i need to show that the function known from the Fresnel integral (wikipedia) converges to the Dirac delta-distribution. This function is defined as $f_{\epsilon}(x) = \sqrt{\frac{a}{i \pi}\frac{1}{\epsilon}} e^{ia\frac{x^2}{\epsilon}} $. More concretely this means that i need to show that $\lim_{\epsilon \rightarrow 0} \; \sqrt{\frac{a}{i \pi}\frac{1}{\epsilon}} e^{ia\frac{x^2}{\epsilon}} = \delta(x)$. I allready know that one criterium for […]

$$\int_{0}^{\infty }\sin(x^{2})dx$$ I’m confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you. I have not done complex analysis (only real analysis as I am a high school student) so how can I evaluate it using elementary functions (without complex analysis)?

This question has been irritating me for awhile so I thought I’d ask here. Are complex substitutions in integration okay? Can the following substitution used to evaluate the Fresnel integrals: $$\int_{0}^{\infty} \sin x^2\, dx=\operatorname {Im}\left( \int_0^\infty\cos x^2\, dx + i\int_0^\infty\sin x^2\, dx\right)=\operatorname {Im}\left(\int_0^\infty \exp(ix^2)\, dx\right)$$ Letting $ix^2=-z^2 \implies x=\pm\sqrt{iz^2}=\pm \sqrt{i}z \implies z=\pm \sqrt{-i} x \implies […]

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ $$C(x)=\int_0^x\cos\frac{\pi z^2}{2}\mathrm dz$$ are the Fresnel integrals. Numerical integration gives an approximate result $0.31311841522422385…$ that is close to $\frac{16\log2-8}{\pi^2}$, so it might be the answer.

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