# Tagged Questions

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

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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### Laplace Transform of $\frac{a}{2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t))$

I'm trying to prove that the Laplace Transform of $$\frac{a} {2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t))$$ is $$\exp(-a\sqrt{s});$$ from the definition of Laplace Transform we should compute ...
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### Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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### Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
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### 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 ...
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### Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
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### If $f$ has a Laplace transform $F$ then $\lim_{s\to\infty}F(s)=0$?

As I know, well-known functions that have Laplace transform vanishes at infinity. Because almost well-known functions has exponential form and the Laplace transform of function has exponential form ...