Binets second loggamma formula states that the error term for stirlings approximation of the loggamma function is $2\int_0^{\infty} \frac{arctan(u/z) }{e^{2\pi u}-1}~\mathrm du$. I remember being able to derive this from the Euler-Maclaurin summation formula, but was wondering if there’s an easier way to show this? I tried using the Abel Plana formula but I don’t think the loggamma formula satisfies the bounds necessary for use. What’s an alternative proof?
2026-03-27 21:44:19.1774647859
Binets second loggamma formula
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The Abel-Plana formula is applicable to $f(x)=1/(x+z)^2$ with $\Re z>0$, giving $$\big(\ln\Gamma(z)\big)''=\sum_{n=0}^{\infty}\frac{1}{(n+z)^2}=\frac{1}{2z^2}+\frac{1}{z}+\int_0^\infty\frac{4zx\ dx}{(z^2+x^2)^2(e^{2\pi x}-1)}.$$ This can be integrated twice, using the main asymptotics to determine the integration constants.