Question

4. (a) Use Cauchy's residue Theorem to provide alternative proofs for Cauchy's integral formula and its extension. (b) Evaluate z+ 4z + 5 dz, son z2 + z where C is the positively oriented circle of radius 2 centered at the origin.

Answer 1

4(a) Proof for Cauchy's Integral formula and its extension by using Cauchy's is residue theoremi of f(2) is analytic within and on a closed curve and it as any point within c, then fcal- f(2) dz 1 zai C 2-a consider +(2)/8(2-a) is analytic at all points within c except at z=a. with a as cente and radius, draw small circke i lying entirely within c. Now +(2)/th-a) being analytic in region enclosed by cidco f (2) dz (42) dz letz-a-re Zea Ja z-a Hatrelo) ir e do éxtio CI сСона, Haldo = if (n) { ** do = anita) --- thus fca) = 1 fel dz Cauchy's Integral formula. zai Jc za f(z) is analytic in region then Extension of canchy's theoreme- I between two simple closed curves of J. F(2) dz = f(2) d2 so prove (2) dz der 다 that Jag fiz)dz + J +2) dz + f treddet ( +12) Azzo D A

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But we get AB and BA integral cancels f fra detta fra) ezzo Reversing the direction of integral around c, and trans, transporting fftz Haldz= f ffaldz 969, (2, (3,.. be number of closed curves with in c, then { feltz- Sa Hajdzt So ffz/ dat [ftz) dz t--- JO2 I- Z=2 z² +42 +5 da 2²+Z 2² +42+5 dz С JC z(2+1) singular pointe z(2+1) = 2-0, -1 uts both points lies inside 2=2² axi [ (Residue at 2=0) + (Residure at 2=-1)] - aki lim z (2²+ 4245) tim (2+1) (2²+ 4275) 2/2+1) 리데 =) I zs 1 2to - ani otot5 (+)² + 4(4) + + 0+) Eki [ 5+ (-2)] = bai

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