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prefacechapter ¢¡ complex number field 1.1 sums and products 1.2 basic algebraic properties 1.3 further properties 1.4 moduli 1.5 conjugates 1.6 exponential form 1.7 products and quotients in exponential form 1.8 roots of complex numbers 1.9 examples 1.10 regions in the complex planechapter ¢¢ analytic functions 2.1 functions of a complex variable 2.2 mappings 2.3 the exponential function and its mapping properties 2.4 limits 2.5 theorems on limits 2.6 limits involving the point at infinity 2.7 continuity 2.8 derivatives 2.9 differentiation formulas 2.10 cauchy-riemann equations 2.11 necessary and sufficient conditions for differentiability 2.12 polar coordinates 2.13 analytic functions 2.14 examples 215 harmonic functionschapter ¢£ elementary functions 3.1 the exponential function 3.2 the logarithmic function 3.3 branches and derivatives of logarithms 3.4 some identities on logarithms 3.5 complex power functions 36 trigonometric functions 3.7 hyperbolic functions 3.8 inverse trigonometric and hyperbolic functionschapter ¢¤ integrals 4.1 derivatives of complex-valued functions of one real variable 4.2 definite integrals of functions w 4.3 paths 4.4 path integrals 4.5 examples 4.6 upper bounds for integrals 4.7 primitive functions 4.8 examples 4.9 cauchy integral theorem 4.10 proof of the cauchy integral theorem 4.11 extended cauchy integral theorem 4.12 cauchy integral formula 4.13 derivatives of analytic functions 4.14 liouville's theorem 4.15 maximum modulus principlechapter ¢¥ series 5.1 convergence of series 5.2 taylor series 5.3 examples 5.4 laurent series 5.5 examples 5.6 absolute and uniform convergence of power series 5.7 continuity of sums of power series 5.8 integration and differentiation of power series 5.9 uniqueness of series representations 5.10 multiplication and division of power serieschapter ¢¦ residues and poles 6.1 residues 6.2 cauchy's residue theorem 6.3 using a single residue 6.4 the three types of isolated singular points 6.5 residues at poles 6.6 examples 6.7 zeros of analytic functions 6.8 uniquely determined analytic functions 6.9 zeros and poles 6.10 behavior of f near isolated singular points 6.11 reflection principlechapter ¢§ applications of residues 7 i evaluation of improper integrals 7.2 examples 7.3 improper integrals from fourier analysis 7.4 jordan's lemma 7.5 indented paths 7.6 an indentation around a branch point 7.7 definite integrals involving sine and cosine 7.8 argument principle 7.9 rouche's theoremchapter ¢¨ conformal mappings 8.1 conformal mappings 82 unilateral functions 8.3 local inverses 84 affine transformations 85 the transformation w = 1/z 8.6 mappings by 1/z 8.7 fractional linear transformations 8.8 cross ratios 8.9 mappings of the upper half plane