DLMF:13.21.E14
type: Item
Digital Library of Mathematical Functions ID: 13.21.E14
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle W_{\kappa,\mu}\left(x\right)=\frac{e^{-\mu\pi% \mathrm{i}}}{\pi}\Gamma\left(\kappa+\mu+\tfrac{1}{2}\right)\*\Gamma\left(% \kappa-\mu+\tfrac{1}{2}\right)\*c(\kappa,\mu)\Psi(\kappa,\mu,x)\*\left(\sin% \left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(\sqrt{\zeta\kappa}\right)-\cos\left% (\kappa\pi-\mu\pi\right)Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)+\mathrm{env}% \mskip-2.0mu Y_{2\mu}\left(\sqrt{\zeta\kappa}\right)O\left(\kappa^{-1}\right)% \right),}}" display="inline"> W κ , μ ⁡ ( x ) = e - μ ⁢ π ⁢ i π ⁢ Γ ⁡ ( κ + μ + 1 2 ) ⁢ Γ ⁡ ( κ - μ + 1 2 ) ⁢ c ⁢ ( κ , μ ) ⁢ Ψ ⁢ ( κ , μ , x ) ⁢ ( sin ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ J 2 ⁢ μ ⁡ ( ζ ⁢ κ ) - cos ⁡ ( κ ⁢ π - μ ⁢ π ) ⁢ Y 2 ⁢ μ ⁡ ( ζ ⁢ κ ) + <mpadded width="-1.1pt" id="p1.1.m1.1.30.2.1.4" xref="p1.1.m1.1.30.2.1.1.cmml"> env </mpadded> ⁢ Y 2 ⁢ μ ⁡ ( ζ ⁢ κ ) ⁢ O ⁡ ( κ - 1 ) ) , <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.34.2.cmml" xref="p1.1.m1.1.34"> <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.m1.1.1.7.2"> <apply id="p1.1.m1.1.1.2.1.1.cmml" xref="p1.1.m1.1.1.2.1.4"> <csymbol cd="dlmf" id="p1.1.m1.1.1.2.1.1.1.cmml" xref="p1.1.m1.1.1.2.1.2">Whittaker-confluent-hypergeometric-W</csymbol> <ci id="p1.1.m1.1.1.2.1.3.1.1.cmml" xref="p1.1.m1.1.1.2.1.3.1.1">𝜅</ci> <ci id="p1.1.m1.1.1.2.1.3.1.3.cmml" xref="p1.1.m1.1.1.2.1.3.1.3">𝜇</ci> </apply> <ci id="p1.1.m1.1.1.5.cmml" xref="p1.1.m1.1.1.5">𝑥</ci> </apply> <apply id="p1.1.m1.1.34.2.1.cmml" xref="p1.1.m1.1.34.2.1"> <times id="p1.1.m1.1.22.cmml" xref="p1.1.m1.1.22"/> <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"> <csymbol cd="ambiguous" id="p1.1.m1.1.3.2.3.cmml" xref="p1.1.m1.1.3.2">superscript</csymbol> <ci id="p1.1.m1.1.3.2.1.cmml" xref="p1.1.m1.1.3.2.1">𝑒</ci> <apply id="p1.1.m1.1.3.2.2.1.cm…
Symbols used: Bessel function of the first kind
Symbols used: Bessel function of the second kind
Symbols used: order not exceeding
Symbols used: gamma function
Symbols used: Whittaker confluent hypergeometric function
Symbols used: the ratio of the circumference of a circle to its diameter
Symbols used: cosine function
Symbols used: Q11065
Symbols used: base of natural logarithm
Symbols used: imaginary unit
Symbols used: sine function
Symbols used: real variable
Symbols used: function (locally)
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile