§33.7 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).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.7

33.7.1 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{\rho^{{\ell+1}}2^{\ell}e^{{i\rho-(\pi\eta/2)}}}{|\mathop{\Gamma\/}\nolimits\!\left(\ell+1+i\eta\right)|}\int _{0}^{1}e^{{-2i\rho t}}t^{{\ell+i\eta}}(1-t)^{{\ell-i\eta}}dt,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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
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33.7.2 $\mathop{{H^{{-}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{e^{{-i\rho}}\rho^{{-\ell}}}{(2\ell+1)!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}\int _{0}^{\infty}e^{{-t}}t^{{\ell-i\eta}}(t+2i\rho)^{{\ell+i\eta}}dt,$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

33.7.3 $\mathop{{H^{{-}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{-ie^{{-\pi\eta}}\rho^{{\ell+1}}}{(2\ell+1)!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}\int _{0}^{\infty}\left(\frac{\mathop{\exp\/}\nolimits\!\left(-i(\rho\mathop{\tanh\/}\nolimits t-2\eta t)\right)}{(\mathop{\cosh\/}\nolimits t)^{{2\ell+2}}}+i(1+t^{2})^{\ell}\mathop{\exp\/}\nolimits\!\left(-\rho t+2\eta\mathop{\mathrm{arctan}\/}\nolimits t\right)\right)dt,$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

33.7.4 $\mathop{{H^{{+}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{ie^{{-\pi\eta}}\rho^{{\ell+1}}}{(2\ell+1)!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}\int _{{-1}}^{{-i\infty}}e^{{-i\rho t}}(1-t)^{{\ell-i\eta}}(1+t)^{{\ell+i\eta}}dt.$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ : 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

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)).