tailieunhanh - Handbook of mathematics for engineers and scienteists part 47

Handbook of mathematics for engineers and scienteists part 47. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 290 Integrals i xf sinx dx nn2 i f sinx dx if f x f -x 00 i xf sin x dx -1 n-1nn i f sin x dx if f -x -f x 00 2n r2n __ i a sin x b cos x dx f Va2 b2 sin x dx 2 f V 00 --sin x- - dx f sin2 x dx if a 1 0 1 2a cos x a2 J J0 r f . oSin2 x-2 dx r f dx if 0 a 1. 0 1 2a cos x a2 0 a2 cos X dx 0 0 . Integrals involving logarithmic functions. d xxf x dx X Ja J A 0 dX I f x g x X dx a b n x dx f a fb f x in g x dx a i f x g x A lnn g x dx dfT i f x g x A dx. aa -I A 0 a . General Asymptotic Formulas for the Calculation of Integrals Below are some general formulas involving arbitrary functions and parameters that may be helpful for obtaining asymptotics of integrals. . Asymptotic formulas for integrals with weak singularity as e 0. 1 . We will consider integrals of the form ia xß 1f x dx 0 x a where 0 a to 3 0 f 0 0 and e 0 is a small parameter. The integral diverges as e - 0 for a 3 that is lim I e to. In this case the leading s 0 term of the asymptotic expansion of the integral I e is given by I r ß p a ß f 0 ß-a O if a ß r a I -f 0 ln O 1 if a ß where r 3 is the gamma function and a min 3 - a 1 0 . . Definite Integral 291 2 . The leading term of the asymptotic expansion as e 0 of the more general integral ra x -lf x dx I e Z k La J0 xk ek a with 0 a oo 3 0 k 0 e 0 and f 0 0 is expressed as I r f 0 v 3 I v n I rfi-ak niS if Zik . 1 e kT a V k J a k j O e if ak 3 I e f 0 In e O 1 if ak where a min ft - ak 1 0 . 3 . The leading terms of the asymptotic expansion as e - 0 of the integral I e Î xa exp -Ex f x dx J a with a 0 ft 0 e 0 and f 0 0 has the form i e 1 f 0 rf 0 2 e a L ft ft J if a -1 if a -1. I e -1 f 0 In e ft 4 . Now consider potential-type integrals n f Hi f 0 d V - Z 2 r2 with z r p being cylindrical coordinates in the three-dimensional space. The function n f is simple layer potential concentrated on the interval z e 1 1 with density f z . If the density is continuous then n f satisfies the Laplace equation AH 0 outside z g 1 1 and .