This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course. This is a revised edition, new examples and exercises have been added, and many minor improvements have been made throughout the text.
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3 Complex Valued Functions
5 Complex Differentiability
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analytic continuation analytic function analytic isomorphism apply assume automorphism boundary bounded calculus Cauchy's formula Cauchy's theorem centered at z0 Chapter closed disc closed path coefficients complex numbers concludes the proof connected open set constant term contained continuous function converges absolutely converges uniformly define deleted derivative differentiable disc of radius end point entire function equal equation Example Exercises exists Figure finite number follows formal power series fractional linear map function f given Hence holomorphic function homologous infinity interval Laurent series Lemma Let f Let f(z Let z0 maximum modulus principle meromorphic open disc open set partition point z0 pole polynomial power series expansion primitive proves the theorem radius of convergence real axis real numbers rectangle residue right-hand side segment sequence series converges Show simply connected Suppose Theorem 1.2 unit circle unit disc upper half plane whence winding number write