## Complex analysis |

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### Contents

Complex Numbers and Functions 1 Definition | 3 |

Polar form | 8 |

Complex valued functions | 12 |

Copyright | |

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absolute value analytic continuation analytic function analytic isomorphism assume bounded calculus Cauchy's theorem Chapter circle of radius closed disc closed path coefficients compact set complex numbers constant term contained continuous function converges absolutely converges uniformly defined derivative differentiable disc of radius elliptic function end point entire function equal equation Example EXERCISES f be analytic Figure finite number function g given harmonic function Hence holomorphic function homologous integral interval inverse lemma Let f Let z0 maximum modulus principle open disc open set point of accumulation point z0 poles polynomial positive integer positive number power series expansion primitive Proof proves the theorem quotient radius of convergence real axis real numbers rectangle residue Riemann mapping theorem right-hand side segment sequence Show shown on Fig simply connected subset Suppose Theorem 2.1 unit circle unit disc upper half plane variables whence write