20.2 稳定问题Green函数的一般性质.(519) 20.3 三维无界空间Helmholtz方程的Green函数.(523) 20.4 圆内Poisson方程第一边值问题的Green函数.(528) *20.5 三维调和函数的均值定理与极值原理.(537) 20.6 波动方程的Gren函数.(539) 20.7 热传导方程的Green函数 .(547） 第二十一章变分法初步.(551) 21.1泛函的概念.(551) 21.2泛函的极值.(553) 21.3 泛函的条件极值.(560) 21.4微分方程定解问题和本征值问题的变分形式.(564) .21.5变边值问题.(568) 21.6 Rayleigh-itz方法.(570) 第二十二章数学物理方程综述.(576) 22.1 二阶线性偏微分方程的分类.(676) 22.2线性偏微分方程解法述评.(582) 22.3非线性偏微分方程问题 .(585 22.4结束语. . .(591 参考书目.(592) 外国人名译名对照表.(594) ·20
Contents Part I Complex Analysis Chapter 1 Complex number and function of a complex variable.(1) 1.1 Complex numbers and complex algebra.(1) 1.2 Geometric representation of complex numbers.(2) 1.3 Complex sequence.() 1.4 Function of a complex variable.(9) 1.5 Limit and continuity.(10) 1.6 Point at infinity. 1.7 Construction of regular heptadecagon.(13) Chapter 2 Analytic functions. .(15) 2.1 Complex differentiability.(15) 2.2 Analytic functions.(.17) 2.3 Elementary functions.(20) 2.4 Multivalued functions.(23 2.5 Conformal property of analytic functions. (30) Chapter 3 Complex integration.(38) 3.1 Complex integration. .(38) 3.2 Cauchy integral theorem for simply connected domain.(40) 3.3 Cauchy integral theorem for multiply connected domain.(45) 3.4 Cauchy integral formula.(47) 3.5 Higher-order deriatives of an analytic function.(50) 3.6 Some consequences of the Cauchy integral formula. "3.7 Poisson integral formula.(55) Chapter 4 Infinite series.(59) 4.1 Complex series.(59) 4.2 Double series. .(63) ·21·
1.3 Series of complex functions. .(65) 4.4 Power series. .(70) 4.5 Analyticity of integrals containing parameter.(73) 4.6 Eutler summation.(76) 4.7 Divergent series and asymptotic series.(80) Chapter 5 Local expansion of an analytic function.(87) 5.1 Expansion in Taylor series.(87) 5.2 Illustractive examples of Taylor serics. (90) 5.3 Expansion in Laurent series.(94) 5.4 Illustractive examples og Iaurent series. .(97) 5.5 Isolated singularities of uniform function.(101) 5.6 Bernoulli numbers and Euler numbers. .(105) 5.7 Holomorphic functions and meromorphic functions.(108) Chapter 6 Solution in power series to linear ordinary differential equation of second order.(109) 6.1 Ordinary and singular points of linear ordinary differential equation of second order.(109) 6.2 Solutions valid in the vicinity of an ordinary point.(111) 6.3 Solutions valid in the vicinity of a regular singular point.(115) 6.4 Solutions to the Bessel equation.(119) .6.5 Solutions valid in the vicinity of an irregular singular point.(131) Chapter 7 Analytic continuation.(137) 7.1 Isolated zeros and identity theorem for analytic functions.(137) 7.2 Analytic continuation.(140) Chapter 8 Residue theorem and its applications.(145) 8.1 Residue theorem.(145) 8.2 Evaluation of definite integrals of rational trigonometric functions. .(150) 8.3 Evaluation of definite integrals with infinite limits.(152) 8.4 Evaluation of definite integrals involving trigonometric function with infinite limits. .(157) 8.5 Evaluation of definite integrals with singularity at real axis.(160) ·22·
8.6 Evaluation of definite integrals involving multivalued functions.(165) 8.7 Application of residue theorem to evaluation of infinite series.(170) 8.8 Other application of residue theorem.(175) Chapter g Gamma function.(177) 9.1 Definition of Gamma function. (177） 9.2 Properties of Gamma function.(179) 9.3 Evaluation of Gamma function.(182) 9.4 si functic0n. .(182 9.5 Beta function.(186) 9.6 Infinite series representation of Gamma function.(188) 9.7 Asymptotic expansion of Gamma function. .(194) *9.8 Corrections of some formulas of special functions.(197) 9.9 Riemann zeta function and Mobius transform.(200) Chapter 10 Laplace transforms.(205) 10.1 Laplace transforms.(205) 10.2 Properties of Laplace transforms.(206) 10.3 Inversion of Laplace transforms.(211) 10.4 Complex inversion formula for Laplace transform.(216) 10.5 Evaluation of infinite series by Laplace transfroms.(223) Chapter 11 Dirac 6 function.(229) 11.I Dirac 6 function.(229) 11.2 Evaluation of definite integrals with 6 function.(234) 11.3 Green function of the initial value problem of ordinary differential equation.(238) 11.4 Green function of the boundary value problem of ordinary differential equation. .(247) Part II Mathematical Physics Equations Chapter 12 Mathematical physics equations,boundary conditions and initial conditions. .(253) 12.1 Equation for transverse vibrations of strings.(254) ·23·
12.2 Equation for longitudinal vibrations of fexible rods.(256) 12.3 Heat conduction equation.(257) 12.4 Time-independent and steady-state problems.(260) 12.5 Boundary conditions and initial conditions.(261) 12.6 Linking condition on interior surface. .(265) 12.7 Well-posed and ill-posed problems.(267) Chapter 13 General solution to linear partial differential equation.(269) 13.1 Superposition principle for linear partial differential equations . .(269) 13.2 General solution to a linear homogeneous partial diffenrential equation with constant coefficients.(271) 13.3 General solution to a linear inhomogeneous partial diffenrential equation with constant coefficients.(273) 13.4 Some linear homogeneous partial diffenrential equations with variable coefficients.(280) 13.5 Travelling waves as solutions to wave equation.(281) 13.6 Dissipation and dispersion in waves.(283) 13.7 Qualitative discussion on heat conduction equation. (287） 13.8 Qualitative discussion on the Laplace equation.(289) Chapter 14 Method of separation of variables.(291) 14.1 Free vibration of a string with fixed ends. .(291 14.2 Steady state problem in a rectangular region.(302) 14.3 Well-posed problem with more than two variables.(306) 14.4 Forced vibration of a string with fixed ends.(310) 14.5 Homogenization of inhomogeneous boundary condition.(320) Chapter 15 Orthogonal curvilinear coordinates.(329) 15.1 Orthogonal curvilinear coordinates. 15.2 Laplacian in orthogonal curvilinear coordinates.(331) 15.3 Invariance of Laplacian under translation,rotation anc reflection. .(334 15.4 Circular region.(339） ·24·