Part Ⅰ Functions of a Complex Variable Chapter 1 Complex Numbers and Complex Functions 1.1 Complex number and its operations 1.1.1 Complex number and its expression 1.1.2 The operations of complex numbers 1.1.3 Regions in the complex plane Exercises 1.1 1.2 Functions of a complex variable 1.2.1 Definition of function of a complex variable 1.2.2 Complex mappings Exercises 1.2 1.3 Limit and continuity of a complex function 1.3.1 Limit of a complex function 1.3.2 Continuity of a complex function Exercises 1.3 Chapter 2 Analytic Functions 2.1 Derivatives of complex functions 2.1.1 Derivatives 2.1.2 Some properties of derivatives 2.1.3 A necessary condition on differentiability 2.1.4 Sufficient conditions on differentiability Exercises 2.1 2.2 Analytic functions 2.2.1 Analytic functions 2.2.2 Harmonic functions Exercises 2.2 2.3 Elementary functions 2.3.1 Exponential functions 2.3.2 Logarithmic functions 2.3.3 Complex exponents 2.3.4 Trigonometric functions 2.3.5 Hyperbolic functions 2.3.6 Inverse trigonometric and hyperbolic functions Exercises 2.3 Chapter 3 Integral of Complex Function 3.1 Derivatives and definite integrals of functions w(t) 3.1.1 Derivatives of functions w(t) 3.1.2 Definite integrals of functions w(t) Exercises 3.1 3.2 Contour integral 3.2.1 Contour 3.2.2 Definition of contour integra 3.2.3 Antiderivatives Exercises 3.2 3.3 Cauchy integral theorem 3.3.1 Cauchy�睪oursat theorem 3.3.2 Simply and multiply connected domains Exercises 3.3 3.4 Cauchy integral formula and derivatives of analytic functions 3.4.1 Cauchy integral formula 3.4.2 Higher�瞣rder derivatives formula of analytic functions Exercises 3.4 Chapter 4 Complex Series 4.1 Complex series and its convergence 4.1.1 Complex sequences and its convergence 4.1.2 Complex series and its convergence Exercises 4.1 4.2 Power series 4.2.1 The definition of power series 4.2.2 The convergence of power series 4.2.3 The operations of power series Exercises 4.2 4.3 Taylor series 4.3.1 Taylor's theorem 4.3.2 Taylor expansions of analytic functions Exercises 4.3 4.4 Laurent series 4.4.1 Laurent's theorem 4.4.2 Laurent series expansion of analytic functions Exercises 4.4 Chapter 5 Residues and Its Application 5.1 Three types of isolated singular points Exercises 5.1 5.2 Residues and Cauchy's residue theorem Exercises 5.2 5.3 Application of residues on definite integrals 5.3.1 Improper integrals 5.3.2 Improper integrals involving sines and cosines 5.3.3 Integrals on [0,2穑? involving sines and cosines Exercises 5.3
Part Ⅱ Mathematical Methods for Physics Chapter 6 Equations of Mathematical Physics and Problems for Defining Solutions 6.1 Basic concept and definition 6.1.1 Basic concept 6.1.2 Linear operator and linear composition 6.1.3 Calculation rule of operator 6.2 Three typical Partial differential equations and problems for defining solutions 6.2.1 Wave equations and physical derivations 6.2.2 Heat (conduction) equations and physical derivations 6.2.3 Laplace equations and physical derivations 6.3 Well�瞤osed problem 6.3.1 Initial conditions 6.3.2 Boundary conditions Chapter 7 Classification and Simplification for Linear Second Order PDEs 7.1 Classification of linear second order Partial differential equations with two variables Exercises 7.1 7.2 Simplification to standard forms Exercises 7.2 Chapter 8 Integral Method on Characteristics 8.1 D'Alembert formula for one dimensional infinite string oscillation Exercises 8.1 8.2 Small oscillations of semi�瞚nfinite string with rigidly fixed or free ends, method of prolongation Exercises 8.2 8.3 Integral method on characteristics for other second order PDEs, some examples162 Exercises 8.3 Chapter 9 The Method of Separation of Variables on Finite Region 9.1 Separation of variables for (1 1)�瞕imensional homogeneous equations 9.1.1 Separation of variables for wave equation on finite region 9.1.2 Separation of variables for heat equation on finite region Exercises 9.1 9.2 Separation of variables for 2�瞕imensional Laplace equations 9.2.1 Laplace equation with rectangular boundary 9.2.2 Laplace equation with circular boundary Exercises 9.2 9.3 Nonhomogeneous equations and nonhomogeneous boundary conditions Exercises 9.3 9.4 Sturm�睱iouville eigenvalue problem Exercises 9.4 Chapter 10 Special Functions 10.1 Bessel function 10.1.1 Introduction to the Bessel equation 10.1.2 The solution of the Bessel equation 10.1.3 The recurrence formula of the Bessel function 10.1.4 The properties of the Bessel function 10.1.5 Application of Bessel function Exercises 10.1 10.2 Legendre polynomial 10.2.1 Introduction of the Legendre equation 10.2.2 The solution of the Legendre equation 10.2.3 The properties of the Legendre polynomial and recurrence formula 10.2.4 Application of Legendre polynomial Exercises 10.2 Chapter 11 Integral Transformations 11.1 Fourier integral transformation 11.1.1 Definition of Fourier integral transformation 11.1.2 The properties of Fourier integral transformation 11.1.3 Convolution and its Fourier transformation 11.1.4 Application of Fourier integral transformation Exercises 11.1 11.2 Laplace integral transformation 11.2.1 Definition of Laplace transformation 11.2.2 Properties of Laplace transformation 11.2.3 Convolution and its Laplace transformation 11.2.4 Application of Laplace integral transformation Exercises 11.2 References