DLMF:5.17.E5 (Q2185)

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DLMF:5.17.E5
No description defined

    Statements

    Defining formula
    Ln ⁡ G ⁡ ( z + 1 ) ∼ 1 4 ⁢ z 2 + z ⁢ Ln ⁡ Γ ⁡ ( z + 1 ) - ( 1 2 ⁢ z ⁢ ( z + 1 ) + 1 12 ) ⁢ Ln ⁡ z - ln ⁡ A + ∑ k = 1 ∞ B 2 ⁢ k + 2 2 ⁢ k ⁢ ( 2 ⁢ k + 1 ) ⁢ ( 2 ⁢ k + 2 ) ⁢ z 2 ⁢ k . asymptotic-expansion multivalued-natural-logarithm Barnes-Gamma 𝑧 1 1 4 superscript 𝑧 2 𝑧 multivalued-natural-logarithm Euler-Gamma 𝑧 1 1 2 𝑧 𝑧 1 1 12 multivalued-natural-logarithm 𝑧 𝐴 superscript subscript 𝑘 1 Bernoulli-number-B 2 𝑘 2 2 𝑘 2 𝑘 1 2 𝑘 2 superscript 𝑧 2 𝑘 {\displaystyle{\displaystyle\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}% z^{2}+z\operatorname{Ln}\Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac% {1}{12}\right)\operatorname{Ln}z-\ln A+\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2% k+1)(2k+2)z^{2k}}.}}
    0 references
    DLMF id
    DLMF:5.17.E5
    0 references
    Symbols used
    Q11295
    Defining formula
    G ⁡ ( z ) Barnes-Gamma 𝑧 {\displaystyle{\displaystyle G\left(\NVar{z}\right)}}
    xml-id
    C5.S17.E1.m1addec
    0 references
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    general logarithm function
    Defining formula
    Ln ⁡ z multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\NVar{z}}}
    xml-id
    C4.S2.E1.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
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1adec
    0 references
    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1cdec
    0 references
    Q11296
    Defining formula
    A 𝐴 {\displaystyle{\displaystyle A}}
    xml-id
    C5.S17.XMD1.m1dec
    0 references
     
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