# Tagged Questions

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

39 views

### Laplace transform of the square of Brownian motion hitting time

Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace ...
25 views

### Finding the inverse laplace of this function: $F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
33 views

### Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
42 views

35 views

### laplace transform of difference between two gamma independent random variables

Knowing that the laplace transform of a Gamma distribution is given by: $$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$ and that for Z = X + Y "Sum of two independent Gamma distribution random variables"...
179 views

### If $\int_{0}^{\infty} f(x) \, dx$ converges, will $\int_{0}^{\infty}e^{-sx} f(x) \, dx$ always converge uniformly on $[0, \infty)$?

I previously asked about sufficient conditions to conclude that $$\lim_{s \to 0^{+}}\int_{0}^{\infty} e^{-sx} f(x) \, dx = \int_{0}^{\infty} f(x) \, dx$$ when $\int_{0}^{\infty} f(x) \, dx$ does not ...
27 views

### What is the inverse Laplace transform of $\frac{p^2}{(p^2+4)^2}$

Given $$f(p)=\dfrac{p^2}{(p^2+4)^2}$$ So $$f(p)=\dfrac{p^2}{(p^2+4)^2}=\dfrac{p^2+4-4}{(p^2+4)^2}=\frac{1}{p^2+4}-\frac{4}{(p^2+4)^2}$$ I know the inverse Laplace transform of the first term but ...
48 views

### Convolution of exponential and rect functions

I have a convolution question in my signals and systems problem set that is puzzling me: $f(t) = e^{-t/2T} u(t)$ and $g(t) = rect(t/2T)$ find the convolution $f \ast g$ and I am assuming $T>0$...
119 views

38 views

56 views

### The Laplace transform of an integrable function is differentiable

let $f\in L^1(0,\infty)$. For x>0, define $g(x)=\int_{0}^{\infty} f(t) e^{-tx} dt$. Prove that $f$ is differentiable for $x>0$ and with derivative $g'(x) = \int_{0}^{\infty} -tf(t) e^{-tx} dt$. ...
5 views

### System response (TF), multiplication vs substition

Let's say I have modeled a system as a transfer function: $H(s) = \frac{Y(s)}{U(s)}$ Given the question: For which values of $a$ is the input signal $e^{a t}$ absorbed by the system $H$? What is ...
31 views

45 views

25 views

### Inverse Laplace transform of the form $F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$

I am trying to solve the inverse Laplace transform of the form $$F(s) = \frac{s^{m}}{(1+a \cdot s)^{n}(1+b \cdot s)^{h}}$$ where, $a$ and $b$ are known constants, $m$, $n$, ...