Why erf(a-b)+erf(a)+erf(a+b) is so close to 3erf(a)?

Question

I am approximating an empirical distribution function with a sum of three gaussians, and noticed that for erf function $erf(a-b)+erf(a)+erf(a+b)$ is numerically very close to $3erf(a)$ for applicable values of $a$ and $b$, $a>>b$. Close, but not quite the same, of course. Could you suggest an analytical explanation why these functions are so close?

Answer 1

Assuming you mean close to $3\, \text{erf}(a)$, set $b=x\, a$ and perform a Taylor expansion at $x=0$.

You should arrive to $$\text{erf}(a-a x)+\text{erf}(a )+\text{erf}(a+ax)=3\, \text{erf}(a)-\frac{4 \left(a^3 e^{-a^2}\right) x^2}{\sqrt{\pi }}+O\left(x^4\right)$$ or, if you prefer, $$\text{erf}(a-b)+\text{erf}(a )+\text{erf}(a+b)=3\,\text{erf}(a)-\frac{4 \left(a e^{-a^2}\right) b^2}{\sqrt{\pi }}+O\left(b^4\right)$$ I suppose that this is clarifying.