Formula:DLMF:25.11:E30

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\HurwitzZeta ⁢ @ ⁢ s ⁢ a = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e a ⁢ z ⁢ z s - 1 1 - e z ⁢ d z \HurwitzZeta @ 𝑠 𝑎 Euler-Gamma 1 𝑠 2 imaginary-unit superscript subscript limit-from 0 𝑎 𝑧 superscript 𝑧 𝑠 1 1 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{\Gamma% \left(1-s\right)}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}\frac{{\mathrm{e}^{az}}z% ^{s-1}}{1-{\mathrm{e}^{z}}}\mathrm{d}z}}}

Constraint(s)

s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}} &

the integration contour is a loop around the negative real axis; it starts at - ∞ {\displaystyle{\displaystyle{\displaystyle-\infty}}} , encircles the origin once in the positive direction without enclosing any of the points

z = ± 2 ⁢ π ⁢ i 𝑧 plus-or-minus 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi\mathrm{i}}}} , ± 4 ⁢ π ⁢ i , … , plus-or-minus 4 imaginary-unit … {\displaystyle{\displaystyle{\displaystyle\pm 4\pi\mathrm{i},\ldots,}}} and returns to - ∞ {\displaystyle{\displaystyle{\displaystyle-\infty}}}


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Assume ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}} , collapse the integration path onto the

real axis, apply
\HurwitzZeta ⁢ @ ⁢ s ⁢ a = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 - e - x ⁢ d x \HurwitzZeta @ 𝑠 𝑎 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑎 𝑥 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{1-{\mathrm{e}% ^{-x}}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-\expe^{-x}} \diff{x} }
and
Γ ⁡ ( z ) ⁢ Γ ⁡ ( 1 - z ) = π / sin ⁡ ( π ⁢ z ) Euler-Gamma 𝑧 Euler-Gamma 1 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z% \right)=\pi/\sin\left(\pi z\right)}}} {\displaystyle \EulerGamma@{z} \EulerGamma@{1-z} = \cpi / \sin@{\cpi z} }

followed by analytic continuation.


Symbols List

& : logical and
ζ 𝜁 {\displaystyle{\displaystyle{\displaystyle\zeta}}}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
π 𝜋 {\displaystyle{\displaystyle{\displaystyle\pi}}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
d n ⁢ x superscript d 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}  : differential : http://dlmf.nist.gov/1.4#iv
ℜ ⁡ z 𝑧 {\displaystyle{\displaystyle{\displaystyle\Re{z}}}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (30), Section 25.11 of DLMF.

URL links

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