Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Question

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following integration "$I$" $$I_n=\int_0^\infty\frac{(1-e^{-t})^n}{t^n}\,dt,n\geq2$$

Answer 1

I guess one way to do this is to differentiate the integral $n$ times, which means that you have to integrate the result $n$ times back.

$$G_n(s) = \int_0^{\infty} dt \left ( \frac{1-e^{-t}}{t}\right )^n e^{-s t}$$

$$\begin{align}(-1)^n \frac{d^n}{ds^n} G_n(s) &= \int_0^{\infty} dt \left ( 1-e^{-t} \right )^n e^{-s t}\\ &= \sum_{k=0}^n (-1)^k \binom{n}{k}\int_0^{\infty} dt \, e^{-(s+k) t}\\ &=\sum_{k=0}^n \frac{(-1)^k}{s+k} \binom{n}{k}\end{align}$$

For example,

$$\frac{d^2}{ds^2}G_2(s) = \frac1{s} - \frac{2}{s+1} + \frac1{s+2}$$ $$\frac{d}{ds} G_2(s) = \log{s} - 2 \log{(s+1)} + \log{(s+2)} + A$$ $$\begin{align}G_2(s) &= s \log{s}-s - 2 (s+1) \log{(s+1)} + 2 (s+1) + (s+2) \log{(s+2)} - (s+2) + A s + B\\ &= s \log{s}- 2 (s+1) \log{(s+1)}+(s+2) \log{(s+2)}+A s+B\end{align}$$

We get $A$ and $B$ by considering the limit as $s\to\infty$; in this limit, both of these constants drop out. Thus,

$$G_2(s) = s \log{s}- 2 (s+1) \log{(s+1)}+(s+2) \log{(s+2)}$$

The integral $I_2 = G_2(0) = 2 \log{2}$.

Higher values of $n$ may be done similarly.