New PDF release: A Course in Complex Analysis: From Basic Results to Advanced

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

ISBN-10: 3834815764

ISBN-13: 9783834815767

ISBN-10: 3834886610

ISBN-13: 9783834886613

This conscientiously written textbook is an creation to the gorgeous innovations 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 collage schooling the content material corresponds to a starting graduate path. The publication provides the elemental effects and techniques of complicated research and applies them to a learn of trouble-free and non-elementary features (elliptic capabilities, Gamma- and Zeta functionality together with an evidence of the leading quantity theorem …) and – a brand new characteristic during this context! – to showing easy proof within the idea of a number of advanced variables.
a part of the booklet is a translation of the authors’ German textual content “Einführung in die komplexe Analysis”; a few fabric used to be further from the by means of now virtually “classical” textual content “Funktionentheorie” written through the authors, and some paragraphs have been newly written for targeted use in a master’s programme.

Content
research within the complicated aircraft - the basic theorems of advanced research - features at the aircraft and at the sphere - vital formulation, residues and purposes - Non-elementary capabilities - Meromorphic services of a number of variables - Holomorphic maps: Geometric aspects

Readership
complicated undergraduates (bachelor scholars) and starting graduate scholars (master's programme)
academics in mathematics
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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Extra info for A Course in Complex Analysis: From Basic Results to Advanced Topics

Example text

A,z0 ] Thus, F (z) − F (z0 ) = f (ζ) dζ, [z0 ,z] and the claim follows by the argument used in the proof of Prop. 2. e. every point a ∈ G has a neighbourhood U such that f U has a primitive. It suffices to take for U a disk that is centred at a and contained in G and then to apply Prop. 3. Exercises 1. Let γ be a path of integration from i + 1 to 2i. Compute the integrals of the following functions over γ: cos(1 + i)z, iz 2 + 1 − 2iz −2 , (z + 1)3 , z exp(iz 2 ). 2. Show that the functions z → Re z and z → |z| do not have primitives on C.

Prop. 3 tells us that the length of the above circle is π |ir eit | dt = 2πr. L (κ(r; z0 )) = −π We thus see that the number π that we defined in terms of the period of the exponential function in the previous section coincides with the number π familiar from geometry. iii. An especially important example is: π dz = z − z0 κ(r;z0 ) −π ir eit dt = 2πi. 4. A parameter transformation is a map h : [c, d] → [a, b] between real intervals with the following properties: i. It is a continuous, piecewise continuously differentiable bijection.

F (z) = n=0 We have a0 = f (z0 ) = limμ f (zμ ) = 0. Suppose now that a0 = . . = an−1 = 0 for some n 1. Then f (z) = (z − z0 )n (an + an+1 (z − z0 ) + . ) = (z − z0 )n F (z), where F is continuous (and in fact holomorphic) at z0 . Since zμ − z0 = 0, we have F (zμ ) = 0 for all μ, and hence an = limμ F (zμ ) = 0. Thus, for all n, f (n) (z0 ) = n! an = 0. c) We now derive ii from iii By Thm. 1, f ≡ 0 in a neighbourhood of z0 . Consider the set M = {z1 ∈ G : f ≡ 0 in a neighbourhood of z1 }. By definition, M is open, and z0 ∈ M .

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


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