I have a Laplace transformed function that I'd like to transform back. It's quite a complex function however, which is why I am stuck:
$$C(x,s) = \left(\frac{m}{s}\right)\exp\left(\lambda^2\frac{\left(v-\sqrt{\frac{4 D s}{\lambda^2} - v^2}\right)}{2 D} x\right)$$
I know the inverse Laplace transform of a less complex, but comparable, function:
$$C(x,s) = \left(\frac{m}{s}\right) \exp\left(\frac{s}{k} x\right)$$
The solution of this would be (derived from results described in the more extensive Laplace transform tables):
$$C(x,t) = m \left(1 - \text{erf}\left(\frac{x}{\sqrt{4 k t}}\right)\right)$$
However, with the more difficult function it won't be that easy. I was trying to rewrite (split and simplify) the power term in order to make it simpler, but it doesn't really work.
I am new to the field of Laplace Transforms, so I am quite blind to the directions I could go. Does any of you have an idea on where to start (e.g., which method to use) to solve it?
I'd prefer to solve it by hand, but I am also interested on how to solve this with, e.g., Matlab. I tried to following piece of code, but it just returns my input:
syms c s k real
F = (c/s)*exp(-x*(sqrt(s/k)))
ilaplace(F,s)