Mathematical Methods for Physics and Engineering - Matematica.NET

24.14 EXERCISES24.14 Prove that, for α>0, the integral∫ ∞t sin αt0 1+t dt 2has the value (π/2) exp(−α).24.15 Prove that∫ ∞cos mx0 4x 4 +5x 2 +1 dx = π (4e −m/2 − e −m) for m>0.624.16 Show that the principal value of the integral∫ ∞cos(x/a)dx−∞ x 2 − a 2is −(π/a)sin1.24.17 The following is an alternative (and roundabout!) way of evaluating the Gaussianintegral.(a) Prove that the integral of [exp(iπz 2 )]cosec πz around the parallelogram withcorners ±1/2 ± R exp(iπ/4) has the value 2i.(b) Show that the parts of the contour parallel to the real axis do not contributewhen R →∞.(c) Evaluate the integrals along the other two sides by putting z ′ = r exp(iπ/4)and working in terms of z ′ + 1 and 2z′ − 1 . Hence, by letting R →∞show2that∫ ∞−∞e −πr2 dr =1.24.18 By applying the residue theorem around a wedge-shaped contour of angle 2π/n,with one side along the real axis, prove that the integral∫ ∞dx0 1+x , nwhere n is real and ≥ 2, has the value (π/n)cosec (π/n).24.19 Using a suitable cut plane, prove that if α is real and 0