# Integral involving erf x

a) $\int_0^{\infty} e^{-\xi^2} \frac{\sin 2 \xi y}{\xi} d \xi = \frac{\pi}{2} erf y$
b) $\int_0^{\infty} e^{-\xi^2} \sin 2 \xi y} d \xi = \frac{\sqrt{\pi}}{2} e^{-y^2} erf y$
where $erf x = \frac{2}{\sqrt{\pi}} \int_0^x e^{-\xi^2} d \xi$