analytic_functions

# analytic_functions - UNIVERSITY OF NOTRE DAME DEPARTMENT OF...

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UNIVERSITY OF NOTRE DAME DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING Professor H.M. Atassi AME-562 113 Hessert Center Mathematical Methods II Tel : 631-5736 Email Analytic Functions of a Complex Variable 1 Definitions and Theorems 1.1 Definition 1 A function f ( z ) is said to be analytic in a region R of the complex plane if f ( z ) has a derivative at each point of R and if f ( z ) is single valued. 1.2 Definition 2 A function f ( z ) is said to be analytic at a point z if z is an interior point of some region where f ( z ) is analytic. Hence the concept of analytic function at a point implies that the function is analytic in some circle with center at this point. 1.3 Theorem If f ( z ) is analytic at a point z , then the derivative f 0 ( z ) is continuous at z . 1.4 Corollary If f ( z ) is analytic at a point z , then f ( z ) has continuous derivatives of all order at the point z . 1

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2 Conditions for a Complex Function to be Analytic 2.1 A necessary condition for a complex function to be analytic Let f ( x, y ) = u ( x, y ) + iv ( x, y ) be a complex function. Since
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