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Questions tagged [fresnel-integrals]

Questions on the Fresnel integrals.

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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
34 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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35 views

How can I calculate Fresnel integral via double improper integral?

So, there are many ways to calculate the integral $\int_{-\infty}^{+\infty} \sin(x^2)\,\mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My ...
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2answers
75 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
41 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) ...
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1answer
124 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
99 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
123 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)]...
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0answers
88 views

Fresnel Integral and his formulas

Below, is how Fresnel approximate the eponymously "Fresnel Integral". In his own Words: Let $i$ and $i+t$ be the narrow limits between which it is proposed to integrate $\ cos(qv^2) \, dv$. ...
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2answers
147 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
131 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(...
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1answer
37 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 ...
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1answer
94 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+\...
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3answers
158 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^{...
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1answer
93 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 ...
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1answer
179 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^{...
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1answer
68 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 ...
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4answers
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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)...
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2answers
319 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 ...
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1answer
228 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 ...
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1answer
99 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 - \...
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1answer
59 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 $$s \in (\frac{-\pi}{2},\frac{\pi}{2})$$ ...
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0answers
29 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 ...
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3answers
122 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 ...
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2answers
237 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}{\...
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0answers
234 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} ...
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5answers
2k 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 ...
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1answer
49 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 ...
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2answers
186 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} \...
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2answers
61 views

Estimate of Fresnel-like integral

Does it correct that if $x\in\mathbb{R}_+$ then $$\left | \int_{x^2}^{(x+1)^2}\frac{\sin t}{\sqrt{t}}dt \right |\le \frac{2}{x}?$$
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1answer
669 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} ...
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1answer
77 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 ...
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1answer
50 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?
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0answers
132 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_{...
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1answer
490 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 ...
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1answer
117 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^...
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3answers
343 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...
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1answer
69 views

What do you call a frequency that varies by a function?

I have a concept that I need to learn more about, but I don't know what it's called so I'm not sure what search terms to use to look for it. I apologize in advance that while I'm comfortable with ...
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1answer
126 views

Uniform convergence of the complex Fresnel integral

Consider the integral $I(\lambda) = \sqrt {\frac {\lambda \mathbb{i}}{\pi}}^n \int_U \mathbb{e}^{-\mathbb{i} \lambda \|x\ - x_0|^2} f(x) \mathbb{d}x, \lambda>0$ and $U$ some open neighbourhood of $...
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2answers
403 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll \infty$....
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2answers
224 views

Updated: Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ using Fresnel Integrals

Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ I've tried substituting $ x^2 = t $ but could not proceed at all thereafter in integration. Any help would be appreciated. I should ...
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3answers
253 views

Show that $\int^\infty_0$ $\int^\infty_0$ sin($x^2$+$y^2$) dxdy value is $\frac{\pi}{4}$

I am trying to show that the value of $\int^\infty_0$$\int^\infty_0$ sin($x^2$+$y^2$) dxdy is $\frac{\pi}{4}$ using Fresnel integrals. I'm having trouble splitting apart the integrand in order to ...
3
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1answer
103 views

the integral of $\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)$ can be written in Fresnel integrals?

$$\int_{0}^{(y^2-x^2)/ 4t}s^{-1/2} \sin(s) \exp\left({-4sxy \over y^2-x^2}\right)\mathrm{d}s=2\int_{0}^{\sqrt{(y^2-x^2)/4t}}\sin(z^2) \exp\left({-4z^2xy \over y^2-x^2}\right)\mathrm{d}z$$ I applied ...
4
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2answers
101 views

Fresnel Integral multiplied with cosine term.

$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$ Can anybody tell me, how to solve this integral ? I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent.
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0answers
306 views

The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] \...
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1answer
1k views

Infinite Series of the asymptotic expansion of Fresnel Integrals

I need to find the infinite series for the asymptotic expansions of the fresnel integrals as $x\rightarrow \infty$ and $x\rightarrow 0$. Now I have computed the asyptotic expansions to be as follows ...
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1answer
399 views

IB Math question: Fresnel integral?

P moves along x-axis such that its velocity, v, at time t is given by $v=\cos(t^2)$. Find the time at which the total distance travelled by P is 1. (all in meters, meters/sec). So the total distance ...
0
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2answers
118 views

Evaluating $\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$

$$\int_0^{\frac{\pi}2}\frac{\sin 2x}{\sqrt{x}}\,dx$$ How to solve this trigonometric integral? I can't find any solutions. Some books suggest to use Fresnel integral. I would be grateful if you could ...
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0answers
244 views

Compute Erf(z) using Fresnel integrals

I have to compute $\operatorname{erf}(z)$ using the Fresnel integrals. I have the relation: $$C(z)+iS(z)=\frac{1+i}{2}\operatorname{erf}\left[ \frac{\sqrt{\pi}}{2}(1-i)z \right].$$ But $\...
15
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1answer
417 views

Need help with $\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx$

Let $$I=\int_0^\infty\left(\pi\,x+\frac{S(x)\cos\frac{\pi x^2}2-C(x)\sin\frac{\pi x^2}2}{S(x)^2+C(x)^2}\right)dx,\tag1$$ where $$S(x)=-\frac12+\int_0^x\sin\frac{\pi t^2}2dt,\tag2$$ $$C(s)=-\frac12+\...