Question

Evaluate the integral. Does Cauchy's theorem apply? Show details
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2 & de 1 6 z dz > ¿ z2+ CZ til: i Z2+1 C: 12-11 Counterclockwile Counter clock wise

Answer 1

Basics: Cauchy's Theorem: If f(t) is analytic function and flcz) is continuous at each point within and on a closed curve c then fézz 42=0 Cauchy integral formula: let f be analytic everywhere inside and on a simple closed contour C, taken in the positive sense. If zo is any point interior onc, then c dz f(zo)= . 1 2 Ti f(z) Z-Zo I= 0 ? dz 1 cilztilal - 0 Z2+) Нех е, Ztil=) is Iz- Zol=8 is positively oriented circle with centre =- ; &r=1 Here Z=- inside the is lies region -

Page NO Date for the cauchy's given formula function we can not , since not analytic apply at 2=-; that is z Z?+) - i ( ? at Z=- - + 0 So, not analytic { Now by Cauchy integral Formula, I= dz z 2271 dz ... (22+1 = (2-1) (2+i)) z (z+i) (2-1) I fcz = where z Z- zti - 6 z I = 275 f(-i) dz where f(2)= 2 - © & Zo=-1 Here f (2) = z z -i So, f(-1) = put in o .. I = 25 i ( 2 ) = Ti Ini this is the required solution

Ex @ 12-11 = 1 I = 6 dz 2²41 dz eti) (2-1) again can not since point in this example we apply cauchy's formula the function to not analytic = í which is inside curve c at the So, by Cauchy integral formula dz I= § 24 (2-1) z -i where zo= i& fcz)= ) ZH; So, f(20) = f(i)= I I= 2 * f (zo) I= ti fi)

Page No. Date so the solution is required I=T

Here in the above examples the functions given in the example are not analytic inside the given curve so we cannot apply cauchys theorem but we can solve them by using Cauchy integral formula.