By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

ISBN-10: 3834815764

ISBN-13: 9783834815767

This rigorously written textbook is an creation to the attractive ideas 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 offers the basic effects and strategies of complicated research and applies them to a research of trouble-free and non-elementary services elliptic features, Gamma- and Zeta functionality together with an explanation of the best quantity theorem ' and ' a brand new function during this context! ' to showing simple evidence within the concept of a number of advanced 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 additional from the through now nearly 'classical' textual content 'Funktionentheorie' written via the authors, and some paragraphs have been newly written for designated use in a master's programme. content material research within the advanced airplane - the elemental theorems of advanced research - capabilities at the aircraft and at the sphere - crucial formulation, residues and functions - Non-elementary capabilities - Meromorphic services of numerous variables - Holomorphic maps: Geometric elements Readership complex undergraduates bachelor scholars and starting graduate scholars master's programme academics in arithmetic 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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**Additional info for A Course in Complex Analysis: From Basic Results to Advanced Topics **

**Sample text**

Where do they take on imaginary values? d tan z = 0 4. Show that the function tan z never takes on the values ±i, and that therefore dz everywhere. Show that the tangent function maps the strip S0 = {z : − π/2 < Re z < π/2} biholomorphically onto C \ {it : t ∈ R, |t| 1}. 5. Show that the radius of convergence of the power series ∞ z ν! f (z) = ν=0 is 1, and that for any ﬁxed α ∈ Q, f (r e2πi α ) is unbounded as r → 1. 5. Integration We will now introduce the most important tool in complex analysis: the integration of complex-valued functions along suitable curves (“paths of integration”) in the complex plane.

In doing so, the proofs and results remain largely unchanged, although the theory of several complex variables eventually leads to questions that do not arise in the theory of one complex variable. Now for the details! Given a point in the n-dimensional complex vector space Cn = {z = (z1 , . . , zn ) : zν ∈ C}, we will denote the real and imaginary parts of its coordinates zν by xν and yν , respectively. Moreover, we equip Cn with the “maximum norm” |z| = max |zν | 1 ν n instead of the euclidean norm.

The connection between π and the circumference of a circle will be derived in the following section. e. the n solutions of the equation Xn = 1 in C. Clearly, ζn = e2πi/n = cos 2π 2π + i sin n n h n − 1. These is such a solution; all the solutions are the powers ζnh , for 0 numbers all lie on the circle S and span a regular n-gon one of whose vertices is 1. They form a cyclic group of order n (under multiplication); in the exercises, the reader is asked to show that all ﬁnite subgroups (and in fact all closed proper subgroups) of S are groups of nth roots of unity.

### A Course in Complex Analysis: From Basic Results to Advanced Topics by Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

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