Fresnel tail expansion ($C=A+P\sin {\lambda }^2-Q\cos {\lambda }^2$)

For $A=\sqrt{\pi /8}$, define real tail functions by $Q+iP=e^{-i{\lambda }^2}{\int }_{\lambda }^{\infty }{e}^{i{t}^2}dt$. Then $C=A+P\sin {\lambda }^2-Q\cos {\lambda }^2$ and $S=A-P\cos {\lambda }^2-Q\sin {\lambda }^2$. Repeated integration by parts giveswhere $(-1)!!=1$. These are asymptotic expansions, rather than convergent infinite sums at a fixed argument.