Dear MOs,

Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function

$$ W_{a,b} (x)= e^{-2 b x}\left( \Phi^2\left(\frac{a b -x}{\sqrt{a}}\right)-\Phi\left(\frac{2 a b -x}{\sqrt{a}}\right)\right), $$

where

$$ \Phi(x): = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2} d y = \frac{1}{2} Erf\left(\frac{x}{\sqrt{2}}+1\right). $$

The question is whether the following equation has only three zeors: $x=0$ and $x=\pm\infty$:

$$ W_{a,b}(x) = W_{a,b}(-x). $$

Plotting the function $W_{a,b}(x)$ suggests that the answer is right. But to find a proof seems quite hard.

Here are some graphs of the functions: $W_{1,-1}(x)$ (the red one), $W_{1,-1}(-x)$ (the blue one), and $W_{1,-1}(x)-W_{1,-1}(-x)$ (the one crossing origin).

alt text http://s11.postimg.org/7o2swl3hv/E_Square.jpg

Thank you very much for any suggestions!

Anand