## Tricks for inverting a Laplace Transform, part V: Pole Decomposition

*This is a continuation of my articles on methods for inverting Laplace transforms. You can find the previous parts here: part I (guesses based on series expansions), part II (products and convolutions), part III, part IV (substitutions).*

## 1. Result

In this post I will explain how to find the inverse of the following Laplace transform:

. (1)

The solution is given in terms of the Jacobi theta function (which can be re-expressed in terms of Jacobi elliptic functions, though I won’t discuss that here in detail):

. (2)

By taking derivatives or integrals with respect to , which are equivalent to multiplication or division of by , one can obtain many other Laplace transform identities, including for example

.

If anyone manages to find a systematic list of those, I’d be very grateful. But for now let’s just see how one obtains (2).

## 2. Derivation

First, we use the classic decomposition of trigonometric functions in infinite products due to Euler:

.

From the second identity, we can obtain a partial fraction decomposition of (following this post on StackExchange):

.

Applying this to the right-hand side of (1), we obtain a sum over simple poles:

The Laplace inverse of a simple pole is just an exponential, . By linearity of the Laplace transform, we can invert each summand individually, and obtain an infinite sum representation for :

This sum can now be evaluated with `Mathematica`

‘s `Sum`

command, or by hand using the representation of theta functions in terms of the nome, for argument and . This finally gives the solution as claimed above:

. (2)

Have fun!

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