Formula:DLMF:25.11:E43

<< Formula:DLMF:25.11:E42

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E44 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{\infty}\frac{\BernoulliB{2k}}{(2k)!}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}a^{1-s-2k}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a\to\infty}}}$ in the sector ${\displaystyle{\displaystyle{\displaystyle|\ph@@{a}|\leq\pi-\delta(<\pi)}}}$, with ${\displaystyle{\displaystyle{\displaystyle s(\neq 1)}}}$ and ${\displaystyle{\displaystyle{\displaystyle\delta}}}$ fixed

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\tilde{}}}}$ : asymptotic equality : http://dlmf.nist.gov/2.1#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ph}}}}$ : phase : http://dlmf.nist.gov/1.9#E7
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (43), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E42

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E44 >>