DLMF:25.11.E44 (Q7718)

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DLMF:25.11.E44
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    Statements

    Defining formula
    ζ ′ ⁡ ( - 1 , a ) - 1 12 + 1 4 ⁢ a 2 - ( 1 12 - 1 2 ⁢ a + 1 2 ⁢ a 2 ) ⁢ ln ⁡ a ∼ - ∑ k = 1 ∞ B 2 ⁢ k + 2 ( 2 ⁢ k + 2 ) ⁢ ( 2 ⁢ k + 1 ) ⁢ 2 ⁢ k ⁢ a - 2 ⁢ k , asymptotic-expansion diffop Hurwitz-zeta 1 1 𝑎 1 12 1 4 superscript 𝑎 2 1 12 1 2 𝑎 1 2 superscript 𝑎 2 𝑎 superscript subscript 𝑘 1 Bernoulli-number-B 2 𝑘 2 2 𝑘 2 2 𝑘 1 2 𝑘 superscript 𝑎 2 𝑘 {\displaystyle{\displaystyle\zeta'\left(-1,a\right)-\frac{1}{12}+\frac{1}{4}a^% {2}-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}a^{2}\right)\ln a\sim-\sum_{k=1% }^{\infty}\frac{B_{2k+2}}{(2k+2)(2k+1)2k}a^{-2k},}}
    0 references
    DLMF id
    DLMF:25.11.E44
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1akdec
    0 references
    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.m2aaqdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2ahdec
    0 references
    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1kdec
    0 references
     
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