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Handbook of mathematics for engineers and scienteists part 140. 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. | 18.3. Sine Integral and Cosine Integral. Fresnel Integrals 941 18.2.3. Logarithmic Integral 18.2.3-1. Integral representations. Definition li x r dL 0 In t lim 0 dt In t if 0 x 1 if x 1. 18.2.3-2. Limiting properties. Relation to the exponential integral. For small x ll x - taiw For large x li x - -. ln x Asymptotic expansion as x 1 li x C ln ln x k 1 lnk x k k Relation to the exponential integral li x Ei ln x x 1 li ex Ei x x 0. 18.3. Sine Integral and Cosine Integral. Fresnel Integrals 18.3.1. Sine Integral 18.3.1-1. Integral representations. Properties. Definition Si x r ïïC dt sux -fœ ÎÜdt si x - n. J0 L Jx L 2 Specific values n Si 0 0 Si œ - si œ 0. Properties Si -x -Si x si x si -x -n lim si x -n. x -œ 942 Special Functions and Their Properties 18.3.1- 2. Expansions as x 0 and x œ. Expansion into series in powers of x as x 0 Si x - V 1 fc 1x2fc 1 f- 2fc - 1 2k - 1 . k-1 Asymptotic expansion as x œ V1 -1 ro 2m z. r2M-n iv-1 D 2m -1 z.-2m si x --cos x z z x2m 1 0xl snx z x2m dx z L m-0 J L m-1 where M N - 1 2 . 18.3.2. Cosine Integral 18.3.2- 1. Integral representations. Definition Ci x -r dt - C ln x r dt J. t Jo t where C - 0.5772 . is the Euler constant. 18.3.2-2. Expansions as x 0 and x œ. Expansion into series in powers of x as x - 0 _ 1 kx2k Ci x C Inx . 2k 2k k 1 Asymptotic expansion as x - œ Ci x cos x E -1 mg -1 xr2M L m 1 J sin x r N-1 _ E . z 2 - A L m 0 J where M N - 1 2 . 18.3.3. Fresnel Integrals 18.3.3-1. Integral representations. Definitions S x dt 1 2 Îsin t2 dt V2n Jo Vt V n Jo C x 4 ix cog dt iTx cos t2 dt. Æ Jo Vt V n Jo 18.4. Gamma Function Psi Function and Beta Function 943 18.3.3-2. Expansions as x 0 and x œ. Expansion into series in powers of x as x 0 S x i 2 x V 1 fcx2fc 1 V n 4k 3 2k 1 k 0 c x xV 1 fcx2fc . V n 4k 1 2k k 0 Asymptotic expansion as x œ S x 2 C x 2 cos x sin x -y P x -- Q x y 2nx y 2nx sin x D cos x r- P x ------Q x V2nx V2nx P x 1x3 - 2x 2 1x3x5x7 2x4 Q x 2x 1x3x5 2x 3 18.4. Gamma Function Psi Function and Beta Function
