Questions tagged [fresnel-integrals]

Questions on the Fresnel integrals.

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38 views

How to prove the positivity of Fresnel-like integrals in the positive real axis

I need to prove the following inequality: $$\int_0^x \sin(t^{\alpha})dt>0$$ for $x>0$ and $\alpha>1$. I used Mathematica to plot the integrals for some values of $\alpha=2,4/3,3$ ($\alpha=2$ ...
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1answer
39 views

Fresnel integral: stationary phase approx.

The following integral describes the propagation of light (in certain cases) $$ U_i(x,y,z) = \frac{e^{ikz}}{i\lambda z} \int_{-\infty}^\infty d\xi \int_{-\infty}^\infty d\eta \; U_0(\xi, \eta,0) \, ...
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0answers
73 views

Fresnel integrals

I read that in order to calculate the Fresnel integrals $\int_{0}^{+\infty} \cos(t^{2}) dt$ and $\int_{0}^{+\infty} \sin(t^{2}) dt$, we could set $A(x) = (\int_{0}^{x} \cos(t^{2}) dt) \times (\int_{0}^...
3
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1answer
80 views

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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23 views

Fresnel Integrals using the fundamental theorem of calculus and chain rule?

For the function $$S(x) = \int_{0}^{x} sin(t^2) dt$$ If we apply the change of variables $u=t^2$, $$S(x) = \int_{0}^{f(x)} sin(u)h(u) du$$ what are the functions $f(x)$ and $h(u)$? And, if we set $$Z(...
2
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1answer
45 views

Using Parseval's identity to find $\sum^{\infty}_{n=1} \frac{S(2\sqrt{n}) \cdot C(2\sqrt{n})}{n^3}$

I would like to find the sum: $$ \sum^{\infty}_{n=1} \frac{S(2\sqrt{n}) \cdot C(2\sqrt{n})}{n^3}$$ where $ S(n)$ and $C(n)$ denote the Sine and Cosine Fresnel Integrals respectively. But somewhere in ...
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75 views

Why do Fresnel-integrals contain $\sqrt{\pi}$?

The non-elementary functions $$ F(x) = \int \sin(x^2)\mathrm dx $$ $$ G(x) = \int \cos(x^2)\mathrm dx $$ will yield $$ F(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)}$$ $$ G(x) =\...
2
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0answers
39 views

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 ...
1
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1answer
123 views

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 ...
4
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2answers
113 views

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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1answer
44 views

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 ...
1
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1answer
50 views

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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1answer
70 views

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 ...
2
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2answers
492 views

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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3answers
115 views

How to find power series for Fresnel integral?

How to get (step-by-step) from this $$ C(x)=\int_{0}^{x} \cos \left(\frac{\pi t^{2}}{2}\right) d t $$ to this? $$ C(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n}\left(\frac{\pi}{2}\right)^{2 n}}{(2 n) !(4 ...
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1answer
55 views

Asymptotic behavior of Fresnel-like integral of an exponential [closed]

Given the integral $$ I(t) = \int_0^t \mathrm{d}x \exp(-[\alpha \cos x + \beta \sin x]),\quad \alpha,\beta\in \mathbb{R}, $$ how can one obtain the asymptotic behavior for $t \to \infty$?
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2answers
153 views

Laplace transforms with Fresnel(?) integrals

I've come into contact with this two part question, and the latter I'm not too sure how to go about; at least to me upon researching, I can't find anything remotely similar to what I've been asked. ...
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0answers
75 views

Problem proving the Fresnel integral

I have shown that $$\int^\infty_0 e^{ix^2}dx = \dfrac{\sqrt\pi}{2},$$ using contour integration on $f(z) = e^{iz^2}$. But since $$\int^\infty_0 e^{ix^2}dx = \int^\infty_0 (\cos x^2+ i \sin x^2) ...
9
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1answer
237 views

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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1answer
507 views

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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1answer
234 views

Generalized Fresnel Integral using Laplace

So I wanted to solve the following integral: $$\int_0^\infty \sin{(x^2) dx}$$ I did it by using the Laplace transform of the function: $$I(t) = \int_0^\infty \sin{(tx^2) dx}$$ $$\mathcal{L} [I(t)]...
4
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2answers
242 views

