DLMF:10.31.E1 (Q3518)

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DLMF:10.31.E1
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    Statements

    test
    K n ⁡ ( z ) = 1 2 ⁢ ( 1 2 ⁢ z ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ ( - 1 4 ⁢ z 2 ) k + ( - 1 ) n + 1 ⁢ ln ⁡ ( 1 2 ⁢ z ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ 1 2 ⁢ ( 1 2 ⁢ z ) n ⁢ ∑ k = 0 ∞ ( ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) ) ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ ( n + k ) ! , modified-Bessel-second-kind 𝑛 𝑧 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 superscript 1 𝑛 1 1 2 𝑧 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 1 2 superscript 1 2 𝑧 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{% -n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln% \left(\tfrac{1}{2}z\right)I_{n}\left(z\right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2% }z)^{n}\sum_{k=0}^{\infty}\left(\psi\left(k+1\right)+\psi\left(n+k+1\right)% \right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!},}}
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    DLMF id
    DLMF:10.31.E1
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    Symbols used
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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