Question

Complex Analysis:

1 + COS Z Define the function 1 f(2)= (z + 1)2(23 +1) (a) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) Let I = 71+72, where yi and 72 are the directed smooth curves parameterized by TT zi(t) = 2i(1 – 2t), 0 < t < 1 z2(t) = 2eit, 277 < t < 2' respectively. Compute ſi f(z)dz. (It is ok to leave terms of the form eil in your final answer without simplifying further.)

Answer 1

Ans i function frane --- dd COS Z (3+1)?(3371) that roots are 3+1 rts of ,W2 f(x) = + tesz First, we know = from (z+1)= (3-2) (3:02) cleasily the pointe singularities from (osz=0 2 pnt) Singulasties Nove at 7=-1 Explicitly eu ? and are z=1, cos is also points of + cos g R=w? are also (3-3)"fcz) is buandeel meer at 23-4 (+1j3 lim always be finite. Ruh (tiga (220) Lacu? Hence, 2= -4 2=-1 is pole wyder 3 At, (2=W) and 2=w², Gleasly (27)*fc2) is bounded near formulat 20 = and Z=0² Hence, 2000, point pole of order to Now the limit the pales a= (antil Henes, @ is the mon- isolated essentiel singularity of fez). 6 Let T=Vitra where, &, and Va are the directed smart curve besamelized by 2, (+)- 22(1-3+) ostsi is : of 327+) = peut

ar 1, COLZ z }et fra fading 741) ܕܙ - - Sfezide= Sri fez, det fra freid? fi (2+1)?(2341) fifah varf(ei- vit ?-1} * at sinas) Midt Sve fel 4+1)* (****) * et corset friet {{sit }dt Sra Qielt dt + -42 Soins - -42+){(21+41+jP+1}* ten (23-404) 22 + dt (پائع cobacitjet dz,= Here, 20² + H)2 (a 7,64) Qz (1-34) = Za(t) = zeit -42dt dz = zielt dt =)