Transform Definition and 1000 Threads

I have a probability distribution as follows: \begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation} where...

Homework Statement Part (a): State inverse Fourier transform. Show Fourier transform is: Part (b): Show Fourier transform is: Part (c): By transforming LHS and RHS, show the solution is: Part(d): Using inverse Fourier transform, find an expression for T(x,t) Homework Equations The Attempt...

Mellin transform of sine Define the Mellin transform as $$\mathcal{MT}\{f(t)\}=\int^\infty_0 t^{z-1} f(t) dt$$ where $$z\in \mathbb{C}$$If the transform exists , it is analytic in some vertical strip $$a<\mathrm{Re}(z)<b $$ in the complex plane Find the vertical strip of $$\mathcal{MT}\{...

Prove the following Suppose that $f$ is piecewise continuous on $$[0,\infty) $$ and of exponential order $c$ then $$\int^\infty_0 e^{-st} f(t)\, dt $$ is analytic in the right half-plane for $$\mathrm{Re}(s)>c$$

Hello! :D I have to find the solution of the equation y'(x)+5y(x)=e^{-x} , 0<x<\infty , y(0)=2 , using the Laplace transform. That's what I have done so far: $$L\{y'(x)+5y(x)\}=L\{e^{-x}\}$$ $$L\{y'(x)\}+5L\{y(x)\}=L\{e^{-x}\}$$ $$sY(s)-y(0)+5Y(s)=\frac{1}{s+1}$$...

I posted this in the homework section, but I haven't received any help, so hopefully putting it in this section won't be an issue. I'm trying to compute the integral $$ \int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt $$ using the Laplace transform. The first thing that catches my eye is the 1...

Homework Statement $$ \int_0^\infty \frac{\sin xt}{x} \, dt $$ Homework Equations The Attempt at a Solution $$ = \int_0^\infty L(\sin xt) \, dp $$ $$ = \int_0^\infty \frac{x}{p^2 + x^2} \, dp $$ $$ = x \int_0^\infty \frac{dx}{p^2 + x^2} \, dp $$ p = x tan theta: $$ = x \int_0^{\pi/2}...

Homework Statement Find $$ L^{-1} \left[ \frac{1}{ (p^2 + a^2)^2} \right] $$ Homework Equations $$ L [ x \cos ax ] = \frac{p^2 - a^2} { (p^2 + a^2) ^2 } $$ The Attempt at a Solution I have no idea. Any thoughts?

Homework Statement Why I am getting wrong answer related to this Laplace Transforms problem? According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get (e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)Homework Equations ∫e^{-st}cos at dt The...

http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf Can someone walk me through this derivation of the Airy integral by Fourier transform? I have tried it but failed

I want to build some software that will transform a formal language from one set of symbols to another. What would be the best software to do that with. Here is one example of what I want to do: Say we have (x'Hy) BC (x'Hz) I need an algorithm to transform that into: ((w'Hp) IDx'-z) A...

Guys, Let me ask you the silliest question of the year. I am looking at the Maxwell equations in their standard form. No 4-dim potential A, no Faraday tensor F, no mentioning of special relativity - just the standard form from a college-level textbook. I know that the eqns are NOT...

I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation So I used the equation for the discrete Fourier transform: ##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}## and I put the Fourier transform into a N-1 by N-1 matrix form...

From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...

Homework Statement Use Laplace transforms to solve the initial value problems. ##y''+4y=0;## ##y(0)=5;## ##y'(0)=0## Homework Equations The Attempt at a Solution $$y''+4y=0$$ $$L(y'')+L(4y)=L(0)\implies s^{2}Y(s)-sy(0)-y'(0)+4(sY(s)-y(0))=0\implies s^{2}Y(s)+4sY(s)-5s-20=0$$...

Homework Statement Find f(t) for the function F(s)=(10s^2+85s+95)/(s^2+6s+5) and apply the initial and final value theorems to each transform pairHomework Equations Initial value theorem: f(0)=lim s->∞ s(F(s)) Final value theorem: f(∞) = lim s->0 s(F(s))The Attempt at a Solution After dividing...

The problem asks to find the standard matrix for the composition of these two linear operations on R2. - A reflection about the line y=x, followed by a rotation counterclockwise of 60o. This is how I proceeded. y=x $\begin{bmatrix}0&1\\1&0 \end{bmatrix}$ counter clockwise 60degs...

x_1(n) = (!/4)^n u(n-1) and x_2(n) = [1- (1/2)^n] u(n) X_1(z) = (1/4)z^-1 / (1-(!/4)z^-1 and X_2(z) = 1/(1-z^-1) + 1/(1-(1/2)z^-1) Y(z) = X_1(z) X_2(z) = (-4/3) /(1-(1/4)z^-1 + (1/3) / (1-z^-1) + 1/(1-(1/2)z^-1

Homework Statement y''+6y=f(t), y(0)=0, y'(0)=-2 f(t)= t for 0≤t<1 and 0 for t≥1 Homework Equations The Attempt at a Solution L{y''}+6L{y}=L{t}-L{tμ(t-1)} where μ(t-1) is Unit Step Y(s)=L{y} sY(s)-y(0)=L{y'} and y(0)=0 s2Y(s)-sy(0)-y'(0)+6Y(s) where y(0)=0 and...

Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?

Homework Statement What is the Fourier transform of a single short pulse and of a sequence of pulses? The Attempt at a Solution In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...

