庫倫函數

庫倫函數(Couloumb function)是下列常微分方程的解

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

此方程的兩個解為：

$F_{\ell }(\eta ,\rho )$ 和 $G_{\ell }(\eta ,\rho )$ , 以合流超幾何函數表示^[1]^[2]

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 )\,.

參考文獻

^ Thompson, I. J., Coulomb Functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR 2723248
^ Abramowitz, Milton; Stegun, Irene Ann (編). 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
.

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