1. ## Integration question

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?

2. Hint

Separate real and imaginary parts. With an adequate substitution you'll obtain the Fresnel integrals.

3. In M.R. Spiegel - Laplace Transforms - Mc Grow-Hill 1965 You can read...

$\displaystyle \mathcal{L} \{e^{-a^{2}\ t^{2}}\} = \int_{0}^{\infty} e^{-a^{2}\ t^{2}}\ e^{-s\ t}\ dt = \frac{\sqrt{\pi}}{2\ a}\ e^{\frac{s^{2}}{4\ a^{2}}}\ erfc (\frac{s}{2a})$ (1)

Setting in (1) $a^{2}= 2\ \pi\ \frac{i}{m}$ and $s=0$ with little computation You [should] obtain...

$\displaystyle \int_{0}^{\infty} e^{-2\ \pi\ i\ \frac{t^{2}}{m}}\ dt = \frac{\sqrt{m}}{4}\ (1-i)$ (2)

Kind regards

$\chi$ $\sigma$

4. Let us see if M.R. Spiegel is right

We have

$I:=\displaystyle\int_0^{+\infty}e^{-2\pi iy^2/m}dy=\displaystyle\int_0^{+\infty}\cos (2\pi y^2/m)dy-i\displaystyle\int_0^{+\infty}\sin (2\pi y^2/m) dy$

using $x=\sqrt{2\pi/m}\;y$ :

$I=\sqrt{m/2\pi}(S(x)-iC(x))$

where $S(x)=C(x)=\sqrt{\pi/8}$ are the Fresnel integrals. So,

$I=\sqrt{m}(1-i)/4$

5. Thanks for this.