Question

4. The rational function 1 p(2) = (z – i)4 + 4 is holomorphic on the domain C\{a1, A2, A3, A4} for some four distinct points d1, A2, A3, and 04. (a) Find the values of a1, A2, A3, and 04. [8] (b) Use one of the Cauchy's formulas to evaluate the integral of p(2) along y, a positively oriented closed contour parametrising a rect- angle with vertices Fi and 4£i. Show that this integral can be expressed in the form of TT è (a + ib) where a, b and c are integers. [8]

Answer 1

4 1 PCZ) = (z-;)4+4 (il Holomorphic in domein - 9,192191 we have to find singulerities - than if a 1 1 then 1 I I (1-1)4+4 first point of of singularity. -4+4 o b 92= -1 then 1 - oll (-1-0)+4 - 4+4 93=1tzi then 1 1 (1+2i-174+4 (1+0)4+4 1 - 1 -4+4 EA --1tzi then 1 1 - (-1+2i-i)4+4 (-1+)4+4 to Go on Bin 4+4 1, t, itzi ,-ltri gubrities of the p (Z). y the

ti & Ati & (11) rectangle 4 Bingulorities of p(2) = 1,-1,1+; -1+zi bo p (2) = (z-c) 4+4 1 Can be written as- 그 = P (2) how بلدية 1' 2 3 (Z-1) (2 72) (Z -(ltri)(Z-6-1+2i) 4ti Bo only z=1 iBin the Coun tour 4-6 d z = ? I (2+1) = 7 (1+i) 8 AA 1 - Bo Sizin -¿) 4+4 16 METS

2 - fres f(4) 2 (+1) (² - (I+)) (Z - (-14 ) ㅗ (2) ( -1 -1) (1+1 -r) 1 (2-12 2-r/ (2+r) . 그 4C(22) (242) 24전 - C2+) 321 - 2조 (2+) = (-주 (2) 322 1. 나. (444) 27.GI) : - 16
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