Inverse Laplace Transform of the following.

Question

What is the inverse laplace transform of $ F(s) = \frac{1}{2}\ln \left ( \frac{s^{^{2}}+b^{2}}{s^{^{2}}+a^{2}} \right ) $ with $ a,b \ \epsilon \ \mathbb{R} $ ?

Answer 1

$$F(s)=\mathcal{L}[f(t)](s)=\int_0^\infty f(t)e^{-st}dt$$ $$F'(s)=-\int_0^\infty tf(t)e^{-st}dt$$

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You have: $$F(s)=\frac12\ln\left(\frac{s^2+b^2}{s^2+a^2}\right)$$ $$F'(s)=\frac{s}{s^2+b^2}-\frac{s}{s^2+a^2}$$ now we know that: $$\mathcal{L}^{-1}\left(\frac{s}{s^2+b^2}-\frac{s}{s^2+a^2}\right)=\cos(bt)-\cos(at)$$ and from what I showed above we can say that: $$-tf(t)=\cos(bt)-\cos(at)$$ $$f(t)=\frac{\cos(at)-\cos(bt)}{t}$$