By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

This conscientiously written textbook is an advent to the attractive techniques and result of advanced research. it really is meant for overseas bachelor and grasp programmes in Germany and all through Europe; within the Anglo-American procedure of college schooling the content material corresponds to a starting graduate direction. The publication offers the basic effects and techniques of advanced research and applies them to a research of easy and non-elementary capabilities elliptic features, Gamma- and Zeta functionality together with an evidence of the best quantity theorem ' and ' a brand new function during this context! ' to displaying easy proof within the thought of numerous complicated variables. a part of the e-book is a translation of the authors' German textual content 'Einfuhrung in die komplexe Analysis'; a few fabric used to be extra from the by means of now nearly 'classical' textual content 'Funktionentheorie' written through the authors, and some paragraphs have been newly written for exact use in a master's programme. content material research within the complicated airplane - the elemental theorems of complicated research - capabilities at the airplane and at the sphere - essential formulation, residues and functions - Non-elementary services - Meromorphic capabilities of a number of variables - Holomorphic maps: Geometric points Readership complex 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, college of Bremen

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Dzn . Γ It is possible to integrate in far greater generality, but this is needed only in the further development of the theory of several complex variables, which is beyond the scope of this book. Exercises 1. Explicitly derive the Cauchy-Riemann equations from the deﬁnition of complex diﬀerentiability. 2. Derive the chain rule (6). 3. A map F : U → V , where U and V are open subsets of Cn , is biholomorphic if it is bijective and holomorphic and its inverse F −1 is holomorphic as well. Show that F is biholomorphic if it is bijective and holomorphic and F −1 is real diﬀerentiable.

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. Show that if f is even (odd), then the values of f on G ∩ iR are real (imaginary). Prove this without using the power series expansion of f . 7. a) Suppose the domain G is symmetric with respect to the real axis and f is continuous on G and holomorphic on G \ R.

Show that: – If f is real-valued on R ∩ D, then all an are real. – If f is an even (odd) function, then an = 0 for all odd (even) n. – If f (iz) = f (z), then an can only be nonzero if n is divisible by 4. In addition: Discuss the equation f (ρz) = μf (z), where ρ, μ ∈ C \ {0} are given. 3. There are only even powers of z in the power series expansion of f (z) = often written in the form 1 cos z about 0. It is most ∞ E2n 2n 1 (−1)n = z . cos z (2n)! n=0 The E2n are called Euler numbers. Determine a recursion formula for the numbers E2n and show that they are all integers.