Mathematical Analysis

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

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By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

This conscientiously written textbook is an advent to the gorgeous thoughts and result of advanced 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 direction. The e-book provides the elemental effects and techniques of advanced research and applies them to a examine of common and non-elementary capabilities elliptic services, Gamma- and Zeta functionality together with an explanation of the leading quantity theorem ' and ' a brand new characteristic during this context! ' to displaying simple evidence within the idea of numerous advanced variables. a part of the e-book is a translation of the authors' German textual content 'Einfuhrung in die komplexe Analysis'; a few fabric was once additional from the through now nearly 'classical' textual content 'Funktionentheorie' written through the authors, and some paragraphs have been newly written for distinct use in a master's programme. content material research within the complicated aircraft - the elemental theorems of advanced research - services at the aircraft and at the sphere - imperative formulation, residues and functions - Non-elementary services - 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 in regards to the authors Professor Dr. Ingo Lieb, division of arithmetic, collage of Bonn Professor Dr. Wolfgang Fischer, division of arithmetic, college of Bremen

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Extra resources for A Course in Complex Analysis: From Basic Results to Advanced Topics

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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 . But M is also closed in G: If z2 ∈ G is a boundary point of M , then there exists a sequence zμ in M \ {z2 } converging to z2 .

To conclude this section, we give several characterizations of the term “holomorphic function”. We leave it to the reader to verify how they follow from the theorems we have proved thus far. 3. Let f : U → C be a function defined on an open set U ⊂ C. The following are equivalent: i. f is holomorphic. ii. iii. iv. v. f f f f is real differentiable and satisfies the Cauchy-Riemann equations. admits a power series expansion about every point in U . has local primitives. is continuous, and for every closed triangle Δ ⊂ U , ∂Δ f (z) dz = 0.

Z 2ν (2ν)! ν=0 ∞ sinh z = z 2ν+1 (2ν + 1)! ν=0 eiz = cos z + i sin z sin2 z + cos2 z ≡ 1 cosh2 z − sinh2 z ≡ 1. It follows from v that, for real z, all of the above functions coincide with the trigonometric and hyperbolic functions familiar from calculus. Formula vi gives the decomposition of ez into its real and imaginary parts (it is also called Euler’s formula): ex+iy = ex (cos y + i sin y). 6. The zeros of the sine function are the real numbers kπ, and the zeros of the cosine function are the numbers π2 + kπ, where k ∈ Z.

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