integral arsing in statistical mechanics

by sangara   Last Updated August 01, 2020 14:20 PM

The integral given below arises in two-dimensional lattice models $$\int_0^{\frac{\pi}{2}} \ln{\left(1+\sqrt{1-a^2\sin^2{x}}\right)} \; dx$$ with $a \leq 1$ The analytical solution seems to be unavailable. By trial and error, this integral was shown to be nearly equal to $$\frac{\pi}{4} \ln{\left(1+\sqrt{1-G^2}\right)}+\frac{\pi\ln{2}}{4}$$ where G denotes the Catalan constant. Is there a method of obtaining an exact analytical solution or justifying the approximate answer given above?

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