Formula:DLMF:25.11:E45

<< Formula:DLMF:25.11:E44

formula in Hurwitz Zeta Function

Formula:DLMF:25.12:E1 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{-2}{a}-\frac{1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\ln a\sim\sum_{k=1}^{\infty}\frac{2\!\BernoulliB{2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-(2k-1)}}}}$

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

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

Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\tilde{}}}}$ : 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

Bibliography

Equation (45), Section 25.11 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:DLMF:25.11:E44

formula in Hurwitz Zeta Function

Formula:DLMF:25.12:E1 >>