Transform Definition and 1000 Threads

Homework Statement I'm trying to derive the result on slide 1 of this link: http://www.physics.ucf.edu/~schellin/teaching/phz3113/lec13-3.pdf Unfortunately, I'm not sure how to integrate the Fourier transform when my u(x,t) function is undefined. Could someone help me get the...

Homework Statement This is an exercise from Taylor & Mann page 468 Q5 : Use the transformation x=au and y=bv to map the region R defined by \frac{x^2}{a^2} + \frac{y^2}{b^2} ≤ 1 onto the uv plane. Evaluate : \int \int_R \frac{x^2}{a^2} + \frac{y^2}{b^2} dxdy with the aid of this transofrm...

The DCT of an even function is comprised of just cosine coefficients, correct? I'm playing around in MATLAB and I came up with a simple even function 1.0000 0.7500 0.5000 0.2500 0 0.2500 0.5000 0.7500 1.0000 0.7500 0.5000 0.2500 0 0 0 0...

Homework Statement find z transform of: x[n] = (1 / n!) *u[n] u[n] is the unit step Homework Equations z transform equationX(z) = Ʃ x[n] * z-n summation is from -∞ to +∞ The Attempt at a Solutioncancel the u[n] by changing the bounds of the summation now it is from 0 to +∞It's at this...

Homework Statement If there is a change of variables: (\vec x(t),t)\to (\vec u=\vec x+\vec a(t),\,\,\,v=t+b) where b is a constant. Suppose I wish to write the following expression in terms of a gradient in (\vec u, v) \nabla_\vec x f(\vec x,t)+{d^2\vec a\over dt^2} How do I do that? Homework...

I know the result: \widehat{H(f)}=i\textrm{sgn}\hspace{1mm}(k)\hat{f} I thought I could use fft, and ifft to compute the transform easily, is there a MATLAB command for sgn? Mat

I am doing a laplace transform as part of a coursework assignment. I have some example transpositions that are relevant to the question I am answering but I can't see how the author has got from one arrangement to the next. [b]2. Homework Equations He has given 1/(s^2(τs+1)) =...

Trying an example from a textbook but I don't understand Legendre transform at all. "construct Legendre transforms of the entropy that are natural functions of (1/T, V, n) and (1/T, V, μ/T)". I don't really understand where to start. An example prior to this exercise just states: A = E - TS =...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution See pdf attached for my attempt at the solution. I'm a little confused as to how to draw the phase spectrum for y(t). Would it simply be a line equation of, -\frac{\pi}{6000}f \pm...

Homework Statement Is the Fourier transform of a even/odd function also even/odd ? Homework Equations The Attempt at a Solution So far this result seems to be true. I can't find a confirmation however... Thanks ahead. Daniel.

I have a question asking for the inverse laplace transform of (e^(-s))/(s^2+pi^2). I split it up to (e^(-s))/s x s/(s^2+pi^2) and got u(t-1)cos(pi(t-1)),but the correct answer is (sin(pi(t-1)/pi)u(t-1). So here it was split up to (e^(-s))/pi x pi/(s^2+pi^2) and I don't understand where the...

Homework Statement Hi there! I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome: 4\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me... Homework...

Homework Statement I am given a Lagrangian, which, per assignment text, describes a single degree of freedom: L= \frac{I}{2}(\dot{q}+\omega)^2-kq^2 I need to find the Hamiltonian. Now, what I am wondering, when performing the Legrende transform...

Homework Statement Consider an infinite line of charge with density λ per unit length lying along the z-axis. If the line of charge is stationary in frame S, use the Lorentz transform to find the current and charge densities in a frame S' which is moving with velocity v parallel to the...

Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency...

Homework Statement Find the cosine Fourier transform of the function f(t)=e-at Homework Equations The Attempt at a Solution F(w)=(2/π)0.5∫dt e-atcos(wt) The integral is from 0 to +∞ Using euler's formula I got the result F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a)...

