Question

TOPIC: COMPLEX VARIABLES

1. Consider the integral from question 2 of the previous homework assignment: | sin ma • dx, x(x2 + a2) and assume that both m and a are positive real numbers. By using an indented contour, evaluate this integral fully. (You are allowed to resubmit material submitted as part of the previous assignment if you wish.]

Answer 1

concider { FC?)dz where f(2) ( 209) The poles of f(2) are given by z (2 tah) =o ie z=0 itzai, zoai simple poles of f(8) are and real - pole 20 lies on the real axis, we take contour c. consisting of the apper half plane ce of a large circle Izl=R the axis from the indented at 2=0 and let or be the rodius of Indentation f(2) is analytic within and on a except at the simple pole at zzai theorem cauchy's residues we get { if(t)oz z 2TIERT withinc . Hence by ar Ce cy e -Y Jo R R -R CR Y foxide + f frejdz + f fixidz &f flake ce = 2Ti ERA where ERt is residue at the simplepole Residue at the simple pole at taai whis c lim (Piai) f(2) نمود

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