Formula:DLMF:25.11:E44

<< Formula:DLMF:25.11:E43

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E45 >>

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

Constraint(s)

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

Note(s)

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

Proof

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Symbols List

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B_{n}}}  : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\mathrm{ph}}}}$ : 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 (44), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E43

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E45 >>