So math stack exchange isn't really helping much with this. 
So initially, I'm proving the inverse laplace transform using contour integration.
This is a good starting point for my research when I eventually need to find the inverse laplace transform when the functions could not be found in tables.
I need to prove that:
$$\DeclareMathOperator{\erfc}{erfc}
\mathscr{L}^{-1} \bigg[ \frac{1}{s}\big(\exp (- \sqrt{s/k}x)\big) \bigg] = \erfc\left(\frac{x}{2\sqrt{kt}}\right)
$$
This inverse laplace transform can be found using a table of Laplace Transforms.
Using the following contour:

![](https://i.sstatic.net/rnGle.png)
Source: https://tex.stackexchange.com/questions/269684/hankel-bromwich-contour-problem

Then, after considering all contributions of this contour to get:
$$
\mathscr{L}^{-1} \bigg[ \frac{1}{s}\big(\exp (- \sqrt{s/k}x)\big) \bigg] =
 1 - \frac{1}{\pi} \int_{0}^{\infty} \exp(-ut) \sin(\sqrt{u/k} x) \frac{du}{u}
$$
Here, we can simplify the integral by letting: $v^{2} = ut$ and $y = x/\sqrt{kt}$ to get:
$$
\mathscr{L}^{-1} \bigg[ \frac{1}{s}\big(\exp (- \sqrt{s/k}x)\big) \bigg] = 1 - \frac{2}{\pi} \int_{0}^{\infty} \exp(-v) \sin(yv) \frac{dv}{v}.
$$
How do I continue from here to eventually get to :
$$ 1 - \erf\left(\frac{y}{2}\right) = 1 - \erf\left(\frac{x}{2\sqrt{kt}}\right)
$$