# Contour integral with two essential singularity

I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating function (basically the Laplace transform where kernel is $$e^{sx}$$ instead of $$e^{-sx}$$, i.e. $$\mathbb E\left[ e^{sx}\right]$$ where $$x$$ is your random variable) using the following formula

$$M(\alpha) = \frac{1}{2\pi i}\Gamma(\alpha)\int_{c-i\infty}^{c+i\infty} L(s)(-s)^{-\alpha} \, ds, \qquad \operatorname{Re}\{\alpha\}>0$$

where $$c\in \mathbb{R}^+$$ is some positive real valued constant. We know that the moment generating function for Gamma variable $$x\sim \Gamma\left(k,\theta\right)$$ is $$\mathbb E\left[ e^{sx}\right] = \frac{1}{(1-\theta s)^k}, \quad \left(1-\theta s\right)> 0$$

where $$\theta , k \in \mathbb{R^+}$$ are positive real valued parameters. Therefore for sum of two independent Gamma variables $$x_1\sim \Gamma(k_1,\theta_1),x_2\sim \Gamma(k_2,\theta_2)$$ we have

$$\mathbb E\left[ e^{s(x_1+x_2)}\right] = E\left[ e^{s(x_1)}\right] E\left[ e^{s(x_2)}\right] = \frac{1}{(1-\theta_1 s)^{k_1} (1-\theta_2 s)^{k_2}}, \quad (1-\theta_1 s)> 0, (1-\theta_2 s)>0$$

Therefore to get the Laplace transform first we have to apply $$s\rightarrow-s$$ to get

$$L(s) = \frac{1}{(1+\theta_1 s)^{k_1} (1+\theta_2 s)^{k_2}}, \quad (1+\theta_1 s)> 0, (1+\theta_2 s)>0$$

and then I have to calculate the following integral

$$M(\alpha) = \frac{1}{2\pi i}\Gamma(\alpha)\int_{c-i\infty}^{c+i\infty} \frac{(-s)^{-\alpha}}{(1+\theta_1 s)^{k_1} (1+\theta_2 s)^{k_2}} \, ds, \qquad \operatorname{Re}\{\alpha\}>0.$$

My question: I don't know how to calculate the last integral! If it was a simple pole or poles with integer power then I simply used partial fraction decomposition and used the Residue theorem, however the power values of $$k_1,k_2$$ are real numbers and the singularity is essential! I would appreciate if you can show me how to deal with this beast!

• Can you elaborate more please? I try to expand using Lorenz series and there is infinite inverse terms around $\frac{1}{\theta_1}$ and $\frac{1}{\theta_2}$. I only know that definition. Commented Jun 28, 2023 at 10:20
• The point is, those functions are multivalued: they change value when you turn around the singularity, like the complex logarithm whose value is augmented by $2\pi$ each time you make a turn around $0$. In order to deal with your integral (not saying that it's possible in an exact manner), you have to specify cuts (like removing $\mathbb{R}_{>0}$ for the logarithm) Commented Jun 28, 2023 at 12:18