I am currently studying the textbook Modeling and Analysis of Stochastic Systems, third edition, by Kulkarni. Chapter 5.1 Exponential Distributions says the following:
The probability density function (pdf) $f_X$ of an $\exp(\lambda)$ random variable is called the exponential density and is given by
$$f_X(x) = \dfrac{d}{dx}F_X(x) = \begin{cases} 0 & \text{if}\, x\leq 0\\ \lambda e^{-\lambda x} & \text{if} \, x \ge 0 \end{cases}$$
The density function is plotted in Figure 5.2. The Laplace Stieltjes transform (LST) of $X \sim \exp(\lambda)$ is given by $$\begin{align}\tilde{F}_X(s) &= E\left( e^{-sX} \right) \\&= \int_0^\infty e^{-sx} f_X(x) \ dx \\&= \dfrac{\lambda}{\lambda + s} \,, \ \text{Re}(s) > - \lambda, \tag{5.2}\end{align}$$ where the $\text{Re}(s)$ denotes the real part of the complex number $s$.
I'm trying to calculate 5.2 myself. I get
$$\begin{align} \int_0^\infty e^{-sx} \left( \lambda e^{-\lambda x} \right) \, dx = \lambda \int_0^\infty e^{-x(s + \lambda)} \, dx \end{align},$$
but I'm unsure of how to proceed. Substitution with $-x(s + \lambda)$ doesn't seem to work here (or, at the very least, I'm not doing it correctly).
How exactly is 5.2 calculated?
The integral $\int e^{ax}dx=\frac{1}{a}e^{ax}$, as you can easily verify by taking the derivative of the right side or making the substitution $u=ax$ on the left.
$$\lambda\int_0^\infty e^{-x(s+\lambda)}dx=\frac{\lambda}{-(s+\lambda)}e^{-x(s+\lambda)}\big|_{x=0}^\infty$$
The first limit is $e^{-\infty(s+\lambda)}$. We can use $e^{b+ic}=e^b(\cos c+i\sin c)$. We have $$\begin{split}e^{-\infty(\Re(s)+\Im(s)i+\lambda)}&=e^{-\infty(\Re(s)+\lambda)-\infty\Im(s)i}\\ &=e^{-\infty(\Re(s)+\lambda)}(\cos(-\infty\Im (s))+i\sin(-\infty\Im(s))\\ &=0\end{split}$$
if $\Re(s)+\lambda>0$ or $\Re(s)>-\lambda$, because the right term is finite and $e^{-\infty a}=0$ if $a>0$. In this case the limit goes to $0$ and we get $$\frac{\lambda}{-(s+\lambda)}\left(0-1\right)=\frac{\lambda}{s+\lambda}$$