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<p><br /></p><p><br /></p><h2>About the book</h2><p>The book covers basic aspects of complex numbers, complex variables and complex functions. It also deals with analytic functions, Laurent series etc.</p><p><br /></p><h2>Contents</h2><p>Introduction 9<br /> Chapter 1. THE COMPLEX VARIABLE AND FUNCTIONS OF A COMPLEX VARIABLE 11</p><p>1.1. Complex Numbers and Operations on Complex Numbers 11<br /> a. The concept of a complex number 11<br /> b. Operations on complex numbers 11<br /> c. The geometric interpretation of complex numbers 13<br /> d. Extracting the root of a complex number 15</p><p>1.2. The Limit of a Sequence of Complex Numbers 17<br /> a. The definition of a convergent sequence 17<br /> b. Cauchy's test 19<br /> c. Point at infinity 19</p><p>1.3. The Concept of a Function of a Complex Variable. Continuity 20<br /> a. Basic definitions 20<br /> b. Continuity 23<br /> c. Examples 26</p><p>1.4. Differentiating the Function of a Complex Variable 30<br /> a. Definition. Cauchy-Riemann conditions 30<br /> b. Properties of analytic functions 33<br /> c. The geometric meaning of the derivative of a function of a complex variable 35<br /> d. Examples 37</p><p>1.5. An Integral with Respect to a Complex Variable 38<br /> a. Basic properties 38<br /> b. Cauchy's Theorem 41<br /> c. Indefinite Integral 44<br /> 1.6. Cauchy's Integral 47<br /> a. Deriving Cauchy's formula 47<br /> b. Corollaries to Cauchy's formula 50<br /> c. The maximum-modulus principle of an analytic function 51</p><p>1.7. Integrals Dependent on a Parameter 53<br /> a. Analytic dependence on a parameter 53<br /> b. An analytic function and the existence of derivatives of all orders 55<br /> Chapter 2. SERIES OF ANALYTIC FUNCTIONS 58</p><p>2.1. Uniformly Convergent Series of Functions of a Complex Variable 58<br /> a. Number series 58<br /> b. Functional series. Uniform convergence 59<br /> c. Properties of uniformly convergent series. Weierstrass' theorems 62<br /> d. Improper integrals dependent on a parameter 66</p><p>2.2. Power Series. Taylor's Series 67<br /> a. Abel's theorem 67<br /> b. Taylor's series 72<br /> c. Examples 74<br /> 2.3. Uniqueness of Definition of an Analytic Function 76<br /> a. Zeros of an analytic function 76<br /> b. Uniqueness theorem 77</p><p>Chapter 3. ANALYTIC CONTINUATION. ELEMENTARY FUNCTIONS OF A COMPLEX VARIABLE 80</p><p>3.1. Elementary Functions of a Complex Variable. Continuation<br /> from the Real Axis 80<br /> a. Continuation from the real axis 80<br /> b. Continuation of relations 84<br /> c. Properties of elementary functions 87<br /> d. Mappings of elementary functions 91<br /> 3.2. Analytic Continuation. The Riemann Surface 95<br /> a. Basic principles. The concept of a Riemann surface 95<br /> b. Analytic continuation across a boundary 98<br /> c. Examples in constructing analytic continuations. Continuation across a boundary 100<br /> d. Examples in constructing analytic continuations. Continuation by means of power series 105<br /> e. Regular and singular points of an analytic function 108<br /> f. The concept of a complete analytic function 111<br /> Chapter 4. THE LAURENT SERIES AND ISOLATED SINGULAR POINTS 113</p><p>4.1. The Laurent Series 113<br /> a. The domain of convergence of a Laurent series 113<br /> b. Expansion of an analytic function in a Laurent series 115</p><p>4.2. A Classification of the Isolated Singular Points of a Single-Valued Analytic Function 118<br /> Chapter 5. RESIDUES AND THEIR APPLICATIONS 125</p><p>5.1. The Residue of an Analytic Function at an Isolated Singularity 125<br /> a. Definition of a residue. Formulas for evaluating residues 125<br /> b. The residue theorem 127</p><p>5.2. Evaluation of Definite Integrals by !