MSO202Lect7

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Unformatted text preview: termination of whose parametric representation may be complicated. In such a case, the possibility of obtaining a curve C2 satisfying the conditions of the corollary and whose parametric representation is simple to obtain, is explored and the integral is evaluated with the help of above corollary. 7 1 dw , where is any anticlockwise w z0 oriented simple closed piecewise smooth curve and z0 is a point lying in the bounded domain D with boundary . Note that direct evaluation of the above integral is not possible, since any explict equation of is not known. However, this integral could be simply evaluated by using the above theorem. Consider any anticlockwise oriented circle Cr :| w z0 | r , with 1 is r small enough so that Cr lies in D . The function w z0 analytic on the curves and Cr and in the domain bounded by these curves. Therefore, by Cauchy Theorem for Multiply connected domains, 2 1 1 1 dw dw it ireit dt 2 i Cr w z0 w z0 0 re since, w(t ) z0 reit , 0 t 2 , is a parametric representation of the circle Cr . Example: Evaluate 8 Cauchy Integral Formula: If f is analytic in a domain G and ________ ________ B (a, r ) G , where B (a, r ) {w : w a r}. Then, for any z {| w a | r} 1 f ( w) f ( z) dw (1) 2 i Cr ( w z ) where, Cr : w(t ) z reit , 0 t 2 . Proof: Consider a circle | w z | * centered at z and having radius * sufficiently small such {| w z | *} {| w a | r}. Then, by Cauchy Theorem of Multiply Connected Domains, f ( w) f ( w) dw dw Cr ( w z ) |w z| * ( w z ) since the integrand is an analytic function in the domain lying between Cr and | w z | * . Now, note that f ( w) f ( w) f ( z ) 1 dw dw f (a ) dw (w z) |w z| * ( w z ) |w z| * |w z| * ( w z ) (*) The second term of (*) 2 i f ( z ) .Therefore, Cauchy Integral Formula follows if we prove that the first term of (*) is zero. For this use continuity of f ( w) at ' z ' , which gives that for every 0, there exists 0 such that | f ( w) f ( z ) | whenever | w z | . Choose * . 9 Then, f ( w) f ( z ) dw | * 2 * 2 (by ML‐Estimate) (w z) |w z| * f ( w) f ( z ) dw 0 since is arbitrary. * (w z) |w z| Note: In view of Cauchy Theorem for multiply connected domains, Cauchy Integral Formula (1) remains valid with Cr replaced by any simple closed piece‐wise smooth curve so that (i) every point enclosed by is in D (ii) encloses the point z . This is because the function f ( w) / ( w z ) is analytic in the domain lying between Cr and . |...
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This document was uploaded on 09/19/2013.

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