An integration related to incomplete gamma function 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.
 A: 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.
A: 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? 
