Exponential Distribution.

For the first case, where $h(x)=x$, we let $x_n$ follow the standard exponential distribution $p_0(x)=\exp(-x)$ for $x>0$ (Section 17.1.2). Then, for ${\rm Re}(\lambda)<1$,

$$e(u,\lambda)= \frac{1}{1-\lambda} \; e^{(\lambda-1)u} \;\;\;$$    for $$\; u>0; \; \frac{1}{1-\lambda}$$    for $$\; u<0.$$

$$c(u,\lambda)= \frac{1}{1-\lambda} \left(1-e^{(\lambda-1)u}\right) \;\;\;$$    for $$\; u>0; \; 0$$    for $$\; u<0.$$