Inverse Laplace transform computation

Question

Calculate the inverse Laplace transform $$\mathcal{L^{-1}} \left\{ s\log \frac{s^2 + a^2}{s^2 - a^2}\right\},$$ where $a\in\mathbb{C}$ is a constant.

I know that is boring but I would really appreciate some help.

Thank's in advance!

Answer 1

I would proceed step by step as follows (using $\risingdotseq$ for the correspondence of the original and image):

$$f\left(x\right)\risingdotseq F=s\log\frac{s^{2}+a^{2}}{s^{2}-a^{2}}$$ $$\int_{0}^{x}f\left(t\right)dt\risingdotseq\frac{F}{s}=\log\frac{s^{2}+a^{2}}{s^{2}-a^{2}}$$ $$-x\int_{0}^{x}f\left(t\right)dt\risingdotseq \frac{d}{ds}\left(\frac{F}{s}\right)=\frac{2s}{s^{2}+a^{2}}-\frac{2s}{s^{2}-a^{2}}$$ $$-x\int_{0}^{x}f\left(t\right)dt\risingdotseq\frac{2s}{s^{2}+a^{2}}-\frac{2s}{s^{2}-a^{2}}$$ inverting the RHS:

$$-x\int_{0}^{x}f\left(t\right)dt=2\cos ax-2\cosh ax \qquad (*)$$ EDIT (thanks to the comment by Fabian): differentiate once with respect to $x$ $$-\int_{0}^{x}f\left(t\right)dt-xf\left(x\right)=-2a\sin ax-2a\sinh2x$$ Now multiply by $x$ and subtract from (*): $$x^2f(x)=2(ax\sin{ax}+ax\sinh{ax}+\cos{ax}-\cosh{ax})$$ $$f(x)=\frac{2}{x^2}(ax\sin{ax}+ax\sinh{ax}+\cos{ax}-\cosh{ax})$$