By Wolfgang Fischer, Ingo Lieb, Jan Cannizzo

ISBN-10: 3834815764

ISBN-13: 9783834815767

This conscientiously written textbook is an creation to the gorgeous options and result of advanced research. it's 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 advanced research and applies them to a learn of user-friendly and non-elementary services elliptic capabilities, Gamma- and Zeta functionality together with an evidence of the leading quantity theorem ' and ' a brand new characteristic during this context! ' to displaying easy evidence within the concept of a number of complicated 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 by means of the authors, and some paragraphs have been newly written for particular use in a master's programme. content material research within the advanced aircraft - the basic theorems of advanced research - capabilities at the airplane and at the sphere - imperative formulation, residues and functions - Non-elementary capabilities - Meromorphic features of numerous variables - Holomorphic maps: Geometric features Readership complex undergraduates bachelor scholars and starting graduate scholars master's programme academics in arithmetic 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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We write: Δν (z0 ) = ∂f (z0 ) = fzν (z0 ). ∂zν If f is holomorphic on all of U , the maps z→ ∂f (z), ∂zν z ∈ U, 6. Several complex variables 33 deﬁne the complex partial derivatives of f as functions on U ; we denote them by fzν = ∂f . +kn f , . . , k1 . 2, namely in terms of the relation n n Δν (z)(zν − zν0 ) + f (z) − f (z0 ) = ν=1 Eν (z)(z ν − z 0ν ), (2) ν=1 where Δν and Eν are continuous at z0 , then as before we conclude: The values Δν (z0 ) and Eν (z0 ) are uniquely determined by f and z0 .

Show that f (z) = f (z) for all z ∈ G. b) Suppose G = Dr (0) and f is holomorphic on G and real-valued on G ∩ R. Show that if f is even (odd), then the values of f on G ∩ iR are real (imaginary). Prove this without using the power series expansion of f . 7. a) Suppose the domain G is symmetric with respect to the real axis and f is continuous on G and holomorphic on G \ R. Show that f is holomorphic on all of G. Hints: Use Morera’s theorem. In splitting up the triangle Δ ⊂ G, one sees that the only problematic case is the one in which an edge of Δ lies on R.

4. Elementary functions 23 Proof: Let us restrict our attention to the cosine function. The equation cos z = 1 iz (e + e−iz ) = 0 2 implies that e2iz + 1 = 0. (12) The only solutions of (12) are the aforementioned ones. In real analysis, one sometimes deﬁnes π via the condition that π/2 is the smallest positive zero of the cosine function. The above theorem shows that our deﬁnition of π gives the same number as this more elementary deﬁnition. The connection between π and the circumference of a circle will be derived in the following section.

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

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