\leans of Residues 130<br /> a. Integrals of the form $\int^{2 \pi}_{0}R (\cos \theta \sin \theta ) d \theta$ 131<br /> b. Integrals of the form $\int^{\infty}_{\infty} f(x)dx$ 132<br /> c. Integrals of the form $\int^{\infty}_{\infty} \exp(iax)f(x)dx$. Jordan's lemma  135<br /> d. The case of multiple-valued functions 141</p><p>5.3. Logarithmic Residue 147<br /> a. The concept of a logarithmic residue 147<br /> b. Counting the number of zeros of an analytic function 149</p><p>Chapter 6. CONFORMAL MAPPING 153</p><p>6.1. General Properties 153<br /> a. Definition of a conformal mapping 153<br /> b. Elementary examples 157<br /> c. Basic principles 160<br /> d. Riemann's theorem 166</p><p>6.2. Linear-Fractional Function 169</p><p>6.3. Zhukovsky's Function 179</p><p>6.4. Schwartz-Christoffel Integral. Transformation of Polygons 181</p><p>Chapter 7. ANALYTIC FUNCTIONS IN THE SOLUTION OF BOUNDARY-VALUE PROBLEMS 191</p><p>7. 1. Generalities 191<br /> a. The relationship of analytic and harmonic functions 191<br /> b. Preservation of the Laplace operator in a conformal mapping 192<br /> c. Dirichlet's problem 194<br /> d. Constructing a source function 197</p><p>7.2. Applications to Problems in Mechanics and Physics 199<br /> a. Two-dimensional steady-state flow of a fluid 199<br /> b. A two-dimensional electrostatic field 211</p><p>Chapter 8. FUNDAMENTALS OF OPERATIONAL CALCULUS 221</p><p>8.1. Basic Properties of the Laplace Transformation 221<br /> a. Definition 221<br /> b. Transforms of elementary functions 225<br /> c. Properties of a transform 227<br /> d. Table of properties of transforms 236<br /> e. Table of transforms 236<br /> 8.2. Determining the Original Function from the Transform 238<br /> a. Mellin's formula 238<br /> h. Existence conditions of the original function 241<br /> c. Computing the Mellin integral 245<br /> d. The case of a function regular at infinity 249<br /> 8.3. Solving Problems for Linear Differential Equations by the Operational Method 252<br /> a. Ordinary differential equations 252<br /> b. Heat-conduction equation 257<br /> c. The boundary-value problem for a partial differential equation 259<br /> Appendix I. SADDLE-POINT METHOD 261<br /> I.1. Introductory Remarks 261</p><p>I.2. Laplace's Method 264<br /> I.3. The Saddle-Point Method 271<br /> Appendix II. THE WIENER-HOPF METHOD 280</p><p>II.1. Introductory Remarks 280<br /> 11.2. Analytic Properties of the Fourier Transformation 284<br /> 11.3. Integral Equations with a Difference Kernel 287<br /> II.4. General Scheme of the Wiener-Hopf Method 292<br /> II.5. Problems Which Reduce to Integral Equations with a Difference<br /> Kernel 297<br /> a. Derivation of Milne's equation 297<br /> b. Investigating the solution of Milne's equation 301<br /> c. Diffraction on a flat screen 305<br /> II.6. Solving Boundary-Value Problems for Partial Differential Equations by the Wiener-Hopf Method 306</p><p>Appendix III. FUNCTIONS OF MANY COMPLEX VARIABLES 310</p><p>III.1. Basic Definitions 310<br /> III.2. The Concept of an Analytic Function of Many Complex Variables 311<br /> III.3. Cauchy's Formula 312<br /> III.4. Power Series 314<br /> III.5. Taylor's Series 316<br /> III.6. Analytic Continuation 317</p><p>Appendix IV. WATSON'S METHOD 320<br /> References 328<br /> Name Index 329<br /> Subject Index 330</p>

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