Show that $\int_0^n\sin x^2dx$ converges

The question Okay. So I'm trying to solve the problem below for a previous exam in real analysis. Thus, only such methods may be used. The integral $\int_0^\infty\sin x^2dx$ is called a Fresnel ...
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1answer
308 views

Numerical solution of generalized Fresnel integral

We need to find an approximate solution for the generalized Fresnel integral: $\int_0^S \cos(as+\frac{bs^2}{2}+\frac{cs^3}{3}+\frac{ds^4}{4})ds$ Our approach is to use the Simpsons rule: $\int_a^bf(...
2
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2answers
55 views

How does Raph Levien's Spiro choose angles for the ends of a path?

I've read Raph Levien's paper on splines (http://www.levien.com/phd/phd.html), and think I mostly understand chapter 8, the nuts and bolts of fitting a piecewise polynomial spiral to a sequence of ...
5
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1answer
114 views

Proving that $\int_{0}^{+\infty}e^{ix^n}\text{d}x=\Gamma\left(1+\frac{1}{n}\right)e^{i\pi/2n}$

I've shown that for all $p \in \mathbb{R}^{*+}$ $$\int_{0}^{+\infty}e^{-x^p}\text{d}x=\Gamma\left(1+\frac{1}{p}\right)$$ And I want to show that $$ \int_{0}^{+\infty}e^{ix^n}\text{d}x=\Gamma\left(1+\...
4
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3answers
165 views

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^{...
2
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1answer
157 views

Fresnel Approximation of $\delta$ Function

According to the German Wikipedia, the $\delta$ distribution can be approximated by sequences of integrable functions $\delta_k$ satisfying $\delta_k(x) \geq 0$ $\int_{\mathbb R} \delta_k(x) \mathrm ...
1
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1answer
272 views

Driving Fresnel Integral $\int_{-\infty}^\infty \sin(x^2)\rm dx$ and $\int_{-\infty}^\infty \cos(x^2)\rm dx$ through polar coordinates

I am trying to derive the Fresnel Integrals $$\int_{-\infty}^\infty \sin(x^2)\rm dx ~~~~and ~~~~~\int_{-\infty}^\infty \cos(x^2)\rm dx$$ through the Gaussian Integral via $I=\int_{-\infty}^\infty e^{...
1
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1answer
71 views

Computing: $\int_0^1\frac{x^2\sin(x^2)+\sin(\frac{1}{x^2})}{x^2}dx$

I would like to compute the exact value of the integral below. $$\int_0^1\frac{x^2\sin(x^2)+\sin(\frac{1}{x^2})}{x^2}dx$$ I have proved the convergence already. but failed to the residues theorem in ...
37
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4answers
2k views

Proof only by transformation that : $ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx $

This was a question in our exam and I did not know which change of variables or trick to apply How to show by inspection ( change of variables or whatever trick ) that $$ \int_0^\infty \cos(x^2) dx ...
3
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2answers
460 views

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 ...
5
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1answer
747 views

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 ...
1
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1answer
101 views

Can you prove the asymptotic formula of the following function?

In the following equation, $$P(x; a)= \frac{\gamma}{2\lambda L \eta} [\frac{1}{π^2N_F(a)\eta(1 - \frac{x}{a\eta})^2} + \frac{1}{π^2N_F(a)\eta(1 + \frac{x}{a\eta})^2} +\frac{2}{π^2N_F(a)\eta(1 - \...
1
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1answer
68 views

Prove Uniform integrability of $ \int_{0}^{\infty} e^{-e^{is}\cdot x^2} dx $

I want to prove Uniform Integrability of the following improper parametric integral (in the parameter $s$: $$ \int_{0}^{\infty} e^{-e^{is}\cdot x^2} dx $$ for $$s \in (\frac{-\pi}{2},\frac{\pi}{2})$$ ...
1
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0answers
31 views

Justify $\frac{d}{ds} \int_{0}^{\infty} e^{-e^{is}\cdot x^2} dx$ = $ \int_{0}^{\infty} \frac{d}{ds} e^{-e^{is}\cdot x^2} dx$ for fresnel integral