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...

Homework Statement What is the inverse laplace transform of (1/(s+s^3))? Homework Equations The Attempt at a Solution I looked it up on wolframalpha and got 1-cos(t), but I don't understand how they got that answer. I looked up a basic laplace transform chart and didn't see anything...

in Fourier transforms of normal baseband sigal , spectral components are replicated on both +ve and -ve sides of frequency axis. i know that both -ve and +ve frequency components contribute to the total power of the signal but i don't know the physical significance of the -ve frequencies...

I am trying to prove the convolution of the Fourier Transform $$ (\widehat{f\star g})(\xi) = 2\pi\hat{f}(\xi)\hat{g}(\xi) $$

Homework Statement a) Find the Fourier transform of the function f(x) defined as: f(x) = 1-3|x| , |x|<2 and 0 for |x|>2 b) Find the values of the inverse Fourier transform of the function F(k) obtained in a) Homework Equations F(k) = \frac{1}{\sqrt{2π}}\int f(t) eikx dx f(x) =...

I've been trying to figure out why it's standard to use complex discrete Fourier transforms instead of just the real version. It's discussed a bit here. http://dsp.stackexchange.com/questions/1406/real-discrete-fourier-transform As far as I can tell there's a hypothetical efficiency...

Homework Statement Find inverse Laplace transform \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]Homework Equations The Attempt at a Solution I try with theorem \mathcal{L}[f(t)*g(t)]=F(s)G(s) So this is some multiple of \mathcal{L}[\sin at*\sin at] So \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]=\propto...

Hello, I am trying to figure out in my notes how my professor did L[(e^-3t)(sin2t)] = 2/(s+3)^2 +4 I think she just did it in her head and wrote it, so I don't actually know how to solve it. I am looking at my table of laplace transforms and there is none for a product of an exponential and...

In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought...

Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...

I'm having trouble with this question. Can anyone please guide me. My Attempt : = 1/(s-1) * L{t^2*e^-t} = 1/(s-1) * (2/(s+1)^3) = 2/((s-1)(s+1)^3)) but that's not the answer , its 2/((s-1) s^3) somehow. The question is attached below.

Homework Statement Find the inverse laplace transform of (2/(s+2)^4) using the given table of identities: Homework Equations Here are the given identities: The Attempt at a Solution Alright, I realize that there is a simple identity that I can use with a factorial symbol, but this...

i have just recently discovered peltier devices which make electricity from the Seebeck effect. one video on you tube showing peltier device producing about 2 volts from a flame. i would like to know how to convert this 2 volts DC (if it is actually dc) into 2 volts AC then put it through a step...

Homework Statement I am looking at finding the Fourier transform of: f(t)=\exp \left[ \frac{-(t-m)^2}{2 \sigma^2}\right] Homework Equations \hat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt The Attempt at a Solution I did it a little differently that my...

I thought it would be obvious, but I can't find a series representation of the Laplace transform. I'm looking for something analogous to the Fourier series and how it can be used to derive the Fourier transform. I though it would simply be f(x) = \sum_{s=-\infty}^{\infty}{C_{s} e^{sx}} , but...

This is a conceptual question on the region of convergence (ROC) and the inverse Laplace transform (ILT). Here the bilateral laplace transform (LT) and the ILT are given by F(s)=L\{f(t)\}=\int_{-\infty}^{+\infty} f(t) e^{-st} dt and f(t)=L^{-1}\{F(s)\}=\frac{1}{i...

I was going to post this in the learning material section but i didnt have access to it for some reason. but i guess i can post it here. its homework after all. so i have noticed that there is almost nothing learning material on fourer transform on the web. like how to transform a function to...

why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...

Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}

How to transform non-linear frequency vibration to constant frequency of vibration of 2 Hz? Such as transform the different frequency of waves to constant frequency and then maintain it..

