# Questions tagged [fresnel-integrals]

Questions on the Fresnel integrals.

58 questions
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### How to show that $\int_0^1 \sin \pi t ~ \left( \zeta (\frac12, \frac{t}{2})-\zeta (\frac12, \frac{t+1}{2}) \right) dt=1$?

I've been trying to prove Fresnel integrals by real methods and encountered an interesting problem. Let's start with the known result: $$\int_0^\infty \sin y^2 dy = \sqrt{\frac{\pi}{8}}$$ Can we ...
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### Derivative of Fresnel integral function with functional limits

How do I find a derivative (with respect to $x$) of a Fresnel integral function with functional limits: $$f(x)=\int_{\sin^2(x^2)}^{e^{2x}}\sin(z^2)\,dz.$$
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### Can Fresnel integrals be expressed as a function of other functions/integrals?

I need to use Fresnel integrals in C++. Is it possible to compute them with any of the new built-in functions of C++17? or do I have to implement my own solver? Said otherwise: can Fresnel integrals ...
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### Approximating $\int_0^1 \cos(x^2)dx$ with power series

I would like to calculate $\int_0^1 \cos(x^2)dx$ with an error smaller than $10^{-6}$ (this error should be proven.). I have a strategy, but I am not quite sure if this is a valid one. Here is what I ...
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### Estimate of Fresnel-like integral

Does it correct that if $x\in\mathbb{R}_+$ then $$\left | \int_{x^2}^{(x+1)^2}\frac{\sin t}{\sqrt{t}}dt \right |\le \frac{2}{x}?$$
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### Diffraction and Fresnel Integrals

Migrated from Physics SE due to mathematical content I am trying to derive the intensity variation function for a single slit diffraction. Sorry for the poor diagram... So I decided to take the ...
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