A double integral for $\frac{\pi}{2} \ln 2$.

Question

Show that \begin{eqnarray*} I=\int_0^1 \int_0^1 \frac{dx \,dy}{\sqrt{1-x^2y^2}} = \frac{\pi}{2} \ln 2. \end{eqnarray*}

My try ... from this question here we have \begin{eqnarray*} \int_0^1 \frac{ \sin^{-1}(x)}{x} \,dx = \frac{\pi}{2} \ln 2 . \end{eqnarray*} And from this question here we have \begin{eqnarray*} \int_0^1 \ln \left( \frac{1+ax}{1-ax} \right) \frac{dx}{x\sqrt{1-x^2}}=\pi\sin^{-1} a,\qquad |a|\leq 1. \end{eqnarray*} It is easy to show \begin{eqnarray*} \int_0^1 \frac{dy}{1+yz}=\frac{1}{z} \ln(1+z). \end{eqnarray*} So we have (with a little tad of algebra) \begin{eqnarray*} \frac{\pi}{2} \ln 2 &=& \int_0^1 \frac{ \sin^{-1}(x)}{x}\, dx \\ &=& \frac{1}{\pi} \int_0^1 \int_0^1 \ln \left( \frac{1+xt}{1-xt} \right) \frac{dt}{xt\sqrt{1-t^2}} x\, dx \\ &=& \frac{2}{\pi} \int_0^1 \int_0^1 \int_0^1 \frac{dx \,dy\, dt}{(1-x^2y^2t^2)\sqrt{1-t^2} } \\ \end{eqnarray*} This suggests we should consider the integral (sub $t=\sin(\theta)$) \begin{eqnarray*} \frac{2}{\pi} \int_0^1 \frac{ dt}{(1-x^2y^2t^2)\sqrt{1-t^2} } \\ = \int_0^{\pi/2} \frac{d \theta }{1- x^2 y^2 \sin^2(\theta)}. \end{eqnarray*} Now it is well known (Geometrically expand, integrate term by term & sum the familiar plum) that \begin{eqnarray*} \frac{2}{\pi} \int_0^{\pi/2} \frac{d \theta }{1- \alpha \sin^2(\theta)}=\frac{2}{\pi} \frac{1}{\sqrt{1-\alpha}} \end{eqnarray*} and using this we have \begin{eqnarray*} \frac{\pi}{2} \ln 2 =\int_0^1 \int_0^1 \frac{dx\, dy}{\sqrt{1-x^2y^2}} . \end{eqnarray*}

The above double integral reminds of \begin{eqnarray*} \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2}=\int_0^1 \int_0^1 \frac{dx\, dy}{1-x^2y^2} = \frac{\pi^2}{8} \end{eqnarray*} which can be evaluated using the substitution $x= \frac{\sin u}{\cos v}$, $y= \frac{\sin v}{\cos u}$.

My solution above used some pretty heavy machinery to establish the result. So my question is: is there an easier method ?

Answer 1

$$\int_0^1 \int_0^1 \frac{dxdy}{\sqrt{1-x^2y^2}}\overset{xy=t}=\int_0^1 \frac{1}{y}\int_0^y \frac{1}{\sqrt{1-t^2}}dtdy=\int_0^1 \frac{\arcsin y}{y}dy $$ $$\overset{IBP}=-\int_0^1 \frac{ \ln y}{\sqrt{1-y^2}}dy\overset{y=\sin x}=-\int_0^\frac{\pi}{2}\ln( \sin x)dx=\frac{\pi}{2}\ln 2$$ See here for the above integral.

Answer 2

Through series expansions: $$ \iint_{(0,1)^2}\frac{dx\,dy}{\sqrt{1-x^2 y^2}}=\sum_{n\geq 0}\frac{\binom{2n}{n}}{4^n}\iint_{(0,1)^2}x^{2n}y^{2n}\,dx\,dy=\sum_{n\geq 0}\frac{\binom{2n}{n}}{4^n(2n+1)^2}=\int_{0}^{1}\frac{\arcsin(x)}{x}\,dx $$ and this is $$ \int_{0}^{\pi/2}x\cot(x)\,dx\stackrel{\text{IBP}}{=}\int_{0}^{\pi/2}\log(\sin x)\,dx $$ which is well known to be $\frac{\pi}{2}\log(2)$. It is possible to exploit symmetry, derivatives of the Beta function, Fourier (or Fourier-Legendre) series and probably much more. For instance, the identity $$ \prod_{k=1}^{n-1}\sin\left(\frac{\pi k}{n}\right)=\frac{2n}{2^n} $$ and just Riemann sums.

