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Part 2 book “Handbook of mathematical functions with formulas, graphs, and mathematical tables” has contents: Coulomb wave functions, hypergeometric functions, weierstrass elliptic and related functions, elliptic integrals, mathieu functions, parabolic cylinder functions, combinatorial analysis, . and other contents. | 14. Coulomb Wave Functions Milton Abramowitz 1 Contents Page Mathematical Properties. 538 14.1. Differential Equation Series Expansions. 538 14.2. Recurrence and Wronskian Relations. 539 14.3. Integral Representations. 539 14.4. Bessel Function Expansions. 539 14.5. Asymptotic Expansions. 540 14.6. Special Values and Asymptotic Behavior. 542 Numerical Methods. 543 14.7. Use and Extension of the Tables. 543 References. 544 Table 14.1. Coulomb Wave Functions of Order Zero .5 n 20 l P 20 . 546 FoGbp Ơo0z p ỡo0bp 1J .5 .5 2O p l l 20 5S Table 14.2. Co ij - T r l iij I. 554 n 0 .05 3 6S The author wishes to acknowledge the assistance of David s. Liepman in checking the formulas and tables. 1 National Bureau of Standards deceased . 537 14. Coulomb Wave Functions Mathematical Properties 14.1. Differential Equation Series Expansions Differential Equation 14.1.1 dp2 p p2 p 0 L a non-negative integer The Coulomb wave equation has a regular singularity at p 0 with indices Z-M and L it has an irregular singularity at p 00. General Solution 14.1.2 w C 17 t pHGƠlGĩ p Cl Cl constants where Fl ti p is the regular Coulomb wave function and GL ii p is the irregular logarithmic Coulomb wave function. Regular Coulomb Wave Function Fl ij p 14.1.3 FL v p CZl rỉ ĩe- M L ỉ-irlí 2Z 2 2ip 14.1.4 CM ri P 14.1.5 n p i Aị rìpk-L- 14.1.6 Al 1 1 k E k-L-V AỈ 2nAi.1-Aị.i k L 2 14.1.12 F L Fl p CM p t p p 14.1.13 7 p - s Irregular Coulomb Wave Function ơt n p 14.1.14 . p c F p ln 2p 1 M p 14.1.15 0L r p Dl ij p l l iị p 14.1.16 p C _ _ 14.1.17 u p aiW k--L 14.1.18 a L l ai i o k L l fc L aỄ 2ijữỉ _1 aỉ-1 2k l pL v AỈ 14.1.19 PlM 1 2 i S 1 8 ƠỊI Ị _ _2y 4-- v rir i i Zy pLứ See Table 6.8. X4.1.7 CM g l i See chapter 6. 14.1.8 ớẵ 2 e2 --l - 14.1.9 72 PtC Cpfi 2Í 2Z ịl 14.1.10 .in L 2L lj 14.1.11 xW_ 1 i 2 4 1 2 . . . Z2 n2 2 2 2L 1 2L 2 14.1.20 r .f y. I - I2fo z 2L l2L P 2L 1 iri-Ddn-L l 2Z-1 2 2 i -Z i - l . . ilj z-l . 2 i 1 14.1.21 ỠL Fah1 2p f p-IFrGbp dp CỈ TỊ Plvu n p 538 COULOMB WAVE .
