Inverse Laplace transform of the logarithmic function $s\log_e\left(\frac{ia-(s+g)}{ia+(s+g)}\right)$.

Question

What is the inverse Laplace transform of the following function? I'm particularly interested in the value of $f(t)$ at $t=0$. $a$ and $g$ are positive real constants. \begin{equation} F(s) = slog_e\left(\dfrac{ia-(s+g)}{ia+(s+g)}\right) \end{equation}

Thanks,

Answer 1

We know that $\text{a}\space\wedge\space\text{q}\in\mathbb{R}^+$:

• $$\text{F}(s)=s\cdot\log_e\left(\frac{\text{a}i-(s+\text{g})}{\text{a}i+(s+\text{g})}\right)=s\ln\left(\frac{\text{a}i-s-\text{g}}{\text{a}i+s+\text{g}}\right)$$

Now, with WolframAlpha I found:

$$f(t)=\pi i\delta(t)-\frac{2i\sin(\text{a}t)e^{-\text{g}t}}{t}$$