DLMF:10.31.E2 (Q3519)

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DLMF:10.31.E2
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    Statements

    Defining formula
    K 0 ⁡ ( z ) = - ( ln ⁡ ( 1 2 ⁢ z ) + γ ) ⁢ I 0 ⁡ ( z ) + 1 4 ⁢ z 2 ( 1 ! ) 2 + ( 1 + 1 2 ) ⁢ ( 1 4 ⁢ z 2 ) 2 ( 2 ! ) 2 + ( 1 + 1 2 + 1 3 ) ⁢ ( 1 4 ⁢ z 2 ) 3 ( 3 ! ) 2 + ⋯ . modified-Bessel-second-kind 0 𝑧 1 2 𝑧 modified-Bessel-first-kind 0 𝑧 1 4 superscript 𝑧 2 superscript 1 2 1 1 2 superscript 1 4 superscript 𝑧 2 2 superscript 2 2 1 1 2 1 3 superscript 1 4 superscript 𝑧 2 3 superscript 3 2 ⋯ {\displaystyle{\displaystyle K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z% \right)+\gamma\right)I_{0}\left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1% +\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{% 1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}+\cdots.}}
    0 references
    DLMF id
    DLMF:10.31.E2
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    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.m2aadec
    0 references
    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.m2aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1adec
    0 references
     
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