The error function is defined as, $$\text{erf}\bigg(\frac{(ax-b)}{\sqrt{2}}\bigg) = \frac{2}{\sqrt{\pi}}\int_{0}^{\frac{(ax-b)}{\sqrt{2}}} e^{-t^{2}/2}dt$$. My question is how to expand the above function. Am I able to write it as, $$\text{erf}\bigg(\frac{(ax-b)}{\sqrt{2}}\bigg) = \text{erf}\bigg(\frac{ax}{\sqrt{2}}\bigg) + \text{some_value}$$. I did a transformation $u = \frac{t}{\sqrt{2}}$, but it didn't help.
Thank you in advance.
To answer your question directly...
\begin{align*} \text{erf}\left(c_1 x + c_2\right) &= \frac{2}{\sqrt{\pi}}\int_0^{c_1x+c_2}e^{-t^2/2}dt \\[1.2ex] &= \frac{2}{\sqrt{\pi}}\int_0^{c_1x}e^{-t^2/2}dt + \frac{2}{\sqrt{\pi}}\int_{c_1x}^{c_1x+c_2}e^{-t^2/2}dt \\[1.2ex] &= \text{erf}(c_1 x) + \frac{2}{\sqrt{\pi}}\int_{c_1x}^{c_1x+c_2}e^{-t^2/2}dt. \end{align*}
This implies that "some_value" is equal to $\frac{2}{\sqrt{\pi}}\int_{c_1x}^{c_1x+c_2}e^{-t^2/2}dt$, which unfortunately can't be simplified in any particularly useful way (at least that I can see).
In fact, if I was given this second term by itself, I would immediately re-write it as $\text{erf}(c_1x+c_2) - \text{erf}(c_1 x)$ which gets us right back to the original expression. Can you give more details on what you are hoping to accomplish?
