Formula:DLMF:25.11:E35: Difference between revisions

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Latest revision as of 08:33, 22 December 2019


n = 0 ( - 1 ) n ( n + a ) s = 1 Γ ( s ) 0 x s - 1 e - a x 1 + e - x d x = 2 - s ( \HurwitzZeta @ s 1 2 a - \HurwitzZeta @ s 1 2 ( 1 + a ) ) superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑎 𝑥 1 𝑥 𝑥 superscript 2 𝑠 \HurwitzZeta @ 𝑠 1 2 𝑎 \HurwitzZeta @ 𝑠 1 2 1 𝑎 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(% n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm% {e}^{-ax}}}{1+{\mathrm{e}^{-x}}}\mathrm{d}x=2^{-s}\left(\HurwitzZeta@{s}{% \tfrac{1}{2}a}-\HurwitzZeta@{s}{\tfrac{1}{2}(1+a)}\right)}}}

Constraint(s)

a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}} , s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
or a = 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}=0}}} , a 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Im{a}\neq 0}}} , 0 < s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


Proof

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Use

\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
\HurwitzZeta @ s 1 2 a = \HurwitzZeta @ s 1 2 a + 1 2 + 2 s n = 0 ( - 1 ) n ( n + a ) s \HurwitzZeta @ 𝑠 1 2 𝑎 \HurwitzZeta @ 𝑠 1 2 𝑎 1 2 superscript 2 𝑠 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\tfrac{1}{2}a}=% \HurwitzZeta@{s}{\tfrac{1}{2}a+\tfrac{1}{2}}+2^{s}\sum_{n=0}^{\infty}\frac{(-1% )^{n}}{(n+a)^{s}}}}} {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2} a} = \HurwitzZeta@{s}{\tfrac{1}{2} a + \tfrac{1}{2}} + 2^s \sum_{n=0}^\infty \frac{\opminus^n}{(n+a)^s} } .


Symbols List

& : logical and
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( - 1 ) 1 {\displaystyle{\displaystyle{\displaystyle(-1)}}}  : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
{\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
ζ 𝜁 {\displaystyle{\displaystyle{\displaystyle\zeta}}}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
z 𝑧 {\displaystyle{\displaystyle{\displaystyle\Re{z}}}}  : real part : http://dlmf.nist.gov/1.9#E2
z 𝑧 {\displaystyle{\displaystyle{\displaystyle\Im{z}}}}  : imaginary part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (35), Section 25.11 of DLMF.

URL links

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