DLMF:33.16.E11 (Q9639)

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DLMF:33.16.E11
No description defined

    Statements

    Defining formula
    h ⁡ ( ϵ , ℓ ; r ) = ( - 1 ) ℓ ⁢ ν ℓ + 1 ⁢ A ⁢ ( ϵ , ℓ ) ⁢ ( sin ⁡ ( π ⁢ ν ) ⁢ ζ ℓ ⁢ ( ν , r ) Γ ⁡ ( ℓ + 1 + ν ) + cos ⁡ ( π ⁢ ν ) ⁢ Γ ⁡ ( ν - ℓ ) ⁢ ξ ℓ ⁢ ( ν , r ) π ) . Coulomb-h italic-ϵ ℓ 𝑟 superscript 1 ℓ superscript 𝜈 ℓ 1 𝐴 italic-ϵ ℓ 𝜋 𝜈 subscript 𝜁 ℓ 𝜈 𝑟 Euler-Gamma ℓ 1 𝜈 𝜋 𝜈 Euler-Gamma 𝜈 ℓ subscript 𝜉 ℓ 𝜈 𝑟 𝜋 {\displaystyle{\displaystyle h\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{% \ell+1}A(\epsilon,\ell)\left(\frac{\sin\left(\pi\nu\right)\zeta_{\ell}(\nu,r)}% {\Gamma\left(\ell+1+\nu\right)}+\frac{\cos\left(\pi\nu\right)\Gamma\left(\nu-% \ell\right)\xi_{\ell}(\nu,r)}{\pi}\right).}}
    0 references
    DLMF id
    DLMF:33.16.E11
    0 references
    Symbols used
    irregular Coulomb function
    Defining formula
    h ⁡ ( ϵ , ℓ ; r ) Coulomb-h italic-ϵ ℓ 𝑟 {\displaystyle{\displaystyle h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right% )}}
    xml-id
    C33.S14.E7.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1hdec
    0 references
    Q12692
    Defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C33.S1.XMD3.m1fdec
    0 references
    Q12693
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C33.S1.XMD6.m1gdec
    0 references
    function (locally)
    Defining formula
    A ⁢ ( ϵ , ℓ ) 𝐴 italic-ϵ ℓ {\displaystyle{\displaystyle A(\epsilon,\ell)}}
    xml-id
    C33.S14.XMD4.m1ddec
    0 references
    parameter (locally)
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C33.S16.XMD2.m1cdec
    0 references
    Q12694
    Defining formula
    ζ ℓ ⁢ ( ν , r ) subscript 𝜁 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\zeta_{\ell}(\nu,r)}}
    xml-id
    C33.S16.XMD3.m1bdec
    0 references
    Q12695
    Defining formula
    ξ ℓ ⁢ ( ν , r ) subscript 𝜉 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\xi_{\ell}(\nu,r)}}
    xml-id
    C33.S16.XMD4.m1bdec
    0 references
     
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