# Laplace inverse of a particular function - trouble following text (Carslaw & Jaeger)

1. Feb 27, 2009

### FiberOptix

Hello all,

I've been following along in Carslaw & Jaeger's book on heat conduction. Section 14.7 p. 366 we're presented with the function:

$$\overline{v} = \frac{sinh(qr)sinhq(a-r')}{4\pi r\cdot r' \cdot k \cdot q sinh(qa)}$$

It's simply stated that the "Inversion Theorem" is used to do the inverse laplace transform to obtain v as:

$$v = \frac{1}{2\pi\cdot a \cdot r \cdot r'} \Sigma_{n = 1}^{\infty} e^{-k\cdot n^2 \cdot \pi^2 \cdot t / a^2}\cdot sin(\frac{n\cdot\pi\cdot r}{a})\cdot sin(\frac{n\cdot\pi\cdot r'}{a})$$

I'm having trouble seeing how this works out. The reason why I ask is because I'm trying to solve a related problem that is of significant importance to me for my own research.

Any help is very greatly appreciated.