DLMF:33.7.E3
type: Item
Digital Library of Mathematical Functions ID: 33.7.E3
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle{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,}}" display="inline"> H ℓ - ⁡ ( η , ρ ) = - i ⁢ e - π ⁢ η ⁢ ρ ℓ + 1 ( 2 ⁢ ℓ + 1 ) ! ⁢ C ℓ ⁡ ( η ) ⁢ ∫ 0 ∞ ( exp ⁡ ( - i ⁢ ( ρ ⁢ tanh ⁡ t - 2 ⁢ η ⁢ t ) ) ( cosh ⁡ t ) 2 ⁢ ℓ + 2 + i ⁢ ( 1 + t 2 ) ℓ ⁢ exp ⁡ ( - ρ ⁢ t + 2 ⁢ η ⁢ arctan ⁡ t ) ) ⁢ d t , <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.22.2.cmml" xref="p1.1.m1.1.22"> <eq id="p1.1.m1.1.2.cmml" xref="p1.1.m1.1.2"/> <apply id="p1.1.m1.1.1.1.cmml" xref="p1.1.m1.1.1.9.2"> <apply id="p1.1.m1.1.1.2.1.1.cmml" xref="p1.1.m1.1.1.2.1.5"> <csymbol cd="dlmf" id="p1.1.m1.1.1.2.1.1.1.cmml" xref="p1.1.m1.1.1.2.1.2">irregular-Coulomb-H</csymbol> <minus id="p1.1.m1.1.1.2.1.3.1.cmml" xref="p1.1.m1.1.1.2.1.3.1"/> <ci id="p1.1.m1.1.1.2.1.4.1.cmml" xref="p1.1.m1.1.1.2.1.4.1">ℓ</ci> </apply> <ci id="p1.1.m1.1.1.5.cmml" xref="p1.1.m1.1.1.5">𝜂</ci> <ci id="p1.1.m1.1.1.7.cmml" xref="p1.1.m1.1.1.7">𝜌</ci> </apply> <apply id="p1.1.m1.1.22.2.1.cmml" xref="p1.1.m1.1.22.2.1"> <times id="p1.1.m1.1.22.2.1.1.cmml" xref="p1.1.m1.1.22.2.1.1"/> <apply id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3"> <divide id="p1.1.m1.1.3.1.cmml" xref="p1.1.m1.1.3"/> <apply id="p1.1.m1.1.3.2.cmml" xref="p1.1.m1.1.3.2"> <minus id="p1.1.m1.1.3.2.1.cmml" xref="p1.1.m1.1.3.2.1"/> <apply id="p1.1.m1.1.3.2.7.cmml" xref="p1.1.m1.1.3.2.7"> <times id="p1.1.m1.1.3.2.7.1.cmml" xref="p1.1.m1.1.3.2.7.1"/> <csymbol cd="dlmf" id="p1.1.m1.1.3.2.2.1.cmml" xref="p1.1.m1.1.3.2.2.2">imaginary-unit</csymbol> <apply id="p1.1.m1.1.3.2.7.2.cmml" xref="p1.1.m1.1.3.2.7.2"> <csymbol cd="ambiguous" id="p1.1.m1.1.3.2.7.2.1.cmml" xr…
Symbols used: normalizing constant for Coulomb radial functions
Symbols used: the ratio of the circumference of a circle to its diameter
Symbols used: differential of x
Symbols used: exponential function
Symbols used: base of natural logarithm
Symbols used: Q10729
Symbols used: hyperbolic cosine function
Symbols used: hyperbolic tangent function
Symbols used: imaginary unit
Symbols used: integral
Symbols used: Q10790
Symbols used: irregular Coulomb radial functions
Symbols used: nonnegative integer
Symbols used: nonnegative real variable
Symbols used: real parameter
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile