I am reading a text and I am curious to know how certain approximations were reached.
The first function approximations is: $ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx \frac{y^2}{2x^2 (1-p^2)}$
,when $y \ll x$. Note that I tried using the approximation $e^x \approx 1+x$, when x is small, but all I got was the conclusion that $1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx 0$.
The second function approximation is: $ 1-e^{\frac{-y}{x}}(1-Q(a,b)+Q(b,a)) \approx \frac{y^2}{x^2 (1-p^2)}$
,when $y \ll x$, where $Q(a,b) = \int_b^\infty e^{-\frac12 (a^2 + u^2)} I_0(au) u \, du$, $b = \sqrt{\frac{2y}{x(1-p^2)}}$, $a = bp$, $I_0$ is a modified Bessel Function of the first kind. It is also a known fact that $ Q(b,0) = 1$ and $ Q(0,b) = e^{-\frac{b^2}{2}}$ I have tried to assume $ a = 0$, since $a = bp$ and $b$ is small, $ p$ is a number between 0 and 1. However, it is not clear how $(1-p^2)$ is in the denominator and not $(1-p^2)^2 $, which would be closer to the traditional $e^x$ approximation.
Any hints on how these approximations were derived would be appreciated. Thanks.