Dear MOs,

I need to calculate the inverse Laplace transform of the following function 

$$
g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0.
$$

I have checked, among many others, the book ["Table of Integral Transform, Vol. I"][1]. In P.267, Eq. (14) is for 

$$
g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}}
$$

which is almost what I need. Other than this formula, I didn't find the one that I need. I have tried mathematica, which couldn't give an answer. I think the hope to find out the solution is quite small.  

EIDT: here is some motivation of the problem.

Suppose the inverse transform gives us a function $f_a(t)$. I want to see the limit $$\lim_{a\rightarrow 0_+} f_a(t)=?$$

Can I simply do this:

$$
\lim_{a \rightarrow 0_+} \mathcal{L}^{-1}\left(g_a\right)(t) \stackrel{?}{=}  \mathcal{L}^{-1}\left(\lim_{a\rightarrow 0_+} g_a\right)(t) = 
\mathcal{L}^{-1}\left(g_0\right)(t) =\frac{1}{\sqrt{\pi t}} +  2 e^{4t} \text{erfc}(-2\sqrt{t})
$$



Thank you very much for any hints!

Anand




  [1]: http://books.google.ch/books?id=IPhQAAAAMAAJ&q=Table+of+Integral+Transforms,+Vol.+I&dq=Table+of+Integral+Transforms,+Vol.+I&hl=en&ei=U0_KT7yxNMv64QSSlXA&sa=X&oi=book_result&ct=book-thumbnail&redir_esc=y