Hi. In Apostol's book "Introduction to analytic number theory", Teorem 3.2(b), Apostol proves (1) \zeta (s) = \lim\limits_{x \to \infty} \left\{ \sum\limits_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} \right\} for s in critical strip. I know this translates to a Mellin transform...

Homework Statement Use Laplace transform to the system: \frac{dy}{dt} + 6y = \frac{dx}{dt}3x - \frac{dx}{dt} = 2\frac{dy}{dt} x(0) = 2 ; y(0) = 3 The Attempt at a Solution I've tried everything on this one. I first solved \frac{dy}{dt} + 6y = 2\frac{dy}{dt} and I got y = 3e^{6t} ...

Homework Statement For a physics problem I must take the inverse Fourier transform of 2 functions. Namely I must compute the integral ##\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk##.Homework Equations Already given. i is the complex number. t is greater or...

Homework Statement Problem 8-19 in Matthews and Walker's book on mathematical physics. A straight wire of radius a is immersed in an infinite volume of liquid. Initially the wire and the liquid have temperature T=0. At time t=0, the wire is suddenly raised to temperature ##T_0## and...

Homework Statement this one stumped me.. d^2y/dt^2 +ωy=ksin((√ω)t) y(∏/4)=0, y'(∏/4)=0 The Attempt at a Solution → (s^2 + ω)U(s)= LT {ksin((√ω)(T+∏/4)} is as far as i can get (i know what to do with the left hand side once i get the LT of the right hand side but i don't know what to do with...

Why is it that the unilateral lateral Laplace transform is used when given initial conditions that are non-zero. Is there a reason that explains why it would be wrong to use the bilateral Laplace transform instead? I know bilateral does not have any input of initial conditions but that does not...

How we get relation \lim_{t\to 0}f(t)=\lim_{p\to \infty}pF(p)? Where ##\mathcal{L}\{f\}=F##.

Homework Statement Evaluate the Laplace transform: L{δ(t-∏)tan(t)} Homework Equations The Attempt at a Solution L{δ(t-∏)tan(t)} = ∫ δ(t-∏)tan(t) dt evaluated from 0 to ∞ =tan(∏)e-∏*s = 0 Could someone check my work on this one? I'm suspicious that my transform is just zero...

Homework Statement y" + y = 4δ(t-2π); y(0)=1, y'(0)=0 Homework Equations L[f(t-a) U(t-a)] = e^{-as} L[f(t)] L[δ(t-c)] = e^{-cs} The Attempt at a Solution My answer is: cos(t) + 4U(t-2π)sin(t-2π). When I used Wolframalpha it gave me 4sin(t)U(t-2π) + cos(t)

Evaluating a "Fourier Transform" Integral Homework Statement Evaluate I = ∫[0,∞] e-ktw2 cos(wx) dw in the following way: Determine ∂I/∂x, then integrate by parts. Homework Equations Possibly? The Attempt at a Solution Since integral limits do not depend on x, the partial with respect...

Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...

Hi All, I'm just trying to practice graphing signals in frequency domain and I came across a stiuation I wasn't familiar with. If the exp() has a constant*t in it I'm not sure how to graph it, I remember that just cos it like a double sided exp(jwt) but with half the magnitude. I've attached a...

I have calculated a k-space function to be f(k) = \frac{1}{2k} I want to Fourier transform this to find f(x), I have found many different Fourier transform equations...can I use this one? f(x) = \frac{1}{\sqrt{2π}}\int\frac{1}{2k}e-ikxdk Limits fo integration -Infinity to Infinity...

Homework Statement Find the Fourier Transform of: f(t)=\frac{cos(\alpha t)}{t^2+\beta^2} Homework Equations F(\omega)=\frac{1}{2\pi}\int^{∞}_{-∞}\frac{cos(\alpha t)exp(i \omega t)}{t^2+\beta^2} The Attempt at a Solution I start with: cos(\alpha t)=\frac{exp(i \alpha t)+exp(-i...