Set Remark Check the result by differentiating the formula in the pre ceding exercise with respect to the parameter Solution VII 2 2 and integrate R Le f z and therefore dx IT x a 20 P I 1 2 a along the contour in Figure VII 2 2 Residue at a double pole at 2 ia where by Rule 2 dr x 1 0 1 d 1 2 Res 52 2 2 10 lim 12 1 ia 22 a iadz 2 ia 2 ia Integrate along 1 and let Roc This gives R da 2 dz 1 17 z S R Integrate along 72 and let Roo This gives 1 0 2 i Remark From Exercise VII 2 1 we have the formula 1 1 7 1 z ia z ia da 0 x a 2 1 2 da Lx 1 9 4 zdr 427 1 1 1 2 d 5 R a Using the Residue Theorem and letting Roo we obtain that that 4 3 R 75 75 TR 0 R zia dr x a 77 a 0 It is allowed to differentiate both sides in the formula because for every compact interval with a 0 both the integrand and its derivative are con tinuous and the primitive of the derivative is uniformly bounded by a a independent function h r such that fh x on every such like interval We start by differentiate the left hand side in the formula with respect to the parameter a a jz dr 2a 2 0 1 6 and differentiate the right hand side in the formula in the same way da 1 I 0 4a3 dr 2a a dr Loo 2