# Questions tagged [laplace-transform]

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### Integral transformation, Laplace-like

Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$\int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt$$ It resembles somewhat the Laplace transformation. ...
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### Bromwich integral transformed to an integral on the real axis

I am new in complex integration and inverse Laplace transforms. I already asked this question on math.se but got no answer. The author of a textbook claims that the inverse Laplace transform has ...
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### Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)

I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...
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### Is harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function: $f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$ is a Bernstein function, ...
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Let $X$ be a non-negative random variable with a CDF $F$. Let $L_X(t)$ denote the Laplace transform of $F$, i.e., \begin{align} L_X(t)=E[ e^{-tX}], \quad t \ge 0 \end{align} It is known that $L_X(... 1answer 167 views ### Can Mellin transform be applied in this function? What's the result? $$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$ 0answers 116 views ### Decay of Laplace (or Mellin) transform beyond region of convergence? Let$f:[0,\infty)\to \mathbb{R}$be a piecewise differentiable function with$f(0)=0$and$f'(t)$of bounded variation. Its Laplace transform$\mathcal{L}f$converges for$\Re s > 0$. Assume it can ... 0answers 120 views ### Diffusion equation solution using Laplace transform [closed] Consider the operator $$L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}$$ with domain$D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value$u(x,0)=g(x), \forall x\in \Bbb R$... 1answer 72 views ### Identity for stable Lévy subordinator I want a proof or a reference for the identity $$\int_0^\infty \frac{s^{n-1}}{\Gamma(n)} p_\beta(s,x)\,ds =\frac{x^{n\beta-1}}{\Gamma(\beta n)},\quad x>0, \,n\in\mathbb N,$$ where$x\mapsto p_\...
How to get some insight in the following integral: $$\mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x$$ where $s$ is real (and the lower integration bound may be set ...