Approximation of $\Phi (p)$

Question

I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe Hulle von n zufaillig gewahlten Punkten they use the relation $$ \varPhi(p) = 1 - \frac{e^{\frac{-p^2}{2}}}{\sqrt{2\pi}p\left(1 + \frac{\theta_p}{p^2}\right)} $$ for $p>1$, where $0< \theta_p < 1$. Can anyone help me understand where this second equation comes from?

Answer 1

Equality $$ \varPhi(p) = 1 - \frac{e^{\frac{-p^2}{2}}}{\sqrt{2\pi}p\left(1 + \frac{\theta_p}{p^2}\right)} $$ for some $\theta_p\in(0,1)$ can be rewritten as $$r_2(p):=\frac1{p+1/p}<r(p):=\frac{1-\varPhi(p)}{e^{-p^2/2}/\sqrt{2\pi}}<r_1(p):=\frac1p,\tag{1}$$ which actually holds for all real $p>0$ and is a special case of a well-known result -- see e.g. formula (1.8) (with $m=1$) in this paper.