This question is cross posted on Physics Stack Exchange https://physics.stackexchange.com/questions/568448/monotonicity-of-sum-of-two-fresnel-integrals
I am studying single-knife edge diffraction of radio waves and came across the following problem, which I could not solve.
I need to show that the function $F$ is monotone decreasing for $x>x_0$, where $x_0\sim-1.2171$. For my purposes it would suffice to show it for $x_0\geq0$, where $F$ is plotted below and given by
$$ F(x)=\frac{|\frac{1-i}{2}-C(x)+iS(x)|}{\sqrt{2}}, $$where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\, dz,\quad C(x)=\int_0^x\cos\frac{\pi z^2}{2}\, dz,$$ are the Fresnel Integrals.
Any ideas?