DLMF:5.11.E1 (Q2117)

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DLMF:5.11.E1
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    Statements

    Defining formula
    Ln ⁡ Γ ⁡ ( z ) ∼ ( z - 1 2 ) ⁢ ln ⁡ z - z + 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) + ∑ k = 1 ∞ B 2 ⁢ k 2 ⁢ k ⁢ ( 2 ⁢ k - 1 ) ⁢ z 2 ⁢ k - 1 asymptotic-expansion multivalued-natural-logarithm Euler-Gamma 𝑧 𝑧 1 2 𝑧 𝑧 1 2 2 𝜋 superscript subscript 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 2 𝑘 1 superscript 𝑧 2 𝑘 1 {\displaystyle{\displaystyle\operatorname{Ln}\Gamma\left(z\right)\sim\left(z-% \tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty% }\frac{B_{2k}}{2k(2k-1)z^{2k-1}}}}
    0 references
    DLMF id
    DLMF:5.11.E1
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    general logarithm function
    Defining formula
    Ln ⁡ z multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\NVar{z}}}
    xml-id
    C4.S2.E1.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
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1dec
    0 references
     
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