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

In summary, the conversation discusses difficulties with finding an antiderivative for a given function, specifically e^(st)/(Sqrt[s]+a). The speaker suggests using Mathematica to find a result in terms of error functions, but is unsure of how to proceed. Another speaker suggests expressing e^(st) in a different form to make it easier to integrate.
  • #1
iiternal
13
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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 

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

What is an Inverse Laplace Transform?

An Inverse Laplace Transform is a mathematical operation that takes a function in the Laplace domain and converts it back into its original form in the time domain.

How is an Inverse Laplace Transform calculated?

An Inverse Laplace Transform is calculated using a set of rules and formulas that involve complex numbers and integration. It can also be calculated using tables or software programs.

Why is the Inverse Laplace Transform important in science?

The Inverse Laplace Transform is important in science because it allows us to analyze and understand systems and phenomena in the time domain. It is used in various fields such as engineering, physics, and mathematics.

What are some common applications of the Inverse Laplace Transform?

The Inverse Laplace Transform is commonly used in electrical engineering to analyze circuit behavior and in control systems to study the response of a system. It is also used in physics to study the behavior of dynamic systems and in mathematics for solving differential equations.

What are the limitations of the Inverse Laplace Transform?

The Inverse Laplace Transform may not exist for every function in the Laplace domain. Additionally, it may be difficult to compute in certain cases and can lead to complex solutions. It also assumes that the system is linear and time-invariant, which may not always be the case in real-world situations.

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