# Inverse laplace transform for unique diffusion type problem

by Groundwater
Tags: diffusion, inverse, laplace, transform, type, unique
 PF Gold P: 162 The most difficult part if you try to do the inverse transform with integration in the complex plane will be this integral along the branch cut $$\int_0^{\infty } \frac{e^{-t \rho } \left(e^{-i x \sqrt{\frac{\rho }{d}}}-e^{i x \sqrt{\frac{\rho }{d}}}\right)}{\beta +(-\alpha -\rho )^2} \, d\rho$$ Maybe you can find that in a table somewhere. I couldn't even find/figure out the case where $\beta = 0$. If you can't find the one with non-zero beta, but perhaps can find $$\int_0^{\infty } \frac{e^{-t \rho } \left(e^{-i x \sqrt{\frac{\rho }{d}}}-e^{i x \sqrt{\frac{\rho }{d}}}\right)}{(-\alpha -\rho )^4} \, d\rho$$ Then if beta is small maybe you could expand as a series in beta?