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Handbook of mathematics for engineers and scienteists part 144

Handbook of mathematics for engineers and scienteists part 144. 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. | 18.13. Elliptic Integrals 969 18.13. Elliptic Integrals 18.13.1. Complete Elliptic Integrals 18.13.1-1. Definitions. Properties. Conversion formulas. Complete elliptic integral of the first kind z x n2 da f1 dx K k . . . Jo V1 - k2 sin2 a Jo a 1 - x2 1 - k2x2 Complete elliptic integral of the second kind p n 2 p 1 E k J V1 - k2 sin2 a da J M1 - k2x2 dx. 1 - x2 The argument k is called the elliptic modulus k2 1 . Notation _ k 1 - k2 K k K k E k E k where kf is the complementary modulus. Properties K -k K k E -k E k K k K k E k E k E k K k E k K k - K k K k n. Conversion formulas for complete elliptic integrals v 1 - k - 1 k Ktki K . T ë K k E TT A E k k K k 1 k 1 k L J k k 1 k K k 1 k E M TTt 2 E k - k 2 K 1 rv J 1 rv 18.13.1-2. Representation of complete elliptic integrals in series form. Representation of complete elliptic integrals in the form of series in powers of the modulus k K k n 1 1 k2 i 3 k4 2 2 2x4 E k n 1 - y ç - f y k -. 21 2 1 2x4 3 2n - 1 12 2n _ 2n -1 2n k2n 2 k2n 2n - 1 970 Special Functions and Their Properties Representation of complete elliptic integrals in the form of series in powers of the complementary modulus k V1 - k2 K k n f k 1 Y k 1 -k 2 k 1X3 y k 1 -k 4 1 k 1 A V kJ A2X4 M kJ K k ln AWIn - k V 2J k 2 1X2 A2 2n-1 2n _ 1 - k 2 1 k 1x3 y kl 4 2 2X4 J k- 1X2 2 3x4 A4 in 4- k 6 2x4x6 k 1x2 3x4 5x6 v 2 E k n 1 k J1 1 - 1dA J J . n - 3 17 Ÿ 4 I 22 V k 2x4 2 1 k L 2n V k E k 1 1 ln 4 1 k 2 1 X3 ln _2 k 4 1 2 V k 1x7 22x4 V k 1x2 3x4 12X32X5 A __ _ __ _ 22x42xX k1 1x2 3x4 5x6 18.13.1-3. Differentiation formulas. Differential equations. Differentiation formulas d K k E k K k d E k E k - K k dk k k 2 - dk k The functions K k and Kz k satisfy the second-order linear ordinary differential equation dkk 1 - k2 dJk k K 0- The functions E k and Ez k - Kz k satisfy the second-order linear ordinary differential equation 1 - k2 dk kitE k E - 18.13.2. Incomplete Elliptic Integrals Elliptic Integrals 18.13.2-1. Definitions. Properties.