Inverse Laplace transform of $\dfrac{e^s}{s(e^s+1)}$

Question

The original problem is to solve $$\mathcal{L}^{-1}\left\lbrace\frac{e^s}{s(e^s+1)}\right\rbrace.$$ Doing partial fractions $$\frac{e^s}{s(e^s+1)}=\frac{1}{s}-\frac{1}{s(e^s+1)}$$ the problem reduces to solve the inverse Laplace transform of the last member. Using Heaviside function the problem is the same. Any ideas?

Answer 1

(See also Laplace transform of a square wave function and inverse of laplace transform of $\frac{1}{s(e^s+1)}$)

Original function: \begin{align} \hat{f}(s)=\frac{1}{s}-\frac{1}{s(e^s+1)}\,. \end{align} Performing the inverse Laplace transformation $f(t)=\mathcal{L}^{-1}\left\{\hat{f}(s)\right\}$ amounts, by the residue theorem, to \begin{align} f(t)&=\sum_{s\in s_{\ell}}\text{Res}\left(\hat{f}(s)\exp(st),s_{\ell}\right)\,, \end{align} where the poles $s_{\ell}=\{s_{0}, s_{j} \}$ of $\hat{f}(s)$ are located at $s_{0}=0$ and $s_{j}=\pi i (2j+1)$ with $k=-2,1,0,1,2,\ldots$ (such that $e^{s_j}+1 =0$).

The pole $s_{0}=0$ gives \begin{align} \text{Res}\left(f(x,s)\exp(s t),0\right)=1-\frac{1}{2}=\frac{1}{2}. \end{align}

To find the residue of $\hat{f}(s)$ at $s_{j}$, we expand \begin{align} e^s+1\overset{s\to s_{j}}=& \frac{d(e^s+1)}{ds}\bigg|_{s=s_{j}}\left(s- s_{j}\right)=-\left(s- s_{j}\right)\\ \end{align} We find \begin{align} \sum_{j=-\infty}^{\infty}\text{Res}\left(\hat{f}\exp(st),s_{j}\right) &=-\sum_{j=-\infty}^{\infty} \text{Res}\left(\frac{\exp(s t)}{s(e^s+1)},s_{j}\right)\\ &=\sum_{j=-\infty}^{\infty}\frac{\exp(s_j t)}{s_j}\\ &=\sum_{j=-\infty}^{\infty}\frac{\exp{\left[\pi i (2j+1)t\right]}}{\pi i (2j+1)}\\ &=\sum_{j=-\infty}^{-1}\frac{\exp{\left[\pi i (2j+1)t\right]}}{\pi i (2j+1)}+\sum_{j=0}^{\infty}\frac{\exp{\left[\pi i (2j+1)t\right]}}{\pi i (2j+1)}. \end{align} In the first sum, taking $j\to-1-k$ gives \begin{align} \sum_{j=-\infty}^{-1}\frac{\exp{\left[\pi i (2j+1)t\right]}}{\pi i (2j+1)}=\sum_{k=\infty}^{k=0}\frac{\exp{\left[-\pi i (2k+1)t\right]}}{-\pi i (2k+1)} \end{align} Hence, \begin{align} \sum_{j=-\infty}^{\infty}\text{Res}\left(\hat{f}\exp(st),s_{j}\right)=&=\sum_{j=0}^{\infty}\frac{\exp{\left[\pi i (2j+1)t\right]}}{\pi i (2j+1)}-\sum_{k=\infty}^{k=0}\frac{\exp{\left[-\pi i (2k+1)t\right]}}{\pi i (2k+1)}\\ &=2\sum_{j=0}^{\infty}\frac{\sin{\left[\pi (2j+1)t\right]}}{\pi (2j+1)} \end{align} Finally, \begin{align} f(t)=\frac{1}{2}+2\sum_{j=0}^{\infty}\frac{\sin{\left[\pi (2j+1)t\right]}}{\pi (2j+1)} \end{align} This turns to out to be a block wave between 1 and 0:

which I produced with the following code:

import numpy as np
import matplotlib.pyplot as plt
def f(t):
    s=0
    for k in range(0,400):
        s+=2np.sin((2k+1)np.pit)/(2*k+1)/np.pi
    return 1/2+s
t  = np.linspace(0,4,100)
fig, ax = plt.subplots()
ax.plot(t,[f(i) for i in t],  'r',lw=2)
ax.set(xlabel=r'$t$', ylabel=r'$f(t)$')