Question

Evaluate Sc(1/22)dz, where C is the line from 1 to 1 + 5i followed by the line from 1+5i to -1 + 5i followed by the line from -1 + 5i to -1. (a) 0 (b) 2 (e) -2 (d) i (e) None of the above. Which of the following integrals is not equal to zero ? a) Sal=10 ze dz b) |--2--(2 – 2)e+/-dz Jul 2 d) cosa 282dz 4) J- 2 e) Si=1 (2–1jadz Which of the numbers below is the value of 2 - (22+ 13 dr ? a) 0 b) 3/4 c) 3/8 d) -37/8 e) 3/4

Answer 1

Cauchy Integral f(3) = Formule is - of Here H2) = Cosz, to 0 and n=1 cost az = 2tif'(o) zz = 2ni (sinal z =0 = 0 Cost dz = 2Ti flo) - 2Ti Cosoo 12=1 - ati 2 =0 by cauchy Integral formule as singularity is in the closed region 12117 z=loeio , OLO LRT de = loi e io do @ zeelde IZ=10 1 e lo eio logio - loie do 80 en ex eo]

2- 12-91=5 It az &z= 2 + 5 eo ocolat dz = sieio do is 5e a treid) sielo do 0 7 where c is lind from 1 to 1751, 115ito-1452, -1 + 51 to 7 ie integral es independent of the path So just compute the line integral from 1 tot yo dy zo z= x+io x dz= do = 1 de = -26 de = does not exist as - does not enist. as 4 7 e = Not finite X2 X2 xox

put a = tand dx = seed do = 21 Seco do 60340 do (seed) = kxts (1,5) = t = 3 + (th 5 = 3 T [ 3 = 5)