库伦函数

库伦函数(Couloumb function)是下列常微分方程的解

Coulomb F function plot

Coulomb G function plot

${\displaystyle {\frac {d^{2}w(\rho )}{d\rho ^{2}}}+(1-{\frac {2\eta }{\rho }}-{\frac {l*(l+1)}{\rho ^{2}}})*w(\rho )=0}$

此方程的两个解为：

${\displaystyle F_{\ell }(\eta ,\rho )}$ 和 ${\displaystyle G_{\ell }(\eta ,\rho )}$, 以 合流超几何函数表示^[1][2]

${\displaystyle F_{\ell }(\eta ,\rho )={\frac {2^{\ell }e^{-\pi \eta /2}|\Gamma (\ell +1+i\eta )|}{(2\ell +1)!}}\rho ^{\ell +1}e^{\mp i\rho }M(\ell +1\mp i\eta ,2\ell +2,\pm 2i\rho )\,.}$

参考文献

1. Thompson, I. J., Coulomb Functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
2. Abramowitz, Milton; Stegun, Irene Ann (编). Chapter 14. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 538. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642.

   LCCN 65-12253

   .

• Bateman, Harry, Higher transcendental functions (PDF) 1, McGraw-Hill, 1953 [2015-02-11], （原始内容存档 (PDF)于2011-08-11）.
• Jaeger, J. C.; Hulme, H. R., The Internal Conversion of γ -Rays with the Production of Electrons and Positrons, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (The Royal Society), 1935, 148 (865): 708–728, Bibcode:1935RSPSA.148..708J, ISSN 0080-4630, JSTOR 96298, doi:10.1098/rspa.1935.0043
• Slater, Lucy Joan, Confluent hypergeometric functions, Cambridge University Press, 1960, MR 0107026.