Question

1 Evaluate ; dz,Cistheellipse + y2 = 1 c 2:+7:+3

Answer 1

Q. Answeron let fa)= 22 +72+3 & ginen inen C is an ellipse x² +42=1 now, Breaking fee), into partial fractione to (27+D (2+3) + B et A (27 +(2+3) 122+1) (2+3) ㅗ A(Z+3) + B.(22+) put, z=-/, then 1 = A(+3) + 0 = 1=5.A = A = 2 put z= -3, then 1 = 0+ B (6+1) -13 fle lolle (2Zf) (Z+3) 7 -/s 17+) (22+1)

now o feidz = dz 22+72+3 С dz - (2Z FJ (213) =20 <dz -2 6 Idz c(27+1) 5, Z+ رف dz -6 I dz (A+12 Z+ Recall: - Cauchy's theoremis if fe) be analytic in a Rigion R & boundary I feidz Cauchy's Integral formula : if ft) be analytic inside and on a simple closed carve c & lete a be ay point preside C then

fa)=16 fel dz (7-00 21TU Ro Carne is an ellibre (. x² + y2 =) 4 Tunz how, I tlet I, - & 1 dz cl2th С "Rez o Z=4 (20) He-va is inside to I, dz ezt fexzmittere 2f,25=[ = 1 xatti sing Cauchy is integral formula dz By Cisir By let In Ip = 6 Idz cZt3 {3))

Clearly z =-3, Ų outside the Curve C. & hence is tic Cinside fonc) + Zt3 d' auchy's Whing Cauchy's theorem Iz (2+3) how from $ c2z²+72+3 da = 1 [I, - Iz (27= I, 250 &I=0 5 2011° Ansi 5