Asymptotic behavior of the maximum of Jacobi's zeta function (Q2047390)

For given modulus \(k\), \(0\leq{k}<1\), let \(Z(\varphi,k)\) be Jacobi's zeta function defined as \[ Z(\varphi,k):=E(\varphi,k)-\frac{E(k)}{K(k)}\,F(\varphi,k), \] where \[ F(\varphi,k):=\int_{0}^{\varphi}\frac{{\operatorname{d}}\theta}{\sqrt{1-k^2\sin^2\theta}} \quad\text{and}\quad E(\varphi,k):=\int_{0}^{\varphi}\sqrt{1-k^2\sin^2\theta}\,{\operatorname{d}}\theta \] and \(K(k):=F(\pi/2,k)\), \(E(k):=E(\pi/2,k)\). There is \(Z(0,k)=Z(\pi/2,k)=0\). For each \(0<k<1\), the function \(Z(\varphi,k)\) is concave in \(\varphi\in\left]0,\pi/2\right[\) and has therefore a maximum point \(\varphi_{\max}(k)\) with corresponding maximum \(Z_{\max}(k)=Z(\varphi_{\max}(k),k)\). In this paper, the author proves that both, \(\varphi_{\max}(k)\) and \(Z_{\max}(k)\), are strictly increasing in \(k\) and that \(Z_{\max}(k)\to1\) as \(k\to1\). Concerning the speed of convergence, it is proved that \[ \lim_{k\to1}\frac{\log1/\sqrt{1-k^2}}{\log\log1/\sqrt{1-k^2}}\Bigl(Z_{\max}(k)-1\Bigr)=-\frac{1}{2}. \]