Formula:DLMF:25.11:E27

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ζ ( s , a ) = 1 2 a - s + a 1 - s s - 1 + 1 Γ ( s ) 0 ( 1 x - 1 - 1 x + 1 2 ) x s - 1 a x x Hurwitz-zeta s a 1 2 superscript a s superscript a 1 s s 1 1 Euler-Gamma s superscript subscript 0 1 superscript x 1 1 x 1 2 superscript x s 1 superscript a x x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{-s}+% \frac{a^{1-s}}{s-1}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{% 0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}% }{e^{ax}}dx}


Constraint(s)


s > - 1 𝑠 1 {\displaystyle\realpart{s}>-1} & s 1 𝑠 1 {\displaystyle s\neq 1} & a > 0 𝑎 0 {\displaystyle\realpart{a}>0}


Proof


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Argue as in
ζ ( s ) = 1 2 + 1 s - 1 + 1 Γ ( s ) 0 ( 1 x - 1 - 1 x + 1 2 ) x s - 1 x x Riemann-zeta s 1 2 1 s 1 1 Euler-Gamma s superscript subscript 0 1 superscript x 1 1 x 1 2 superscript x s 1 superscript x x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{2}+\frac{1}{s% -1}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}dx} .



Symbols List


ζ 𝜁 {\displaystyle\zeta}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
Γ Γ {\displaystyle\Gamma}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
{\displaystyle\int}  : integral : http://dlmf.nist.gov/1.4#iv
{\displaystyle e}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
a b 𝑏 𝑎 {\displaystyle\diff[a]{b}}  : differential : http://dlmf.nist.gov/1.4#iv
ζ 𝜁 {\displaystyle\zeta}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
a 𝑎 {\displaystyle\Re{\,\hskip 0.0pta}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography


Equation (6), Section 25.5 of DLMF.

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