# Integral of the product of two error functions

I would like to know whether this integral could be solved analytically

$\int_{-\frac{L}{2}}^{\frac{L}{2}} \text{erf}\left(\frac{a-\text{x}}{k}\right) \text{erf}\left(\frac{L+2 \text{x}}{2 k}\right) \, d\text{x}$

where $\text{erf}(\cdot)$ is the error function, k and L are constants, and $a\in[-L/2,L/2]$.

Edit: I think this integral can be written in the form

$\int_{u}^{v} \text{erf}\left(\frac{z-\text{x}}{k}\right) \text{erf}\left(\frac{z+ \text{x}}{k}\right) \, d\text{x}$

• Have you tried using the definition of error function and changing the order of integration? – Yuriy S Jun 23 '17 at 12:37
• Yes, I also tried with Mathematica without success. (it seems that Mathematica is not very efficient in evaluating integrals involving erf, even when the analytical solution is known to exist) – m137 Jun 23 '17 at 12:46
• Ok, I'll see what I can do. Meanwhile, $k$ has no meaning here, you can get rid of it easily – Yuriy S Jun 23 '17 at 12:47