I have a numerical evidence of $\int_0^{1/2} \frac{1}{\sqrt{2\pi}\sigma_0x}\exp\left(-\frac{(\mu_0x-y)^2}{2\sigma_0^2x^2}\right)dx \approx 1+\cos(2\pi y),$ where $\sigma_0^2=\frac{2}{\pi^2}-\mu_0^2$ and $\mu_0=\frac{4}{\pi^2}.$ The latter should be the mean and the variance of the function $h(x)= \pi(1-x)\sin(\pi(1-x)).$ Could anybody suggest a proof turning the numerical approximate equality above into an equality?
Proof of a gaussian integral turning into a cosine
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calculus
normal-distribution
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0Actually there is no complete oscillation since the period is twice the integration domain. The resulting integral approximated by $1+cos(2\pi y)$ is more like an S-function thus it could have something to do with $erf$'s or similar. – 2012-12-03