I have a Laplace tranform in the form given below


which is an multiplication of two stretched exponential decay function

where $A,B >0$

I need to find the inverse Laplace transform of this.

$\textit{I know that this cannot be solved symbolically using elementary functions.}$

Can some one suggest and/or refer me to some solution?

Here, $\alpha$ and $\beta$ can take values like $3,4,5...$

Any approximation is also welcome


Hint: Try to expand the exponential function as a power sum.


You will need the following formula for inverting the powers of s.

$$\mathcal{L}^{-1}\left[ s^\alpha\right]=\frac{t^{-\alpha-1}}{\Gamma(-\alpha)}$$

Where $\Gamma$ is the Gamma function.

  • $\begingroup$ Thanks for your hint. Frankly speaking, I am not a math person. I just came across this problem and do not know how to solve this. Would you please give me some solution. $\endgroup$ Mar 30 '16 at 8:31
  • $\begingroup$ The solution is an infinite series. You just expand the series and apply the inverse Laplace transform for every individual term. $\endgroup$
    – MrYouMath
    Mar 30 '16 at 8:34
  • $\begingroup$ Should there not be something else added to your final formula...perhaps an $\mathcal{L}^{-1}$? $\endgroup$ Mar 30 '16 at 9:30
  • 1
    $\begingroup$ Yeah, you are right :D. Btw how did you do the curly L? $\endgroup$
    – MrYouMath
    Mar 30 '16 at 12:05
  • 2
    $\begingroup$ \mathcal{L} is how $\endgroup$
    – Moo
    Mar 30 '16 at 12:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.