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I have no clues about the following equation, expecting some help from anyone. $\int_\beta^\infty e^{-x \theta}\frac{1}{\theta}(\frac{\theta}{\beta}-1)^{-\alpha}d \theta=\Gamma(1-\alpha)\Gamma(\alpha,\beta x)$, where $\Gamma(z)$ and $\Gamma(z,\alpha)$ are the gamma function and incomplete gamma functions.

I may also wondering what is the Laplace transform to the gamma function. Any references?

Thanks in advance.

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For $\operatorname{Re}(\alpha) < 1$ and $\beta > 0$ the function $$ f \colon \mathbb{R}^+ \to \mathbb{C}\, , \, f(x) = \int \limits_\beta^\infty \mathrm{e}^{- x \theta} \frac{1}{\theta} \left(\frac{\theta}{\beta}-1\right)^{-\alpha} \, \mathrm{d} \theta \, , $$ is well-defined and differentiable. Using the change of variables $\theta = \beta \left(1 + \frac{\phi}{\beta x}\right)$ we find for $x > 0$ $$ f'(x) = - \int \limits_\beta^\infty \mathrm{e}^{- x \theta} \left(\frac{\theta}{\beta}-1\right)^{-\alpha} \, \mathrm{d} \theta = - \beta^{\alpha} x^{\alpha-1} \mathrm{e}^{-\beta x} \int \limits_0^\infty \phi^{-\alpha} \mathrm{e}^{- \phi} \, \mathrm{d} \phi = - \beta^{\alpha} x^{\alpha-1} \mathrm{e}^{-\beta x} \, \Gamma(1-\alpha) \, .$$ Since $\lim_{x \to \infty} f(x) = 0$ , we obtain \begin{align} f(x) &= - \int \limits_x^\infty f'(y) \, \mathrm{d} y = \Gamma(1-\alpha) \beta^{\alpha} \int \limits_x^\infty y^{\alpha-1} \mathrm{e}^{- \beta y} \, \mathrm{d} y = \Gamma(1-\alpha) \int \limits_{\beta x}^\infty z^{\alpha-1} \mathrm{e}^{- z} \, \mathrm{d} z \\ &= \Gamma(1-\alpha) \Gamma(\alpha, \beta x) \, \end{align} for $x > 0$ .

The Laplace transform of the gamma function does not exist, since $\Gamma$ grows faster than any exponential function.

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Many thanks.

A further one is how to calculate the Laplace transform of a three parameters Pareto distribution. The density function: $$f(x)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)\Gamma(k)}\frac{\lambda^\alpha x^{k-1}}{(\lambda+x)^{\alpha+k}}, \ x>0,$$ where all parameters are positive. Now we no longer have the integer power and your differentiation method cannot be applied. How can I calculate the Laplace transform of it?

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