I have a problem about Inverse Laplace Transform, I would be appreciated to get your help for solving this problem (It took me about several hours to think but didn't come up with any solution).
Please find the inverse laplace transform of : $$\ln\left(1+\frac{a^2}{s^2}\right)$$ where "$a$" is an constant.
Suppose that there is a function $f$ such that $$ \ln\left(1+\frac{a^2}{s^2}\right)=\int_{0}^{\infty}f(t)e^{-st}dt. $$ Then, differentiating with respect to $s$, would give $$ -\frac{1}{1+\frac{a^2}{s^2}}\frac{2a^2}{s^3}=-\int_{0}^{\infty}f(t)te^{-st}dt \\ \frac{2a^2}{s(s-ia)(s+ia)}=\int_{0}^{\infty}f(t)te^{-st}dt \\ \left[\frac{2}{s}-\frac{1}{s-ia}-\frac{1}{s+ia}\right]=\int_{0}^{\infty}f(t)te^{-st}dt \\ \int_{0}^{\infty}\left[2-e^{iat}-e^{-iat}\right]e^{-st}dt =\int_{0}^{\infty}f(t)te^{-st}dt \\ 2-e^{-iat}-e^{iat}=f(t)t \\ 2\frac{1-\cos(at)}{t}=f(t). $$