DLMF:33.16.E11 (Q9639)

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DLMF:33.16.E11
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    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
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
     
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