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 ...
Daerst's user avatar
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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 ...
Nebula's user avatar
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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 ...
Yihu's user avatar
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1 answer
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Dealing with definite integrals of $\cos(x^p)$ and $\sin(x^p)$ for $p > 1$ [duplicate]

I was reading an interesting book on evaluating some integrals(Inside Interesting Integrals by Paul J. Nahin) and came across the Fresnel integrals: $\int_{0}^{\infty} \cos(x^2) \text{ dx}= \int_{0}^{\...
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Integrals of the Fresnel integrals

I find in an engineering book the Integrals of the Fresnel integrals and i don't know how to prove the expressions given bellow. $$C_I(t)=\int_{0}^{1}C(u)du=tC(t)-\frac{1}{\pi}sin(\frac{\pi t^2}{2})$$ ...
Alexandru Harai's user avatar
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Evaluating $\int_{\mathbb{R}}\sin(x^2)\ dx$ via complex numbers

I found a rather interesting integral in my favourite maths book of all time "Les Mathématiciens de A à Z" under the name of Fresnel's integral. It is the one in the title of this post and I ...
Jacques Tarr's user avatar
2 votes
1 answer
247 views

Using $e^{iz^2}$ vs. $e^{-z^2}$ to calculate Fresnel Integrals

I have been doing some research and found out that the most used function to contour integrate when it comes to fresnel integrals is $e^{iz^2}$ on the boundary of $|z| \leq R$ and $0 \leq \arg(z) \leq ...
Sebastian Giraldo Gomez's user avatar
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How to calculate this integral to find the range of u?

I have a integral $$ f(u)= \int_{-n/2}^{n/2} \int_{-n/2}^{n/2} e^{j u \pi ((x^2+y^2)- (ax+by)^2) } d_x d_y,$$ where $n>0,u>0$, and $a , b \in [0,1]$. I want to find the range of $u$ to make $ |f(...
Kris Prokins's user avatar
2 votes
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143 views

Is there a (relatively simple) function bounding the two Fresnel integrals?

I just discovered the two Fresnel integrals and it really seems like they are bounded by two functions (maybe exponentials?). Here is an image explaining what I mean: As you can see, I just drew the ...
Josef Zoller's user avatar
3 votes
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204 views

Fresnel-Kirchhoff’s Diffraction Formula Vs. Angular Spectrum Method

I tried to work through this lecture and first of all, I'm curious, what is the reason for this: We would like to use Green’s theorem with $v(x) = G(x − x0)$ As far as I understand, Green's theorem ...
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1 vote
1 answer
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Generalized Fresnel Integral [duplicate]

This a follow-up to my previous question. I suspected that the following identity, $\int_0^{\infty} e^{-ax^n} \, \text{d}x = \frac1{n} \cdot a^{-\frac1{n}} \Gamma{\left(\frac1{n}\right)} = a^{-\frac1{...
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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 ...
Mailbox's user avatar
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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 ...
matteogost's user avatar
1 vote
2 answers
103 views

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)\...
matteogost's user avatar
4 votes
1 answer
119 views

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 ...
UtilityMaximiser's user avatar
6 votes
1 answer
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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 $a,b,c>0$...
matteogost's user avatar
1 vote
1 answer
170 views

Closed form of $\int e^{i\csc^2(x)}dx=\int \cos\left(\csc^2(x)\right)dx+i\int \sin\left(\csc^2(x)\right)dx$

This will be an experimental integral with the final result using the Kampé de Fériet function with the goal integrals being: $$\int \cos\left(\csc^2(x)\right)dx,\int\sin\left(\csc^2(x)\right)dx\...
Тyma Gaidash's user avatar
3 votes
1 answer
252 views

asymptotic approximation of Fresnel integrals with complex argument

It turns out that SciPy's Fresnel values are wrong for complex arguments and large enough absolute value. I'm trying to fix that. The implementation is based on Zhang/Jin, Computation of special ...
Nico Schlömer's user avatar
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185 views

Does this Auxiliary Fresnel Sum=$\frac1{2\sqrt2\pi}\int \limits_0^\infty \frac{\vartheta_3\left(e^{-\frac{\pi x}2}\right)\sqrt x}{x^2+1}dx +\frac14 $?

$$\large{\text{Motivation:}}$$ Here is a related Fresnel Integral sum for a seventh in a series of a sum of just a single function: On $$\mathrm{\sum\limits_{n=0}^\infty \left(C(n)-\frac{\sqrt\pi}{2\...
Тyma Gaidash's user avatar
2 votes
2 answers
384 views

integral from fresnel integral

I came across an exercise that says: Find $\displaystyle \int S(x) dx$, where $S(x)$ is $\displaystyle \int_0^x \sin(\pi t^2/2)dt$ I understand what I should get $\displaystyle\int\left( \int_0^x \sin(...
Luis Alexandher's user avatar
3 votes
1 answer
910 views

Proof of $\int_{0}^{\infty}\sin(x^2)dx=\int_{0}^{\infty}\cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$

If I want to prove that \begin{equation*} \int_{0}^{\infty}\sin(x^2)dx=\int_{0}^{\infty}\cos(x^2)dx=\frac{\sqrt{2\pi}}{4} \end{equation*} First method: It is possible to approach it by the method in ...
p-adic-manimanito's user avatar
2 votes
1 answer
194 views

Contour integration mistake: Fresnel Integral computes to zero

Summary : I have "convinced myself", via contour integration and residue theorem, that the Fresnel integral : $$ s(\infty)=\overset{\infty }{\underset{0}{\int }}dx \sin \left(\frac{\pi x^2}...
elscan's user avatar
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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 $\alpha=2,4/3,3$ ($\alpha=2$ ...
J.Yang's user avatar
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0 votes
1 answer
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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) \, ...
Semoi's user avatar
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2 votes
0 answers
248 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}^...
JackEight's user avatar
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4 votes
1 answer
132 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 ...
Aadhaar Murty's user avatar
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50 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(...
Jayden Rice's user avatar
2 votes
1 answer
85 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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179 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) =\...
Rickard Martensson's user avatar
2 votes
0 answers
46 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 ...
tommsch's user avatar
  • 180
2 votes
1 answer
667 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 ...
Zaragesh's user avatar
4 votes
2 answers
187 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,...
user3760593's user avatar
0 votes
1 answer
59 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 ...
Integrand's user avatar
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1 vote
1 answer
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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 ...
Reynan Henry's user avatar
0 votes
1 answer
85 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 ...
David Lingard's user avatar
2 votes
2 answers
2k 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 ...
Ali Arsalan's user avatar
0 votes
3 answers
301 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 ...
user avatar
2 votes
1 answer
98 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 $t \to \infty$?
Marduk's user avatar
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4 votes
2 answers
255 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. ...
Frankie S. Palmer's user avatar
1 vote
0 answers
135 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) ...
Stijn D'hondt's user avatar
9 votes
1 answer
400 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 ...
Yuriy S's user avatar
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1 vote
1 answer
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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: $$f(x)=\int_{\sin^2(x^2)}^{e^{2x}}\sin(z^2)\,dz.$$
weekens's user avatar
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1 vote
1 answer
443 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)]...
Villa's user avatar
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4 votes
2 answers
916 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 ...
Laplace's Demon's user avatar
0 votes
1 answer
559 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(...
Nils H's user avatar
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2 votes
2 answers
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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 ...
Anton Sherwood's user avatar
5 votes
1 answer
123 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+\...
Atmos's user avatar
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4 votes
3 answers
187 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^{...
Yulia23576's user avatar