0
votes
1answer
22 views

Are FT and LT both isomorphic?

As the following diagram:(from a textbook) Note: 1. L2: L2 space, H2: H2 space 2. The upper one is in t-domain; the lower one, f-domain 3. : the Laplas transform operator : the fourier tansform ...
2
votes
1answer
39 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
0
votes
1answer
63 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
1
vote
0answers
36 views

Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
1
vote
0answers
17 views

Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
0
votes
1answer
83 views

Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is ...
0
votes
0answers
45 views

s-plane and fourier transform, together in 3d space.

I dont understand how can varying the real part in the s-plane make the amplitude in the fourier plane go to infinity. Lets say the pole is at -3 + -j for example.. Then the laplace transform is the ...
4
votes
1answer
124 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
1
vote
1answer
76 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
1
vote
1answer
78 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
1
vote
1answer
67 views

Laplace question

How do you express Laplace transform $\mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt$ with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ...
0
votes
1answer
78 views

Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
1
vote
1answer
86 views

Laplace Transform and Fourier Transform of a function

I have this transfer function: $$ h(t)= -\frac{1}{16}te^{-2t} $$ and the Laplace Transform is: $$ H(s) = \frac{-\frac{1}{16}}{(s+2)^{2}} $$ I know that to find the Fourier Transform, I would ...
4
votes
3answers
201 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
2
votes
1answer
173 views

Fourier transform and Laplace transform to solve differential equation

generally we know that both Fourier transform and Laplace transform both is used to solve differential equation,first of all let us recall both form,first Fourier transform: some times instead ...
0
votes
1answer
42 views

Derive Fourier transforms from Fourier expansion. How are they related?

I am just trying to relate Fourier Series expansion to Fourier Transforms. If someone could show how one value on the middle of the table is derived (from expansion) as opposed to deriving their ...
1
vote
2answers
2k views

How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
1
vote
2answers
48 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
0
votes
1answer
41 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
3
votes
1answer
161 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
0
votes
0answers
47 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
2
votes
3answers
190 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
2
votes
1answer
92 views

inverse Radon transform

I'm having a trouble understanding a proof in regards to the inversion of the Radon transform in $\mathbb{R}^3$. The statement is as follows: if $f \in \mathcal{S}(\mathbb{R^3})$, then ...
4
votes
2answers
132 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
6
votes
0answers
285 views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
0
votes
1answer
45 views

How to distinguish between the different frequency domains?

Sometimes the terms 'Fourier domain', 'complex frequency domain', 'Frequency domain' and 's domain' are used interchangeably. Take those answers here for example: ...
4
votes
2answers
4k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
4
votes
0answers
127 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
-3
votes
1answer
100 views

Can you do this Laplace Transform

Can you do this? This is part of my final year EE work. I need to solve this in order to figure out how my sensor is behaving. Please help, and stop down voting. If it is too difficult for you, just ...
0
votes
1answer
112 views

Laplace Transform of this function

Find $L\{F(t)\}$ if $$F(t) = \begin{cases} \sin t & \text{between }0 < t < \pi \\ 0 & \text{between } \pi < t < 2\pi \end{cases}$$ Really stumped by this one. Please can you ...
1
vote
1answer
114 views

Is this a correct way of thinking of Fourier transforms

I am working on my understanding of various transforms and I have been thinking about the Fourier transform, what i "does" to the function it is applied to. The way I see it: The function $f$ that ...
1
vote
0answers
258 views

square root of s domain transfer function

I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in ...
10
votes
1answer
315 views

Laplace transform identity

Is there a function equal to its Laplace transform? I mean $$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$ Of course I know $f(t)=0 $ satisfy the equation. For the case of the Fourier transform, I ...
7
votes
2answers
2k views

Why the Fourier and Laplace transforms of the Heaviside (unit) step function do not match?

The Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$. The Laplace transform of the same function is $\dfrac{1}{s}$. I remember the proof came from derivatives and ...
15
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
2
votes
1answer
127 views

Find the probability of certain measurement for a Laplace Operator on a state function

Let $H$ be the operator $ -\frac{d^{2}}{dx^{2}} $ and let its domain be $$\{f\in L^{2}(\mathbb{R},d\lambda)\text{ }:\int_{-\infty}^{\infty}|xF[f(x)]|^{2}dx<\infty\} $$ where $F$ is the Fourier ...