A Course in Complex Analysis: From Basic Results to Advanced by Prof. Dr. Wolfgang Fischer, Prof. Dr. Ingo Lieb (auth.)

By Prof. Dr. Wolfgang Fischer, Prof. Dr. Ingo Lieb (auth.)

This conscientiously written textbook is an creation to the attractive ideas and result of advanced research. it really is meant for foreign bachelor and grasp programmes in Germany and all through Europe; within the Anglo-American process of college schooling the content material corresponds to a starting graduate path. The ebook provides the basic effects and strategies of complicated research and applies them to a examine of user-friendly and non-elementary services (elliptic services, Gamma- and Zeta functionality together with an evidence of the leading quantity theorem …) and – a brand new characteristic during this context! – to showing uncomplicated evidence within the thought of numerous complicated variables.
a part of the publication is a translation of the authors’ German textual content “Einführung in die komplexe Analysis”; a few fabric was once additional from the through now virtually “classical” textual content “Funktionentheorie” written via the authors, and some paragraphs have been newly written for specified use in a master’s programme.

Content
research within the advanced airplane - the basic theorems of complicated research - services at the airplane and at the sphere - crucial formulation, residues and functions - Non-elementary services - Meromorphic services of numerous variables - Holomorphic maps: Geometric aspects

Readership
complex undergraduates (bachelor scholars) and starting graduate scholars (master's programme)
academics in mathematics
in regards to the authors
Professor Dr. Ingo Lieb, division of arithmetic, collage of Bonn
Professor Dr. Wolfgang Fischer, division of arithmetic, collage of Bremen

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N (tn )) defines a mapping of the n-dimensional cube Qn = {(t1 , . . , tn ) : 0 tν 1} ⊂ Rn ν n. Then (7) 6. Several complex variables 35 into Cn . We call Γ a parametrized surface of integration in Cn ; its image Γ(Qn ) is the trace of Γ. For a continuous function f on the trace of Γ, we define ··· f (z1 , . . , zn ) dz1 . . dzn = γn Γ f (z1 , . . , zn ) dz1 dz2 . . dzn . γ2 (8) γ1 Thus one first fixes the variables z2 , . . , zn and integrates the function z1 → f (z1 , z2 , . . 5) in the variables z2 , .

Let f be holomorphic in a neighbourhood of the closed disk D ⊂ C. Show that f (ζ) dζ ζ−z z→ ∂D determines a holomorphic function on C \ D. Which function is this? 3. Let f be continuous on the closed disk D ⊂ C and holomorphic on D. Show that 1 2πi ∂D f (ζ) dζ = f (z) ζ−z for z ∈ D (cf. Ex. 2). 4. Check whether the following functions can be holomorphically extended to z0 = 0. z cot2 z; z 2 cot2 z; z (ez − 1)−1 ; z 2 sin(1/z). 4. Power series expansions of holomorphic functions In Chapter I, we saw that a convergent power series determines a holomorphic function in its disk of convergence.

5 (and can be seen as a generalization of Prop. 5). 7. Let α and β be two paths of integration, and suppose the function f is continuous on Tr α × Tr β. Then f (z, w) dw dz = α β f (z, w) dz dw. β α Moreover, the theory of integration tells us that the following statement can be viewed as a special case of Prop. 8. Let aνμ , where (ν, μ) ∈ N × N, be complex numbers. If there exists a bound M , independent of n and m, for which m n |aνμ | M, μ=0 ν=0 ∞ ν=0 the series ∞ μ=0 aνμ and ∞ μ=0 ∞ ν=0 aνμ are convergent, with the same sum.

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