# Questions tagged [laplace-transform]

The tag has no usage guidance.

160 questions
Filter by
Sorted by
Tagged with
34 views

### Inverse Laplace transform dependent on a parameter

I have to evaluate the inverse Laplace transform of a function of the type $F(s-a)/F(s)$. Clearly, if $a=0$ the solution is the impulse function. If $a$ is not equal to zero, I can compute the ...
• 31
129 views

• 259
1k views

### What is the intuition behind applying the Mellin Transform to prime distribution?

I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...
• 103
119 views

1 vote
151 views

### Laplace transform

\begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
• 21
3k views

### Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
142 views

• 1
368 views

### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
• 61
372 views

### How to evaluate inverse Laplace transform of $e^{- \sqrt{s}}$ using series?

I tried to find an inverse Laplace transform by series as follows $$f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
• 431
245 views

• 720
44 views

• 545
1 vote
36 views

### Solving an equation containing Laplace transform

Consider the equation $$\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),$$ where $\mathcal{L}$ is the ...
• 47
### What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?
A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
### Inverse Laplace transform of $\frac{1}{s^a + 1}$ with $0 < a \leq 1$
Problem I am looking for the following inverse Laplace transform, $$f(t) = \mathcal{L}^{-1}\left[\frac{1}{s^a + 1}\right] \;\;\;\;\; \text{with} \;\;\;\;\; 0 < a \leq 1.$$ What I understand ...