What is Transform: Definition and 1000 Discussions
In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable
t
{\displaystyle t}
(often time) to a function of a complex variable
s
{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral
Question on my study guide:
An optical systems consists of 4 lenses spaced apart. Each lens has a focal length f. Each lens is located a distance "z" away from each plane as shown. The total length of the system is 8z.
Find the distance z needed to satisfy a FOURIER TRANSFORM condition...
Hello,
I was wondering if one can give meaning to the Fourier transform of the linear function:
\int_{-\infty}^{+\infty} x e^{ikx}\, dx
I found that it is \frac{\delta(k)}{ik} , does this make sense?
Okay, I am trying to determine the Fourier transform of cos (2\pix)=f(x)
Where F(k)=^{\infty}_{\infty}\intf(x)exp^{-ikx} dx,
So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as...
Laplace transform initial value problem--need help!
Looking at the solutions to these initial value problems, I am very confused as to how the highlighted steps are derived (both use heaviside step functions). I know the goal is to get the fractions in a familiar form so that one can look them...
Homework Statement
Homework Equations
sinc(x) = \frac{sin(x)}{x}
The Attempt at a Solution
bit unsure how to get started?? i know transform of rectangular pulse pτ(t)=τ*sinc(τω/2∏)
also that sin(ωt)= ejωt-e-jωt / (2)
I could also probably sketch sinc(t/2∏), if that helps.
Dear all,
I have recently come across the following Fourier transform (FT):
I=\int_{-\infty}^{\infty} dx \, e^{-\imath x t} \frac{(1-x^2)}{(1+x^2)^{3/2} (a^2+x^2)}.
The integrand contains two branch points on the imaginary axis, plus two poles also residing on the imaginary...
The typical way to evaluate $ \displaystyle \int_{0}^{\infty} \frac{\cos mx}{a^{2}+x^{2}} \ dx$ is by contour integration.
In a recent thread I evaluated that integral using the Laplace transform.
http://mathhelpboards.com/analysis-50/advanced-integration-problem-9129.html#post42551My...
Hello,
As of recently, I've been working with Laplace transforms and have a question about their relationship to solving differential equations.
I know the definition of the laplace transform and I know that a function is essentially being transformed from the time domain to complex...
I understand that the Fourier transform maps one function onto another. So it is a mapping from one function space onto another.
My question is, why is it often referred to as a mapping from time domain to the frequency domain? I don't understand why the image of the Fourier transform...
Hi Guys,
I have an expression that i am struggling to manipulate into a laplace transform. This expression should fit one or a combination of the common transform pairs. I believe the transform the expression should be fitting is either a unit step 1/s a unit ramp 1/s^2 an exponential 1/s+a...
Hi,
I would like to find the inverse Laplace transform for
11/(s^2+16)^2
I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work
C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2
Does anyone have any suggestions?
Homework Statement
Hi I would like to know how to expand 20/2s(s^2+9) in order to find the inverse Laplace Transform of the function K(s) to gt k(t). The (s^2+9) term in the denominator is throwing my calculations off for the constants because of the s^2 term.
Homework Equations...
Hello,
I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like
Us(t-2)
From tables, the Laplace transform for a regular units step...
Hi guys,
I've been using this site for a while now, but this is going to be my first post. I want to pick your brains to get some insight on this problem I'm tackling.
I'm trying to take a Fourier Transform of a function. My function is a function of (r,phi) and it is a piecewise function...
Homework Statement
Using the Schwarzschild Metric and the Contravariant Position Vector 1x4 ##x^k## with 4 vector:
$$x^{k'} = \left[ \begin {array}{c}r \\ \theta \\ \phi \\ t \end {array} \right]$$
where
##x^1## = r = 1 per unit distance
##x^2## = ##\theta## = 50 Degrees
##x^3##...
Given the transfer function
\[
H(s) = \frac{Y(s)}{U(s)} = \frac{1}{s + 1}
\]
and
\[
u(t) = 1(t) + 1(t - 1).
\]
How do I find U(s)? I know I take the Laplace transform of u(t) but with the two step functions how can this be done?
The Laplace transform of the step function is \(\frac{1}{s}\)...
I can't seem to part of an inverse Laplace transform correct.
\begin{align*}
f(t) &= \frac{6}{5}\mathcal{L}^{-1}\bigg\{\frac{1}{s + 2}\bigg\} +
\frac{3}{5}\mathcal{L}^{-1}\bigg\{\frac{3s - 1}...
Homework Statement
Find the Laplace transform of
f(t) = t \forall 0≤t≤T, 0 otherwise
Homework Equations
The Attempt at a Solution
I write the function as
tu(t)-t*u(t-T)
That is turn on the function t at t=0 and turn the function t off at t=T. It seems to be right to me...
Hi~
I recently solve a Laplace transform problem as following
L[int{t,0}cosh(t'-1)U(t'-1)dt']=? U(t'-1) is the unit step function(=1 for t'>1, =zero otherwise)
According the standard Laplace formula :
(1)L[cosh(t-1)U(t-1)]=exp(-s)*s/(s^2-1);
(2)L(int{t,0}f(t')dt')=F(s)/s...
Hello,
I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as
a_p = \int_0^1 x(s)...
Hi,
I hope somebody can help me with this one.
Homework Statement
Compute the Fourier Transform of the distribution x-a
Homework Equations
The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.The Attempt at a Solution
See...
I have an exercise with a function of the form:
h(t) = f(t)g(t)
and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...
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}\{ \sin(t)\}...
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
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
The exercise is a) in the attached trial. I have attached my attempt at a solution, but there are some issues. First of all: Isn't the example result wrong? As I demonstrate you get a delta function which yields the sum I have written (as far as I can see), not the sum...
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...
Homework Statement
Hi
I'm trying to understand how symmetry transform works.
Suppose a lagrangian L = q^{-2}
(actually it had another kinetic member, but I don't need it for my question here)
The matching action S = \int dt q^{-2}
We were told that it has the next symmetry
t...
I am going through my professors notes on generating functions. I come across the following equation:
\frac{\partial}{\partial t} \frac{\partial F}{\partial \xi^k} = \frac{\partial}{\partial t} \left( \gamma_{il} \frac{\partial \eta^i}{\partial \xi^k}\eta^l - \gamma_{kl}\xi^l \right ).
Here...
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...
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...