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How to evaluate the ratio of integrals

$$ I(u)=\frac{\int_0^{\theta_0}\sin^2\theta\sin^2(u\sin\theta)d\theta}{\int_0^{\theta_0}\sin^2(u\sin\theta)d\theta} $$

for the case when $\theta_0 \to 0$ and the case when $\theta_0 \to \frac{\pi}{2}$, where $u$ is a real parameter.

It seems this problem is closely related to the integral representation of Bessel function when $\theta_0 \to \frac{\pi}{2}$, but I don't know how to proceed. Hope you can help, thanks.

ecook
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    What have you tried? – Ninad Munshi Oct 23 '23 at 00:58
  • To connect with the Bessel integral $\cos 2 x = 1 - 2 \sin^2 x$ might help – userrandrand Oct 23 '23 at 01:49
  • OBSERVATION : Numerator Integral & Denominator Integral are "Same" except for weighting. The weighting is tending to $0^2$ in one Case , hence that limit seems to be $0$. The weighting is tending to $1^2$ in other Case , hence that limit seems to be $1$. The weighting is not "Constant" , of course , hence that "Heuristic" is not so simple to use here ! – Prem Oct 23 '23 at 04:54
  • @userrandrand I have tried to write both terms of the integrand of the numerator with this triogometric identity. This will split the integral into four part, but some of which are difficult to solve. – ecook Oct 25 '23 at 03:23
  • @Prem I have tried to approximate the integrand using Taylor expansion, only keeping the first 2 terms, but it is still hard to evaluate. – ecook Oct 25 '23 at 03:24
  • @ecook I did not check the details but with the identity mentioned before, it seems to me that the numerator is related to the second derivative of the denominator with respect to $u$. If that is true, then you only need to find the expression of the denominator in terms of Bessel functions and then maybe use the recurrence relations in https://en.wikipedia.org/wiki/Bessel_function#Recurrence_relations. As for the $\theta_0 \rightarrow0$ case, my mind went to l'hôpital 's rule https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule – userrandrand Oct 25 '23 at 04:15

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