DLMF:10.23.E17 (Q3471)

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DLMF:10.23.E17
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    Statements

    test
    Y n ⁡ ( z ) = - n ! ⁢ ( 1 2 ⁢ z ) - n π ⁢ ∑ k = 0 n - 1 ( 1 2 ⁢ z ) k ⁢ J k ⁡ ( z ) k ! ⁢ ( n - k ) + 2 π ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ J n ⁡ ( z ) - 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( n + 2 ⁢ k ) ⁢ J n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) , Bessel-Y-Weber 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 𝜋 superscript subscript 𝑘 0 𝑛 1 superscript 1 2 𝑧 𝑘 Bessel-J 𝑘 𝑧 𝑘 𝑛 𝑘 2 𝜋 1 2 𝑧 digamma 𝑛 1 Bessel-J 𝑛 𝑧 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 𝑛 2 𝑘 Bessel-J 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}% }{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+% \frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_% {n}\left(z\right)-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}% \left(z\right)}{k(n+k)},}}
    0 references
    DLMF id
    DLMF:10.23.E17
    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.m2aldec
    0 references
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aadec
    0 references
     
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