Question

12. (a) Show that 1y dt By letting R o, deduce that the residue of f ) at t 0. at zoo by the equation f (z) dz is given by 2πί times (b) When zoe is an isolated singular point, define the residue of f () Show that (e)d2miRes () Coo (c) Use the above result to evaluate the integral Ca2 + 22 z where C is any positive contour enclosing the points z 0, tia, and check the result by calculating the residues at those poles. f (z) dz 0. (d) Show that if 22f (z) is analytic at zoo, then

Answer 1

@Shao he 2 Sol 2 下 101) t2 dt lie

2 Poly ale Ciy insile a c Pole, 2 a 2 simpu po zゥ。 (z-«a)2 dt a

azL ci2 22