Proving the Inverse Laplace Transform of 1/(sqrt(s)+a)

Click For Summary
SUMMARY

The discussion focuses on the challenge of proving the inverse Laplace transform of the function 1/(sqrt(s)+a). The user employs Mathematica to derive a result expressed in terms of error functions but struggles to prove it analytically. The integral representation using the contour integral method is identified, specifically the integral from gamma - i∞ to gamma + i∞. The user suggests that expressing e^(st) as e^((sqrt(s)^2)*t) may facilitate integration and lead to a solution involving error functions.

PREREQUISITES
  • Understanding of inverse Laplace transforms
  • Familiarity with contour integration techniques
  • Knowledge of error functions and their properties
  • Proficiency in using Mathematica for symbolic computation
NEXT STEPS
  • Research the properties of error functions and their applications in inverse Laplace transforms
  • Explore contour integration methods in complex analysis
  • Learn how to manipulate expressions in Mathematica for symbolic integration
  • Study the relationship between exponential functions and error functions in integrals
USEFUL FOR

Mathematicians, engineers, and students involved in advanced calculus, particularly those working with Laplace transforms and complex analysis.

iiternal
Messages
13
Reaction score
0
Hi, all.
I am doing an inverse Laplace transform and meeting some difficulty.
InverseLaplaceTransform[1/(Sqrt+a)].
Using Mathematica I found the result, however, I failed to prove it. I know it should be like
\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \frac{1}{\sqrt{s}+a}ds
and I can take \gamma =0.
But, then, I cannot continue. The sqrt on the denominator killed me.
Thank you very much.
 
Physics news on Phys.org
From what I can tell, there is no good antiderivative of e^(st)/(Sqrt+a). If you can find one, I'd love to know.
 
timthereaper said:
From what I can tell, there is no good antiderivative of e^(st)/(Sqrt+a). If you can find one, I'd love to know.


Well, by using Mathematica, I got a result expressed in terms of error functions. So I guess I have to transform it into error functions. But, how?
 
Well, e^(st) can possibly be expressed as e^((Sqrt^2)*t), which when integrated with respect to s is similar to the error function. That might be the way to attack this.
 

Similar threads

Undergrad Why is the heaviside function in the inverse Laplace transform of 1?
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Inverse Laplace transform of a rational function
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)
  • · Replies 7 ·
Replies
7
Views
4K
MHB Solving PDE using laplace transforms
  • · Replies 4 ·
Replies
4
Views
3K
MHB Confused about Laplace and Inverse Laplace Transform of Various Functions?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad On convolution theorem of Laplace transform: Schiff
  • · Replies 4 ·
Replies
4
Views
2K
What is the Inverse Laplace Transform of e^(-sx^2/2)?
  • · Replies 10 ·
Replies
10
Views
3K
MHB Simplifying the inverse Laplace Transform using the inverse shift formula
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Inverse Laplace transform. Bromwitch integral
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Laplace transform using differential equations
  • · Replies 2 ·
Replies
2
Views
2K