Inverse Laplace Transformation of arctan (s/2)

[nileszoso]

The Title pretty much says it all. I'm trying to learn how to solve the Inverse Laplace Transformation of Arctan(s/2). An equation of this sort was not explicitly covered in class and I'm having difficulty figuring where to start to solve it. If anyone could give me a general idea that would point me in the right direction that would be greatly appreciated.

Thanks in advance

 

[keyns]

The derivative of arctan(t)=1/(t^2+1). Differentiation in time-domain is the same as multiplication by s in Laplace-domain. See http://en.wikipedia.org/wiki/Laplace_transform for a table. You should be able to derive the correct answer from this.

 

[nileszoso]

Thank you! This helped, although I'm still not sure what exactly is meant by "Differentiation in time-domain is the same as multiplication by s in Laplace-domain. " But like I said this got my started on the way to solve the problem.

 

[JJacquelin]

Hi nileszoso!

A classical method to compute the inverse Laplace transform is the use of the Bromwich integral. In the case of arctan(s/2) this leads to ardous calculus with special functions.
Moreover, the inverse Laplace Transform of arctan(s) doesn't appears in some extended tables. So, it is questionable whether there is really a solution which can be expressed with not too complicated combinations of standard functions.
Where this problem is coming from ?

 

[blue_raver22]

Hello,

The inverse Laplace transformation of arctan(s/2) can be solved by using the properties of Laplace transforms and the inverse Laplace transform. One possible approach is to use the Laplace transform of arctan(x) which is given by 1/sqrt(s^2 + 1). Using this, we can rewrite arctan(s/2) as 1/2 * arctan(s/2) = 1/2 * arctan(s/2) * 1/2 * 2 = 1/2 * arctan(s/2) * 1/2 * 1/2 * 4 = 1/2 * arctan(s/2) * 1/2 * 1/2 * 1/2 * 8 = 1/2 * arctan(s/2) * 1/2 * 1/2 * 1/2 * 1/2 * 16 = ... = 1/2^n * arctan(s/2) * 1/2^n * 1/2^n * ... * 1/2^n * 2^n = 1/2^n * arctan(s/2) * 2^n.

Using the inverse Laplace transform property, we can now find the inverse Laplace transform of 1/2^n * arctan(s/2) * 2^n which is given by 1/2^n * arctan(t/2^n) * 2^n. Therefore, the inverse Laplace transform of arctan(s/2) is equal to 1/2^n * arctan(t/2^n) * 2^n.

I hope this helps point you in the right direction. Remember to always check your answer by taking the Laplace transform of your solution to ensure it matches the original function. Good luck with your studies!