DLMF:10.15.E1 (Q3147)

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DLMF:10.15.E1
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    Statements

    test
    ∂ ⁡ J ± ν ⁡ ( z ) ∂ ⁡ ν = ± J ± ν ⁡ ( z ) ⁢ ln ⁡ ( 1 2 ⁢ z ) ∓ ( 1 2 ⁢ z ) ± ν ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ ψ ⁡ ( k + 1 ± ν ) Γ ⁡ ( k + 1 ± ν ) ⁢ ( 1 4 ⁢ z 2 ) k k ! , partial-derivative Bessel-J plus-or-minus 𝜈 𝑧 𝜈 minus-or-plus plus-or-minus Bessel-J plus-or-minus 𝜈 𝑧 1 2 𝑧 superscript 1 2 𝑧 plus-or-minus 𝜈 superscript subscript 𝑘 0 superscript 1 𝑘 digamma plus-or-minus 𝑘 1 𝜈 Euler-Gamma plus-or-minus 𝑘 1 𝜈 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial J_{\pm\nu}\left(z\right)}{\partial% \nu}=\pm J_{\pm\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}% z)^{\pm\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma% \left(k+1\pm\nu\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!},}}
    0 references
    DLMF id
    DLMF:10.15.E1
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
     
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