Answer 3

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ $\ds{\bbox[15px,#ffd]{I \equiv \int_{0}^{1}\int_{0}^{1}{\dd x\,\dd y \over \root{1 - x^{2}y^{2}}} = {\pi \over 2}\ln\pars{2}}:\ {\large ?}}$. \begin{align} I & \equiv \int_{0}^{1}\int_{0}^{1}{\dd x\,\dd y \over \root{1 - x^{2}y^{2}}} = \sum_{n = 0}^{\infty}{-1/2 \choose n}\int_{0}^{1}\int_{0}^{1} \pars{-x^{2}y^{2}}^{n}\,\dd x\,\dd y \\ & = \sum_{n = 0}^{\infty}{-1/2 \choose n}{\pars{-1}^{n} \over \pars{2n + 1}^{2}} = \sum_{n = 0}^{\infty}{-1/2 \choose n}\pars{-1}^{n}\ \overbrace{\bracks{-\int_{0}^{1}\ln\pars{x}x^{2n}\,\dd x}} ^{\ds{1 \over \pars{2n + 1}^{2}}} \\[5mm] & = -\int_{0}^{1}\ln\pars{x} \sum_{n = 0}^{\infty}{-1/2 \choose n}\pars{-x^{2}}^{n}\,\dd x = -\int_{0}^{1}{\ln\pars{x} \over \root{1 - x^{2}}}\,\dd x \\[5mm] & \stackrel{x\ =\ \sin\pars{\theta}}{=}\,\,\, \underbrace{-\int_{0}^{\pi/2}\ln\pars{\sin\pars{\theta}}\,\dd\theta} _{\ds{\ =\ I}}\ =\ -\,{1 \over 2} \int_{0}^{\pi/2}\ln\pars{\sin\pars{\theta}\cos\pars{\theta}}\,\dd\theta \label{1}\tag{1} \\[5mm] & = -\,{1 \over 4}\int_{0}^{\pi}\ln\pars{\sin\pars{\theta} \over 2}\,\dd\theta \\[5mm] & = {1 \over 4}\,\pi\ln\pars{2} - {1 \over 4}\int_{0}^{\pi/2}\ln\pars{\sin\pars{\theta}}\,\dd\theta - {1 \over 4}\int_{-\pi/2}^{0}\ln\pars{-\sin\pars{\theta}}\,\dd\theta \\[5mm] & = {1 \over 4}\,\pi\ln\pars{2}\ \underbrace{- {1 \over 2}\int_{0}^{\pi/2}\ln\pars{\sin\pars{\theta}}\,\dd\theta} _{\ds{\ =\ {1 \over 2}\, I}} \label{2}\tag{2} \end{align} See lines \ref{1} and \ref{2}: $\ds{I = {1 \over 4}\,\pi\ln\pars{2} + {1 \over 2}\,I \implies I = \bbox[15px,#ffd,border:1px solid navy]{{\pi \over 2}\,\ln\pars{2}}\ \approx 1.0888}$.

Answer 4

Why not go head-on? It seems to work here. That is, write the integral as $$\int_0^1\int_0^1\frac {\mathrm d x \mathrm d y}{\sqrt{1-x^2y^2}}=\int_0^1\int_0^1\frac {\mathrm d x \mathrm d y}{y \sqrt{\frac{1}{y^2}-x^2}}.$$ Setting $x=\frac1y\sin\phi$ as usual helps us evaluate the first integral, or you can simply note that it has the form of an $\arcsin.$ If you make this substitution, the first integral becomes $$\int_0^{\arcsin y}\frac1y\mathrm d \phi=\frac{\arcsin y}{y}.$$ Thus, the integral reduces to $$\int_0^1\frac{\arcsin y}{y}\mathrm d y,$$ which may be done by parts. Forgetting limits for now, this begins as $$\arcsin y\log y-\int\frac{\log y}{\sqrt{1-y^2}}\mathrm dy,$$ etc.