1Lecture 7Cauchy Theorem:Letfbe analytic inside and on a simple, closed, piecewisesmooth curve C. Then,( )0Cfz dz.Definitions:Let( ),z tatb, be parametric representation ofthe curve C.Simple Curve: The curve C is said to besimple,if it does nothave any self‐intersections(i.e.12()()z tz twhenever1212(,)ttattb).Closed Curve: The curve C is said to beClosed,if end point of thecurve is the same as its initial point(i.e.( )( )z az b).Piece‐wise smooth Curve: The curve C is said to bePiece‐wisesmooth,if( )z tis piece‐wise differentiable (i.e. differentiable forall except finitely manyt) and( ) (( ))dz tdenoted as z tdtis piece‐wise continuous in the interval[ , ]a b
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