I'm trying to integrate the function exp((-2*pi*i*y^2)/m) dy from 0 to inf.
I've tried using the substitution u=y/sqrt(m) but just cannot do it.
Is this integral even possible to evaluate?
In M.R. Spiegel - Laplace Transforms - Mc Grow-Hill 1965 You can read...
$\displaystyle \displaystyle \mathcal{L} \{e^{-a^{2}\ t^{2}}\} = \int_{0}^{\infty} e^{-a^{2}\ t^{2}}\ e^{-s\ t}\ dt = \frac{\sqrt{\pi}}{2\ a}\ e^{\frac{s^{2}}{4\ a^{2}}}\ erfc (\frac{s}{2a}) $ (1)
Setting in (1) $\displaystyle a^{2}= 2\ \pi\ \frac{i}{m}$ and $\displaystyle s=0$ with little computation You [should] obtain...
$\displaystyle \displaystyle \int_{0}^{\infty} e^{-2\ \pi\ i\ \frac{t^{2}}{m}}\ dt = \frac{\sqrt{m}}{4}\ (1-i)$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Let us see if M.R. Spiegel is right
We have
$\displaystyle I:=\displaystyle\int_0^{+\infty}e^{-2\pi iy^2/m}dy=\displaystyle\int_0^{+\infty}\cos (2\pi y^2/m)dy-i\displaystyle\int_0^{+\infty}\sin (2\pi y^2/m) dy$
using $\displaystyle x=\sqrt{2\pi/m}\;y$ :
$\displaystyle I=\sqrt{m/2\pi}(S(x)-iC(x))$
where $\displaystyle S(x)=C(x)=\sqrt{\pi/8}$ are the Fresnel integrals. So,
$\displaystyle I=\sqrt{m}(1-i)/4$