# Inverse Laplace Transform to arrive at Error Function

I hope this question is not redundant and allowed, but I couldn't find the answer via search. I'm reading a paper where the authors perform an inverse Laplace transform of the following:

$$C_{0}exp(\gamma )\times L^{-1}\left ( \frac{exp\left ( -\sqrt{\alpha +\beta s} \right )}{s+k}\right)+C_{0} L^{-1}\left ( \frac{1}{s+k} \right )$$

to arrive at this result:

$$-\frac{C_{0}}{2} exp(-kt)erfc\left ( \frac{\beta - 2\gamma t}{2\sqrt{\beta t}} \right ) - \frac{C_{0}}{2} exp(2\gamma -kt)erfc\left ( \frac{\beta + 2\gamma t}{2\sqrt{\beta t}} \right ) + C_{0}exp(-kt)$$

When deriving this solution, the authors mention using an inverse Laplace transform formula (Abramowitz and Stegun, 1970). I went looking for the specific formula and the closest I found seems to be this: $$L^{-1}\left (\frac{e^{-x\sqrt{p/\kappa}}}{p} \right ) = erfc\left ( \frac{x}{2\sqrt{\kappa t}} \right )$$

Can anyone provide some insight? Any and all help is appreciated!

Here's the link to the paper: http://jast.modares.ac.ir/article_5035_6c0982d81fc1c899f700ea96e88f2b24.pdf

• Can you edit the question, with the link of the paper?! – Jan Feb 9 '17 at 12:05
• Will do! Give me a quick second. – WazzaGunner Feb 9 '17 at 12:25