By Prof. Dr. Wolfgang Fischer, Prof. Dr. Ingo Lieb (auth.)
This conscientiously written textbook is an advent to the attractive recommendations and result of complicated research. it's meant for overseas bachelor and grasp programmes in Germany and all through Europe; within the Anglo-American procedure of college schooling the content material corresponds to a starting graduate direction. The booklet provides the elemental effects and techniques of advanced research and applies them to a learn of uncomplicated and non-elementary services (elliptic capabilities, Gamma- and Zeta functionality together with an evidence of the top quantity theorem …) and – a brand new characteristic during this context! – to showing uncomplicated proof within the conception 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 was once 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 airplane - the basic theorems of advanced research - capabilities at the aircraft and at the sphere - imperative formulation, residues and functions - Non-elementary services - Meromorphic features of a number of variables - Holomorphic maps: Geometric aspects
complex undergraduates (bachelor scholars) and starting graduate scholars (master's programme)
teachers in mathematics
in regards to the authors
Professor Dr. Ingo Lieb, division of arithmetic, college of Bonn
Professor Dr. Wolfgang Fischer, division of arithmetic, college of Bremen
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Additional resources for A Course in Complex Analysis: From Basic Results to Advanced Topics
With centres xkλ in V . As fν (xkλ ) converges, there is an index ν0 such that fν (xkλ ) − fμ (xkλ ) < ε for λ = 1, . . , and μ, ν ν0 . Now let x ∈ V be arbitrary. Choose λ with x ∈ Dδ (xkλ ) and let ν, μ fν (x) − fμ (x) ν0 . Then fν (x) − fν (xkλ ) + fν (xkλ ) − fμ (xkλ ) + fμ (xkλ ) − fμ (x) < 3ε by the choice of δ and ν0 . Hence the Cauchy criterion for uniform convergence on V is satisﬁed. Proof of Thm. 3: Local uniform boundedness of holomorphic functions fν implies local equicontinuity: If, on Dr (z0 ) ⊂⊂ G, we have |fν (z)| M0 for all ν, the Cauchy inequalities (Prop.
If f is continuous on a domain G ⊂ C and satisﬁes f (z) dz = 0 γ for every closed path of integration γ in G, then f has a primitive on G. Proof: Let a ∈ G be ﬁxed. For every z ∈ G, choose a path of integration γz in G from a to z and set f (ζ) dζ. F (z) = γz Let us show that F is complex diﬀerentiable at every point z0 ∈ G and satisﬁes F (z0 ) = f (z0 ). If z is a point that is suﬃciently close to z0 , then [z0 , z] ⊂ G, and γz0 + [z0 , z] − γz is a closed path in G. By assumption, we have f (ζ) dζ − f (ζ) dζ + γz 0 [z0 ,z] f (ζ) dζ = 0, γz 1.
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. Exercises 1.