DLMF:10.45.E8 (Q3665)

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DLMF:10.45.E8
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    Statements

    Defining formula
    K ~ 0 ⁡ ( x ) = K 0 ⁡ ( x ) = - ln ⁡ ( 1 2 ⁢ x ) - γ + O ⁡ ( x 2 ⁢ ln ⁡ x ) , modified-Bessel-second-kind-imaginary-order 0 𝑥 modified-Bessel-second-kind 0 𝑥 1 2 𝑥 Big-O superscript 𝑥 2 𝑥 {\displaystyle{\displaystyle\widetilde{K}_{0}\left(x\right)=K_{0}\left(x\right% )=-\ln\left(\tfrac{1}{2}x\right)-\gamma+O\left(x^{2}\ln x\right),}}
    0 references
    DLMF id
    DLMF:10.45.E8
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
    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
    modified Bessel function fo the second kind of imaginary order
    Defining formula
    K ~ ν ⁡ ( x ) modified-Bessel-second-kind-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)}}
    xml-id
    C10.S45.E2.m3aedec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1gdec
    0 references
     
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