Transform Fresnel integrals into each other

Question

Let $S,C$ be given by $$ S = \int _{0}^{\infty} \sin(x^2)\,dx,\,\,C = \int _{0}^{\infty} \cos(x^2)\,dx $$I know you can show they're equal to each other using complex contour integration, and I've seen the posts on here using partial fractions and the like. I'm looking for a transformation, something like $x=f(z)$ to turn $S$ into $C$ or vice versa.

Answer 1

I think I can explain it. Start with $$ \Gamma(z) = \int_{0}^{\infty}y^{z-1}e^{-y}\,dy $$Using Euler's formula, we have $$ C + i S = \int_{0}^{\infty} e^{i t^2}\,dt $$ $$ =\frac{e^{\pi i/(4)}}{2}\int _{0}^{\infty}s^{1/2-1} e^{-s}\,ds $$ $$ = \frac{e^{\pi i/(4)}}{2}\Gamma(1/2)= {e^{\pi i/(4)}}\Gamma(3/2) $$ This recovers $\displaystyle{S=C=\sqrt{\frac{\pi}{8}}}$. In fact, this is a general result: with $T\in \{\sin,\cos\}$ and $\Re(n)>1$, we have $$ \Gamma(1+1/n)T(\pi/(2n))= \int _{0}^{\infty} T(t^n)\,dt $$