# Tagged Questions

The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

34k views

### What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
4k views

### Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
6k 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 ...
2k views

### Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$F(z) = e^{-\sqrt{z}}.$$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
3k views

### Physical interpretation of Laplace transforms

One may define the derivative of $f$ at $x$ as $\lim\limits_{h\to0}\cdots\cdots\cdots$ etc., and show that that has certain properties, but it also has a "physical" interpretation: it is an ...
565 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 ...
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 ...
2k views

500 views

### Is the Laplace transform a functor?

I may be oversimplifying, as I know very little about category theory, but: Does the Laplace transform, which—to my limited recollection—is a morphism between differential equations and algebraic ...
648 views

### Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
522 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, ...
192 views

### Evaluate $\int_{0}^{\infty}\frac{1-e^{-t}}{t}e^{-st}\;dt$

This is laplace transform of $\dfrac{1-e^{-t}}{t}$ and the integral exists according to wolfram Do i get any help/hints about how to work this ? I have been trying integration by parts with different ...
2k views

### How to find the Laplace transform of $\frac{1-\cos(t)}{t^2}$?

$$f(t)=\frac{1-\cos(t)}{t^2}$$ $$F(S)= ?$$
76 views

### Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty e^{-st}\cos\left(t\right)...
188 views

967 views

### How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?

How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
622 views