By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

This conscientiously written textbook is an advent to the gorgeous thoughts and result of complicated research. it truly is meant for overseas bachelor and grasp programmes in Germany and all through Europe; within the Anglo-American method of collage schooling the content material corresponds to a starting graduate path. The booklet provides the basic effects and strategies of advanced research and applies them to a examine of basic and non-elementary capabilities elliptic capabilities, Gamma- and Zeta functionality together with an explanation of the leading quantity theorem ' and ' a brand new characteristic during this context! ' to showing uncomplicated evidence within the concept of numerous complicated variables. a part of the booklet is a translation of the authors' German textual content 'Einfuhrung in die komplexe Analysis'; a few fabric used to be further from the by way of now virtually 'classical' textual content 'Funktionentheorie' written via the authors, and some paragraphs have been newly written for specific use in a master's programme. content material research within the advanced airplane - the basic theorems of complicated research - capabilities at the airplane and at the sphere - crucial formulation, residues and purposes - Non-elementary capabilities - Meromorphic capabilities of a number of variables - Holomorphic maps: Geometric facets Readership complicated undergraduates bachelor scholars and starting graduate scholars master's programme teachers in arithmetic concerning the authors Professor Dr. Ingo Lieb, division of arithmetic, college of Bonn Professor Dr. Wolfgang Fischer, division of arithmetic, collage of Bremen

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4 (The standard estimate). Let f be continuous on the trace of the path of integration γ. Then max |f (z)| · L(γ). f (z) dz z∈Tr γ γ Proof: Since Tr γ is compact, |f | is bounded on it, say by M . 1 then implies b b f (z) dz = |f (γ(t))| |γ (t)| dt f (γ(t))γ (t) dt γ a a b |γ (t)| dt = M L(γ). 5. Let fν be a sequence of continuous functions that converges uniformly to a function f on the trace Tr γ of a path of integration γ. Then f is continuous, and f (z) dz = lim fν (z) dz. ) Proof: We already know that f is continuous.

Show that the radius of convergence of the Taylor series z→z1 expansion of f about 0 is exactly r. b) Determine the radii of convergence of the Taylor series expansions about 0 of tan z, 1 , cos z z , sin z z . ez − 1 5. Convergence theorems, maximum modulus principle, open mapping theorem 55 6. a) Suppose the domain G is symmetric with respect to the real axis and that f is holomorphic on G and real-valued on G ∩ R. Show that f (z) = f (z) for all z ∈ G. b) Suppose G = Dr (0) and f is holomorphic on G and real-valued on G ∩ R.

2 shows their relationship with the partial derivatives of f with respect to the real coordinates xν and yν : ∂f ∂f −i ∂xν ∂yν ∂f ∂f +i ∂xν ∂yν ∂f 1 = ∂zν 2 ∂f 1 = = ∂z ν 2 fzν = fz ν (4) . 1. A function f : U → C is holomorphic if and only if it is real diﬀerentiable and satisﬁes the diﬀerential equations ∂f = 0, ∂z ν 1 ν n. ). The system (5) is called the system of Cauchy-Riemann equations in n variables. A holomorphic function f in n variables is clearly holomorphic in each variable; more generally, if one ﬁxes k variables, then f is holomorphic in the remaining n − k variables.