Set Remark Check the result by differentiating the formula in the pre ceding exercise with respect to the parameter Solution VII

Question

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

Kunduz · Accepted Answer

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