Inverse Laplace Transformation of arctan (s/2)

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Discussion Overview

The discussion revolves around the Inverse Laplace Transformation of the function arctan(s/2). Participants explore methods and challenges related to this transformation, noting that it was not explicitly covered in class. The scope includes theoretical understanding and mathematical reasoning.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in starting the Inverse Laplace Transformation of arctan(s/2) and seeks guidance.
  • Another participant mentions that the derivative of arctan(t) is 1/(t^2+1) and suggests that differentiation in the time-domain corresponds to multiplication by s in the Laplace-domain, referencing a Wikipedia article for further information.
  • A participant acknowledges the help received but expresses confusion regarding the relationship between differentiation in the time-domain and multiplication by s in the Laplace-domain.
  • Another participant introduces the Bromwich integral as a classical method for computing the inverse Laplace transform, noting that applying it to arctan(s/2) may involve complex calculus and special functions.
  • This participant also points out that the inverse Laplace Transform of arctan(s) does not appear in some extended tables, raising questions about the existence of a solution expressible in simple terms.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a straightforward solution for the inverse Laplace transformation of arctan(s/2), and multiple viewpoints regarding methods and challenges remain present.

Contextual Notes

There are limitations regarding the assumptions made about the applicability of standard functions and the complexity of the calculus involved in the Bromwich integral method. The discussion also highlights the potential absence of solutions in extended tables.

nileszoso
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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
 
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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.
 
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.
 
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 ?
 

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