DLMF:10.44.E5 (Q3653)

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DLMF:10.44.E5
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    Statements

    Defining formula
    K 0 ⁡ ( z ) = - ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ I 0 ⁡ ( z ) + 2 ⁢ ∑ k = 1 ∞ I 2 ⁢ k ⁡ ( z ) k , modified-Bessel-second-kind 0 𝑧 1 2 𝑧 modified-Bessel-first-kind 0 𝑧 2 superscript subscript 𝑘 1 modified-Bessel-first-kind 2 𝑘 𝑧 𝑘 {\displaystyle{\displaystyle K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z% \right)+\gamma\right)I_{0}\left(z\right)+2\sum_{k=1}^{\infty}\frac{I_{2k}\left% (z\right)}{k},}}
    0 references
    DLMF id
    DLMF:10.44.E5
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    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2acdec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1ddec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1cdec
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