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Complex Variables Problem set: 6 1. Evaluate the following integrals (a) Z ∞ 0 dx (x 2 + a 2 ) 2 , a 2 > 0 (b) Z ∞ 0 dx (x 2 + a 2 )(x 2 + b 2 ) , a 2 ,b 2 > 0 2. Evaluate the integral Z ∞ -∞ x sin x x 2 + a 2 dx, a 2 > 0 3. Consider a rectangular contour C R with corners at (±R, 0) and (±R, a). Show that I C R e -z 2 dz = Z R -R e -x 2 dx - Z R -R e -(x+ia) 2 dx + J R =0 where J R = Z a 0 e -(R+iy) 2 i dy - Z a 0 e -(-R+iy) 2 i dy Show lim R→∞ J R = 0, whereupon we have R ∞ -∞ e -(x+ia) 2 = R ∞ -∞ e -x 2 = √ π, and consequently, deduce that R +∞ -∞ e x 2 cos 2ax dx = √ πe -a 2 . 4. Evaluate the following definite integral using complex integration (a) Z ∞ -∞ e iax x 2 - b 2 dx, a, b > 0 (b) Z ∞ 0 sin x x(x 2 + 1) dx Note: The poles may fall on the x axis, therefore choose the contour of integration accordingly. 1
