Formula:10229

Digital Library of Mathematical Functions ID 33.6.E5

${\displaystyle{\displaystyle{H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}\left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}\left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}}k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2\mathrm{i}\rho)^{a-k}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle{\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)}}$: phase of Coulomb functions
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle{H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)}}$: irregular Coulomb radial functions
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle\rho}}$: nonnegative real variable
• ${\displaystyle{\displaystyle\eta}}$: real parameter
• ${\displaystyle{\displaystyle a}}$: variable
 

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