By Prof. Dr. Wolfgang Fischer, Prof. Dr. Ingo Lieb (auth.)
This rigorously 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 approach of collage schooling the content material corresponds to a starting graduate path. The e-book provides the basic effects and techniques of advanced research and applies them to a research of undemanding and non-elementary capabilities (elliptic features, Gamma- and Zeta functionality together with an explanation of the top quantity theorem …) and – a brand new characteristic during this context! – to displaying uncomplicated proof within the idea of numerous advanced variables.
a part of the publication is a translation of the authors’ German textual content “Einführung in die komplexe Analysis”; a few fabric used to be further from the by way of now virtually “classical” textual content “Funktionentheorie” written via the authors, and some paragraphs have been newly written for specific use in a master’s programme.
research within the complicated aircraft - the elemental theorems of advanced research - capabilities at the aircraft and at the sphere - indispensable formulation, residues and purposes - Non-elementary features - Meromorphic services of numerous variables - Holomorphic maps: Geometric aspects
complex 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, college of Bremen
Read or Download A Course in Complex Analysis: From Basic Results to Advanced Topics PDF
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Extra info for A Course in Complex Analysis: From Basic Results to Advanced Topics
To be more explicit: F (z1 , . . , zn ) = (w1 , . . , wm ), where wμ = fμ (z1 , . . , zn ), 1 μ m. Let all fμ be holomorphic. Then if f is a function that is holomorphic in a neighbourhood of a point w0 = F (z0 ), one obtains for the function f ◦ F in a neighbourhood of z0 : Chain rule. The function f ◦ F is holomorphic at z0 , and m ∂f ∂fμ ∂(f ◦ F ) (z0 ) = (F (z0 )) (z0 ). ∂zν ∂w ∂zν μ μ=1 (6) Formula (6) is typically written more concisely as m ∂f ∂wμ ∂(f ◦ F ) = . ∂zν ∂w μ ∂zν μ=1 (6 ) The proof is obtained as in the case of one variable.
Then f (z) dz = 0. ∂Δ Holomorphic functions thus satisfy the conditions of Prop. 3. 2. The Cauchy integral theorem 41 Δ21 Δ Δ31 Δ41 Δ11 Figure 1. Subdivision of a triangle Proof: Let us subdivide Δ into four subtriangles Δ11 , . . , Δ41 by connecting the midpoints of the edges of Δ (see Fig. 1). Every segment that joins two midpoints belongs to the boundary of exactly two subtriangles; thus, if we form the sum 4 f (z) dz, k=1 ∂Δk 1 the integrals over these segments cancel, and we have 4 f (z) dz = f (z) dz, k=1 ∂Δk 1 ∂Δ so that f (z) dz 4 max f (z) dz .
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 , . . , zn , say F (z2 , . . , zn ). One then integrates the function z2 → F (z2 , . . , zn ), where z3 , . . , zn are ﬁxed, over γ2 , and so on. By Prop. 7, the order in which one integrates does not matter. For the reader who has studied diﬀerential forms: (8) gives the integral of the complex (n, 0)-form f (z1 , . . , zn ) dz1 ∧ .
A Course in Complex Analysis: From Basic Results to Advanced Topics by Prof. Dr. Wolfgang Fischer, Prof. Dr. Ingo Lieb (auth.)