# Questions tagged [fresnel-integrals]

Questions on the Fresnel integrals.

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### Can we evaluate the Fresnel integral of a quadratic in general using real methods?

After tackling about the Fresnel integral in the post, I want go further with its quadratic as $$\int_{-\infty}^{\infty} \sin \left(a x^2+b x+c\right) d x$$ where $a,b$ and $c$ are real. Starting ...
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### Finding the angular acceleration to fit a curve of constant curvature through a 2D point

Given is a point $(x, y)$ in Cartesian 2D space and a parametric curve of which we know the following: The curve starts a $(0, 0)$ and extends in positive $x$ direction. The curve has an angular ...
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### How to evaluate the limit: $\lim_{t \to \infty} \int_0^1 \frac{e^{it^2(1+y^2)}}{1+y^2} \, dy$

I was evaluating the Fresnel integral: $$\int_{-\infty}^{\infty} e^{ix^2}dx$$ After some calculations, I successfully evaluated it correctly. However, there is one problem that I don't know how to ...
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### Fresnel Integral principle

I want to use clothoid obtain the next point, i found the formulars in the figure and source code, but it is difficult for me to understand it , who can help me to tackle the problem? enter image ...
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### Fresnel Integral Proof (without rigorous complex analysis)

I’d like to present my dodgy proof of the Fresnel Integrals, which I wrote before I knew anything about complex analysis. It takes some… liberties; yet, it still managed to produce the right value for ...
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### Computing $\int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau$ -- part 2

Problem I have a problem with the following integral \begin{equation*} I_2(t)\triangleq \int_0^t \tau\,\cos(c+b\tau+a\tau^2)\text{ d}\tau \end{equation*} where $t,a,b,c$ are given parameters. For ...
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### Quadratic-trigonometric integral -- part 4

References I still have problems with this funny integral \begin{equation*}\int_0^t \cos(c+b\tau+a\tau^2)\text{ d}\tau\end{equation*} This post is the continuation of these other 3 posts, where you ...
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### Quadratic-trigonometric integral -- part 3

Problem This is a continuation of these other two posts: Quadratic-trigonometric integral, Quadratic-trigonometric integral -- part 2. I'm studying the following integral \begin{equation*} I_1(t)\...
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### How to show positivity of Fresnel C integral?

The Fresnel $C$-integral is defined as follows. $$C(x) = \int_0^x \cos(t^2) \, dt$$ From the plot found on Wikipedia it seems to be non-negative for all $x \geq 0$ however it is not obvious to me why ...
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### Quadratic-trigonometric integral -- part 2

Problem I need to compute the following integral \begin{equation*}\int_{t_\text{s}}^{t_\text{e}} \cos(a+b\tau+c\tau^2)\text{ d}\tau\end{equation*} where $t_{\text{s}}<t_{\text{e}}$ and $a,b,c>0$...
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### An interesting identity for Dirichlet-like integrals [duplicate]

I was looking for a proof of the following identity $$\int_{0}^{\infty} \sin(x^{a})dx = \Gamma\left(1 + \frac{1}{a}\right) \cdot \sin\left(\frac {\pi}{2a}\right)$$ I have tried using the Legendre ...
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### Monotonicity of sum of two Fresnel integrals

This question is cross posted on Physics Stack Exchange https://physics.stackexchange.com/questions/568448/monotonicity-of-sum-of-two-fresnel-integrals I am studying single-knife edge diffraction of ...
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### Calculating Clothoid between two tangents

I am trying to connect two points with a Clothoid (Euler-Spiral) https://en.wikipedia.org/wiki/Euler_spiral . It is mandatory to connect the points with the correct start and endHeading of the ...
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### Sum of squared Fresnel sine integral

I'm trying to find the following sum: $$\sum_{n=0}^{\infty} \frac{S\left(\sqrt{2n}\right)^2}{n^3}$$ where $S(n)$ is the fresnel sine integral, however, I think I made a mistake somewhere. To start,...
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### Transform Fresnel integrals into each other

Let $S,C$ be given by $$S = \int _{0}^{\infty} \sin(x^2)\,dx,\,\,C = \int _{0}^{\infty} \cos(x^2)\,dx$$I know you can show they're equal to each other using complex contour integration, and I've ...
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### Extended Fresnel integral [duplicate]

I want to see the method for computing the following integrals: $$\int_0^\infty\ln(x)\sin(x^2)dx$$ and $$\int_0^\infty\ln(x)\cos(x^2)dx$$ I believe I have seen these in this forum before, but I cannot ...
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### Proof of $\sum_{n=1}^\infty \frac{\cos \ n}{n}=-\frac{\ln\left(2-2 \cos(1)\right)}{2}$

The series $\sum_{n=1}^\infty \frac{\cos \ n}{n}$ convergences. Mathematica gives as limit $$\sum_{n=1}^\infty \frac{\cos \ n}{n}=-\frac{\ln\left(2-2 \cos(1)\right)}{2}$$ What are the proofs of this ...
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### How to calculate length of Clothoid segment?

I want to calculate the length of a clothoid segment from the following available information. initial radius of clothoid segment final radius of clothoid segment angle (i am not really sure which ...
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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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### Show that $\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$ for $n\ge 5.$
Question is to show that $$\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$$ when $n\geq5$, $x \in \mathbb R$ and $i$ is an imaginary unit. My effort: |\int_{-n}^{n}e^{iy^2}dy|\leq \int_{-n}^{n}|e^{...