\[S\left(z\right)=-\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}%
\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}+\sin\left(\tfrac{1}{2}%
\pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1%
)}z^{4n+1}.\]