Is it possible to calculate the inverse Laplace transform of the following?

Question

I have a Laplace tranform in the form given below

$\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$

Can some one help me to find the inverse Laplace transform of it?

Here, $\alpha$ can take values like 1,2,3,4,5...

$P$ and $\lambda$ are constants.

Note: for $\alpha=4$, there exists a closed-form solution. How about for general values of $\alpha$.

Answer 1

Let me give you some solutions for $\alpha$. For general values of $\alpha$ is impossible to solve.Maybe someone smarter to solve this in the future. :)

Now we simplify to: $\mathcal{L}_I(s)=\text{exp}(-{A} s^{2/\alpha})$

where $A = \pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}$ and $A > 0$

for $\alpha = 1$ is Infinity.

for $\alpha = 2$ is $0$.

for $\alpha = 3$

where $\text{AiryAi}$ is Airy function,$\text{AiryAiPrime}$ is derivative of the Airy function.

$$\frac{2 A e^{-\frac{2 A^3}{27 t^2}} \left(\sqrt[3]{3} A \text{Ai}\left(\frac{A^2}{3 \sqrt[3]{3} t^{4/3}}\right)-3 t^{2/3} \text{Ai}'\left(\frac{A^2}{3 \sqrt[3]{3} t^{4/3}}\right)\right)}{3\ 3^{2/3} t^{7/3}}$$

for $\alpha = 4$

$$\frac{A e^{-\frac{A^2}{4 t}}}{2 \sqrt{\pi } t^{3/2}}$$

for $\alpha = 5$

where $_1F_3$ is the generalized hypergeometric function. $$\frac{A \left(120 \sqrt{5+\sqrt{5}} t^{6/5} \Gamma \left(\frac{7}{5}\right) \, _1F_3\left(\frac{7}{10};\frac{2}{5},\frac{3}{5},\frac{4}{5};-\frac{4 A^5}{3125 t^2}\right)+A \left(A \left(5 \sqrt{5+\sqrt{5}} A \Gamma \left(\frac{13}{5}\right) \, _1F_3\left(\frac{13}{10};\frac{6}{5},\frac{7}{5},\frac{8}{5};-\frac{4 A^5}{3125 t^2}\right)-24 \sqrt{5-\sqrt{5}} t^{2/5} \Gamma \left(\frac{6}{5}\right) \, _1F_3\left(\frac{11}{10};\frac{4}{5},\frac{6}{5},\frac{7}{5};-\frac{4 A^5}{3125 t^2}\right)\right)-48 \sqrt{5-\sqrt{5}} t^{4/5} \Gamma \left(\frac{4}{5}\right) \, _1F_3\left(\frac{9}{10};\frac{3}{5},\frac{4}{5},\frac{6}{5};-\frac{4 A^5}{3125 t^2}\right)\right)\right)}{240 \sqrt{2} \pi t^{13/5}}$$

for $\alpha = 6$

$$\frac{A \text{Ai}\left(\frac{A}{\sqrt[3]{3} \sqrt[3]{t}}\right)}{\sqrt[3]{3} t^{4/3}}$$

Mathematica code:

for $\alpha \in integers$ and $\alpha > 0$

 \[Alpha] = 4;
 f[s_] := Exp[-A*s^(2/\[Alpha])]; 
 F = Assuming[{A > 0, t > 0, {t, A} \[Element] Reals},Integrate[f[s]*Exp[s*t] /. s -> I*y, {y, 0, Infinity}]/Pi]; 
 FullSimplify[ComplexExpand@Re[F], {t > 0, A > 0, {t, A} \[Element] Reals}]

A similar solution I'm found in the Book, page: 1012, example: 15.12.2.