Questions on the Fresnel integrals.

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Compute Erf(z) using Fresnel integrals

I have to compute $\operatorname{erf}(z)$ using the Fresnel integrals. I have the relation: $$C(z)+iS(z)=\frac{1+i}{2}\operatorname{erf}\left[ \frac{\sqrt{\pi}}{2}(1-i)z \right].$$ But ...
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How to solve $\int_0^\infty \sin(x^2)\,{\rm d}x$

I was trying to solve this Fresnel integral using Feynman integration(differentiation under the integral sign) $$\int_{-\infty}^\infty \sin(x^2)\,{\rm d}x$$ So we define, $$F(b)= \int_0^\infty ...