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MSO202Lect7

# MSO202Lect7 - 1 Lecture 7 Cauchy Theorem Let f be analytic...

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1 Lecture 7 Cauchy Theorem : Let f be analytic inside and on a simple, closed, piecewise smooth curve C. Then, ( ) 0 C f z dz . Definitions: Let ( ), z t a t b , be parametric representation of the curve C. Simple Curve : The curve C is said to be simple, if it does not have any self intersections (i.e. 1 2 ( ) ( ) z t z t whenever 1 2 1 2 ( , ) t t a t t b ). Closed Curve : The curve C is said to be Closed, if end point of the curve is the same as its initial point (i.e. ( ) ( ) z a z b ). Piece wise smooth Curve : The curve C is said to be Piece wise smooth, if ( ) z t is piece wise differentiable (i.e. differentiable for all except finitely many t ) and ( ) ( ( )) d z t denoted as z t dt is piece wise continuous in the interval [ , ] a b

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