# inordinatum

Physics and Mathematics of Disordered Systems

## 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 $P(x)$ of the following Laplace transform:

$\displaystyle \int_0^\infty \mathrm{d}x\, e^{-sx}P(x) = \frac{\tanh \sqrt{2s}}{\sqrt{2s}}$.   (1)

The solution is given in terms of the Jacobi theta function $\theta_2$ (which can be re-expressed in terms of Jacobi elliptic functions, though I won’t discuss that here in detail):

$\displaystyle P(x) = \frac{1}{2}\theta_2 \left(0; e^{-x\pi^2/2}\right)$.   (2)

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

$\displaystyle \int_0^\infty \mathrm{d}x\, e^{-sx}\frac{1}{2x}\theta_2 \left(0; e^{-x\pi^2/2}\right) = \ln \cosh \sqrt{2s}$.

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:

$\displaystyle \begin{array}{rl} \sinh z &= z \prod_{n=1}^{\infty}\left(1+\frac{z^2}{\pi^2 n^2}\right) \\ \cosh z &= \prod_{n=1}^{\infty}\left(1+\frac{z^2}{\pi^2 \left(n-\frac{1}{2}\right)^2}\right) \end{array}$.

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

$\displaystyle \begin{array}{rl} \tanh z = & \partial_z \ln \cosh z = \partial_z \sum_{n=1}^{\infty} \ln \left(1+\frac{z^2}{\pi^2 \left(n-\frac{1}{2}\right)^2}\right) \\ = & \sum_{n=1}^{\infty} \frac{\frac{2z}{\pi^2 \left(n-\frac{1}{2}\right)^2}}{1+\frac{z^2}{\pi^2 \left(n-\frac{1}{2}\right)^2}} \\ = & 2z \sum_{n=1}^{\infty} \frac{1}{\frac{\pi^2(2n-1)^2}{4}+z^2}. \end{array}$.

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

$\displaystyle \int_0^\infty \mathrm{d}x\, e^{-sx}P(x) = \frac{\tanh \sqrt{2s}}{\sqrt{2s}} = 2\sum_{n=1}^{\infty} \frac{1}{\frac{\pi^2(2n-1)^2}{4}+2s}$

The Laplace inverse of a simple pole is just an exponential, $\int_0^\infty \mathrm{d}x\, e^{-sx}\,e^{-p x}=\frac{1}{p+s}$. By linearity of the Laplace transform, we can invert each summand individually, and obtain an infinite sum representation for $P(x)$:

$\displaystyle P(x) = \sum_{n=1}^{\infty} \exp\left(-\frac{\pi^2(2n-1)^2}{8}x\right)$

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 $z=0 \Leftrightarrow w=1$ and $q=e^{-\pi^2 x/2}$. This finally gives the solution as claimed above:

$\displaystyle P(x) = \frac{1}{2}\theta_2 \left(0; e^{-x\pi^2/2}\right)$.   (2)

Have fun!

Written by inordinatum

April 15, 2013 at 10:27 pm

Posted in Maths, Technical Tricks