DLMF:25.11.E43 (Q7717)

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DLMF:25.11.E43
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    Statements

    Defining formula
    ζ ⁡ ( s , a ) - a 1 - s s - 1 - 1 2 ⁢ a - s ∼ ∑ k = 1 ∞ B 2 ⁢ k ( 2 ⁢ k ) ! ⁢ ( s ) 2 ⁢ k - 1 ⁢ a 1 - s - 2 ⁢ k . asymptotic-expansion Hurwitz-zeta 𝑠 𝑎 superscript 𝑎 1 𝑠 𝑠 1 1 2 superscript 𝑎 𝑠 superscript subscript 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 Pochhammer 𝑠 2 𝑘 1 superscript 𝑎 1 𝑠 2 𝑘 {\displaystyle{\displaystyle\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}% {2}a^{-s}\sim\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{% 1-s-2k}.}}
    0 references
    DLMF id
    DLMF:25.11.E43
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1ajdec
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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.m2akdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aapdec
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
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    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
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