In this (central) chapter, we are concerned with the construction of analytic functions. We will meet three different construction principles.
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(1)
We first study in detail a classical function using methods of the function theory of one complex variable, namely the Gamma function.
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(2)
We treat the theorems of WEIERSTRASS and MITTAG-LEFFLER for the construction of analytic functions with prescribed zeros and respectively poles with specified principle parts.
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(3)
We prove the REIMANN Mapping Theorem, which claims that an elementary domain D ≠ C can be conformally mapped onto the unit disk E. In this context we will once more review the Cauchy Integral Theorem, prove more general variants of it, and gain different topological characterizations of elementary domains as regions “without holes”.
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© 2009 Springer-Verlag Berlin Heidelberg
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Freitag, E., Busam, R. (2009). Construction of Analytic Functions. In: Complex Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93983-2_5
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DOI: https://doi.org/10.1007/978-3-540-93983-2_5
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-93982-5
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