\rho,\eta

\rho

§33.7 Integral Representations

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , integral representations
Notes:: See Hull and Breit (1959, pp. 413–416). For (33.7.1) see also Lowan and Horenstein (1942), with change of variable $\xi=1-t$ in the integral that follows Eq. (8). For (33.7.2) see also Hoisington and Breit (1938). For (33.7.3) see also Bloch et al. (1950). For (33.7.4) see also Newton (1952).
Permalink:: http://dlmf.nist.gov/33.7
See also:: Annotations for Ch.33

33.7.1		$F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1}2^{\ell}e^{\mathrm{i}\rho-(% \pi\eta/2)}}{\|\Gamma\left(\ell+1+\mathrm{i}\eta\right)\|}\int_{0}^{1}e^{-2% \mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-\mathrm{i}\eta}\,\mathrm{d% }t,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter Referenced by: §33.7, §33.7 Permalink: http://dlmf.nist.gov/33.7.E1 Encodings: TeX, pMML, png See also: Annotations for §33.7 and Ch.33

33.7.2		${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{e^{-\mathrm{i}\rho}\rho^{-\ell}}{(2% \ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}e^{-t}t^{\ell-\mathrm{i}% \eta}(t+2\mathrm{i}\rho)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t,$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.3.1 (modified) Referenced by: §33.7 Permalink: http://dlmf.nist.gov/33.7.E2 Encodings: TeX, pMML, png See also: Annotations for §33.7 and Ch.33

33.7.3		${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{-\mathrm{i}e^{-\pi\eta}\rho^{\ell+1% }}{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}\left(\frac{\exp\left(% -\mathrm{i}(\rho\tanh t-2\eta t)\right)}{(\cosh t)^{2\ell+2}}+\mathrm{i}(1+t^{% 2})^{\ell}\exp\left(-\rho t+2\eta\operatorname{arctan}t\right)\right)\,\mathrm% {d}t,$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.3.3 Referenced by: §33.7 Permalink: http://dlmf.nist.gov/33.7.E3 Encodings: TeX, pMML, png See also: Annotations for §33.7 and Ch.33

33.7.4		${H^{+}_{\ell}}\left(\eta,\rho\right)=\frac{\mathrm{i}e^{-\pi\eta}\rho^{\ell+1}% }{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{-1}^{-\mathrm{i}\infty}e^{-\mathrm% {i}\rho t}(1-t)^{\ell-\mathrm{i}\eta}(1+t)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t.$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.3.2 Referenced by: §33.7, §33.7 Permalink: http://dlmf.nist.gov/33.7.E4 Encodings: TeX, pMML, png See also: Annotations for §33.7 and Ch.33

Noninteger powers in (33.7.1)–(33.7.4) and the arctangent assume their principal values (§§4.2(i), 4.2(iv), 4.23(ii)).

\rho