# Formula:DLMF:25.11:E44

$\displaystyle {\displaystyle \HurwitzZeta'@{-1}{a} - \frac{1}{12} + \frac{1}{4} a^2 - \left( \frac{1}{12} - \frac{1}{2} a + \frac{1}{2} a^2 \right) \ln@@{a} \sim -\sum_{k=1}^\infty \frac{\BernoulliB{2k+2}}{(2k+2)(2k+1)2k} a^{-2k} }$

## Constraint(s)

$\displaystyle {\displaystyle a \to \infty}$ in the sector $\displaystyle {\displaystyle |\ph@@{a}| \leq \tfrac{1}{2} \cpi-\delta (<\tfrac{1}{2}\cpi)}$

## Note(s)

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

## 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

$\displaystyle {\displaystyle \zeta}$  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
$\displaystyle {\displaystyle \mathrm{ln}}$  : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
$\displaystyle {\displaystyle \tilde}$  : asymptotic equality : http://dlmf.nist.gov/2.1#E1
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle B_{n}}$  : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
$\displaystyle {\displaystyle \mathrm{ph}}$  : phase : http://dlmf.nist.gov/1.9#E7
$\displaystyle {\displaystyle \pi}$  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4