions2.1Exercises in the text2.1.Proof.xxxxxyyyif = i . u + iv = v + iu, f = u + iv .So if = fif and only if Cauchy-Riemann equations hold.xy2.2.22z2z2Proof. Since x = 1 (z + z) and y = i (z z), we have x = 1 and y = i . Soz2 x2 y2 x2 y2xy2yx f = . 1

10、 u + i u + i . 1 v + i v = 1 . u v + i . u + v .zTherefore f = 0 if and only if Cauchy-Riemann equations hold.2.3.Proof. Apply Cauchy-Riemann equations. 2.4.Proof. Omitted since the proofs are based on the definition and are similar to those for functions of real variables.122.5.Proof. Let z and z b

11、e any two distinct points in the complex plane. Define f (z) = exp i 2z(z1 +z2) .z2 z1Then f (z) is analytic with f (z1) = f (z2) = 1. But f j(z) = 2i exp .i 2z(z1 +z2) 0, z C. So thez2z1mean value theorem does not apply to f (z). (This example is from 8.)2.6.z2z1Proof. Suppose f (z) = u(x, y) + iv(x, y). Then by Cauchy-Riemann equations, f j(z) = 0 in tt implies all the partial derivatives of u and v with respect to x and y equal to 0 in tt. Using the r