Hello, We were introduced to Laplace transforms in my ODE class a few days ago, so naturally I went online to try to figure out what this transform actually is, rather than being satisfied with being able to compute simple Laplace transforms. The Wikipedia page for the transform says it...

I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...

Homework Statement Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). 1) f(t)=t Homework Equations The Attempt at a Solution Seems pretty easy... Question is, I don't understand the directions exactly.. Am I...

I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...

Hello, Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it. According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...

Homework Statement What is the laplace transform of H(-t-17) Homework Equations Shifting theorem: L(H(t-a)) = (e^-as)/s The Attempt at a Solution This is the only part of the problem that I can not get (this part is from a larger differential equation I'm trying to solve). I'm can't seem...

Fourier transform of RF signal with a "prism"? We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process. How about RF signals? There is no fundamental difference...

Hi guys, I'm not sure how they got from first step to the second. Did they use integration by parts? I tried but I didn't arrive at the same result..

\mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r} in spherical polar coordinates \mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r) Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...

The windowed Fourier transform on R Definition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula) http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html

Homework Statement Determine the Laplace Transform of ∫(from 0 to t) (t-τ)cos(2(t-τ))e-4τ dτ using Laplace Transform tables. Homework Equations I know the basic convolution theorem is (f*g)(t) = ∫f(τ)g(t-τ)dτ The Attempt at a Solution I'm not sure if this is double convolution...

I have 2 sets of known 3x1 vectors A = [ x y 1 ] and B =[ x' y' 1 ] which represent points on two coordinates calculated by some MATLAB algorithm. I was wondering what I could do to find the 2x3 transform matrix that turns the x y set into the x' y' set. [x'] [ ] [x] [y']=[?] [y] [1]...

Homework Statement find the range of values for s: f(t)=e^(-t/2)u(t)Homework Equations The Attempt at a Solution what I did initially was to take the integral from 0 to infinity of e^(-st)e^(-t/2)u(t) dt this gave me 2/(2s+1) which when I sub in -.5 to find a divide by 0 or discontinuity then...

Ok, so I was playing around with some Z transforms. I'm sorry about the long derivation, but I'm a bit unsure of the mathematical rigor, and want to make sure every step is clear. I started with the recurrence relation defining the factorial: $$n!: u_{n+1}=(n+1)u_n=u_n+nu_n $$ $$ u_0 = 1 $$...

I'm working on some research with a professor, and we're looking at data collected by an x-band radar array looking at ocean waves as they approach the coast (the radar is on land, and we can see about 3 miles out). What we're trying to do is perform an fft on the signal using Matlab, and...

I'm given a Gaussian function to apply a Fourier transform to. $$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$ Not the most appetizing integral... $$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$...

Please refer to the attachment. For part a) so far I have: $e^x = 1 + \frac{x}{1!} + ...+ \frac{x^n}{n!}$ So $S^\frac{-1}{2}e^\frac{-1}{S} = S^\frac{-1}{2}(1 -\frac{1}{S} + \frac{1}{2!S^2} - \frac{1}{3!S^3} + \frac{1}{4!S^4} + ... - ...)$ I don't think my $S^\frac{-1}{2}$ on the outside is...

I made a few excercises with Fourier series, Fourier integrals and Fourier transforms. But i am getting stuck at a few questions, most of the time a Fourier transform needs to be calculated in part a, and than part b ask to solve an intergal with the help of your aswer by part a. i...

I understand the Fourier transform conceptually, but I am unable to reproduce it mathematically; I am very familiar with calculus and integration, but I am taking a QM course and I need to know how to apply it. No websites or videos are able to give me a good explanation as to how I can use it...

How do you find the laplace transform of this without expanding it? $ L(t^2+1)^2 $

When the rod is infinite or semi-infinite, I was taught to use Fourier transform. But I don't know when should the full Fourier transform or sine/cosine transform be used. how's the B.C. related to the choice of the transform ?

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

If you have resistors in series ---\/\/\/\/---\/\/\/\/\/--- then the equivalent resistance of this system will merely be the sum of the individual resistors in the series. But the proof of this result is based on the fundamental assumption that the currents going through the two resistors (as...

Homework Statement $$L\{ { e }^{ -t }*{ e }^{ t }cost\}$$ Homework Equations The Attempt at a Solution $$L\{ { e }^{ -t }*{ e }^{ t }cost\} \\ =L\{ \int _{ 0 }^{ t }{ { e }^{ -\tau }{ e }^{ t-\tau }cos(t-\tau )d\tau } \} \\ =\frac { L\{ { e }^{ t }cost\} }{ s } \\ =\frac {...

Homework Statement $$L\{ 1*{ t }^{ 3 }\}$$ Homework Equations $$L\{ \int _{ 0 }^{ t }{ f(\tau )d\tau } \} =\frac { L\{ f(t)\} }{ s }$$ The Attempt at a Solution $$L\{ 1*{ t }^{ 3 }\} \\ =L\{ \int _{ 0 }^{ t }{ { (t-\tau ) }^{ 3 }d\tau } \} \\ =\frac { L\{ { t }^{ 3 }\} }{ s } \\ =\frac...

I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...

Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it. Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...

1 Problem Statement: 2 Relevant Equations KVL KCL Ohm's Law 3 Attempt at Solution I solved Problem 4.2, which was just mesh analysis on the original problem. I got the correct answers, indicated by F.A. The problem I am having with 4.3 is how to redraw the circuit with the...

Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...