I want to know how to get the following integration.
How can we get it?
\documentclass{article}\begin{document}
\[\int_0^{\pi}\int_0^\pi\frac{\sin\theta_1\sin\theta_2}{\sqrt{(a+b\cos(\theta_1+\theta_2))(a+b\cos(\theta_1-\theta_2))}(2a+b(\cos(\theta_1+\theta_2)+\cos(\theta_1-\theta_2))}d\theta_1d\theta_2,\]
where $a>b>0$.
\end{document}
Same answer as to another question: write the numerator as a sum of functions of θ_1±θ_2, notice that the denominator factorizes into expressions only of u=θ_1+θ_2 and v=θ_1-θ_2 and compute the two one-dimesnsional integrals numerically (or, eventually, using contour integration).
It will be useful to substitutions cos(theta1) = (1/2) [ exp (i * theta1 ) + exp ( -i * theta1) ] and sin (theta_1 ) = (1/2 i) [ exp (i * theta1 ) + exp ( -i * theta1) ] .
Attempt to simplify the integrand using above substitutions. Then the numerator and denominator might be expressed as ratio of polynomials in x and y , where x = exp ( i * theta1) and y= exp (i * theta2) .
Thank you for answering. I used u=θ_1+θ_2 and v=θ_1-θ_2 , but I had no idea since then. I cannot know the method of trapezoidal rule. I am going to try to handle exponential functions.
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