MATH502_StudyGuide

MATH502_StudyGuide - Serge Ballif Gamma function Complex...

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Serge Ballif Complex Analysis Facts and Formulas Spring 2008 Gamma function Γ( z ) = Z 0 t z - 1 e - t d t Zeta function ζ ( s ) = X n =1 1 n s = Y p 1 1 - p - s Path Integral Z γ f ( z )d z := Z b a f ( γ ( t )) γ 0 ( t )d t LM Estimate | R γ f ( z ) | d z M Length( γ ) for M ≥ | f ( z ) | on γ . Coefficients c n of a power series are given by c n = 1 2 πi Z ∂D ( a,r ) f ( z ) ( z - a ) n +1 d z Morera’s Theorem If a continuous function f on an open subset Ω of C has R ∂T f ( z )d z = 0 for all triangles T in Ω, then f is holomorphic. Principle of Isolated Zeroes Let Ω C be open and connected. If f : Γ C is holo- morphic and not identically zero, then the set of zeroes of f has no limit point. Maximum Principle Let f be a nonconstant holomorphic function on a connected open set Ω. Then | f | does not attain a local maximum anywhere on Ω. Minimum Principle | f | attains a local minimum only at the zeroes of f . Liouville Theorem
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MATH502_StudyGuide - Serge Ballif Gamma function Complex...

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