I want to compute the Fresnel integrals $$\int_{-\infty}^{\infty} e^{\pm i\cdot x^2}, \hspace{5mm} dx \int_{-\infty}^{\infty} \cos(x^2)dx, \hspace{5mm} \int_{-\infty}^{\infty} \sin(x^2)dx$$ using real ...
0
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3answers
242 views

Determine whether $\int_0^\infty\frac{\sin x}{x^{3/2}}dx$ is divergent or convergent

Determine whether $\displaystyle\int_1^\infty\dfrac{\sin x}{x^{3/2}}\,\mathrm dx$ is divergent or convergent This was one of the questions on my Technical maths exam, I have tried finding a solution ...
1
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2answers
357 views

Fresnel function converging to delta distribution

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}{\...
5
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0answers
254 views

Tricky integral

I am trying to do the following integral for a few days now, but getting nowhere. \begin{align} \int_0^x \mathrm dy\,y^2 \cos(y^2) C^2 \!\!\left(\!\dfrac{\sqrt{2}\,y}{\sqrt\pi}\!\right)\! \end{align} ...
8
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5answers
4k views

Proof of Fresnel's Integral

So, my teacher wants us to prove the Fresnel's integral: $$\int_0^\infty\cos(x^2)dx=\sqrt{\frac{\pi}{8}}$$ The problem is that we cannot use complex analysis to prove that and we should do that ...
0
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1answer
67 views

What's about the convergence of $\sum_{k=1}^\infty\frac{C(2\sqrt{k})}{k^{5/2}},$ where $C(x)$ is the Fresnel C integral?

One can calculate a closed-form of (this or different examples) $$\sum_{k=1}^\infty\frac{C(2\sqrt{k})}{k^{5/2}},$$ where $C(x)$ is the Fresnel C integral from the Fourier expansion of the fractional ...
1
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2answers
419 views

Expressing $\int dx\, \cos(ax^2 + 2bx + c)$ in terms of Fresnel integrals

I have difficulties with this integral, which is is related to Fresnel integrals ($a>0$): $$ \int dx\, \cos(ax^2 + 2bx + c) = \\ =\sqrt{\frac{\pi}{2a}} \left[ \cos(\frac{ac-b^2}{a}) \mathscr{C} \...
3
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2answers
168 views

$\int_0^\infty \frac{\sin(x^2+ax)}{x}\,dx$ converges

I want to show that the following converges for $a\ge 0$ $$\int_0^\infty \frac{\sin(x^2+ax)}{x}\,dx$$ We can show that $\lim_{x\to 0} \frac{\sin(x^2+ax)}{x} = a$, and with that fact it's ...
1
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2answers
68 views

Estimate of Fresnel-like integral

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

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
1
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1answer
91 views

Integral similar to fresnel integrals

$$\int_{0}^{+\infty} \frac{e^{-r^2}}{r^2-i\gamma^2} dr = ?$$ I tried the normal semicircular contour integrals, but there is always a problem with the exponential when I close the contour. This post ...
1
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1answer
54 views

Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
2
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0answers
178 views

Multidimensional Gaussian-Fresnel integral related to a matrix inverse

How would you evaluate the following multidimensional Gaussian-Fresnel integral? $$ I_{k,N;\lambda,\epsilon}(\mathbf{H})=\int_{\mathbb{R}^N} dx_1\cdots dx_N\ x_k^2\exp\left[-\frac{\mathrm{i}}{2}\sum_{...
3
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1answer
634 views

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 ...
1
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1answer
229 views

Integration of sine with quadratic polynomial

I need to integrate: $$\int_{-\infty}^{+\infty}\sin\left(\alpha x^2+2\beta x+\gamma\right)dx$$ So, my steps: Change $\alpha=a^2$, $\beta=ab$, $\gamma=b^2+c$. $$\int_{-\infty}^{+\infty}\sin\left(a^2x^...
3
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3answers
421 views

Differentiation under the integral sign — where is my mistake?

I'm trying to find $$\int_0^\infty \sin \left( x^2 \right)\,dx$$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to $t$ on $A(t)$ — ...