Integrating Fresnel Integrals with Cauchy Theorem?

In regards to the above proof, I'm a little confused as to how the last conclusion was made --

How does the fact that

$$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}$$

to conclude that:

$$\int_0^{\infty}\cos{x^2} + i\sin{x^2}dx = \int_0^{\infty}e^{ix^2}dx = \frac{\sqrt{2\pi}}{4} + i\frac{\sqrt{2\pi}}{4}?$$

• They're using the $\sqrt{\pi}$ equality to finish the computation of the integral on the line above. – Josh Keneda Sep 15 '14 at 1:22
• $\int_{\infty}^{\infty}$ should be $\int_{\color{red}{-}\infty}^{\infty}$ – mike Sep 15 '14 at 1:23
• So $ie^{i\pi / 4}\int_R^0 e^{-u^2}(-1/R)du = (-1/R)ie^{i\pi / 4} \int_R^0 \sqrt{\pi} du = \pi i e^{i\pi / 4}$, if I've done that correctly. How are the $\frac{\sqrt{2\pi}}{4}$ terms derived? – Ryan Yu Sep 15 '14 at 1:47
• Be careful: they're not saying that $e^{-u^2} = \sqrt{\pi}$. They're saying that the integral of $e^{-u^2}$ over the real line is $\sqrt{\pi}$. I've posted a full answer below. – Josh Keneda Sep 15 '14 at 3:20
• possible duplicate of Some way to integrate $\sin(x^2)$? – leo Jul 23 '15 at 16:11

There's a typo in their parametrization of $\gamma_3$. They have $dz = (-iRe^{i \frac{\pi}{4}}) dt$ instead of $dz = (-Re^{i \frac{\pi}{4}}) dt$. With this correction, their line after showing that the integral over $\gamma_2$ drops out should read: $$\lim_{R\rightarrow \infty} \int_0^R e^{iz^2} dz = \lim_{R\rightarrow \infty} R e^{i \pi/4} \int_0^1 e^{i(Re^{i \pi/4}(1-t))^2} dt. \quad\quad\quad (*)$$
Now, if we do their $u$-substitution, the right hand side becomes \begin{align*}\lim_{R \rightarrow \infty} R e^{i\pi/4} \int_R^0 e^{-u^2} \frac{-1}{R} du &= \lim_{R\rightarrow \infty} R e^{i\pi/4} \int_0^R e^{-u^2} \frac{1}{R} du\\ &= \lim_{R \rightarrow \infty} e^{i \pi/4} \int_0^R e^{-u^2} du\\ &= e^{i\pi/4} \int_0^\infty e^{-u^2} du\\&=e^{i\pi/4}\frac{\sqrt{\pi}}{2},\end{align*}
where the last equality follows from the fact that $e^{-u^2}$ is even and so $$\int_0^\infty e^{-u^2} du = \frac{1}{2}\int_{-\infty}^\infty e^{-u^2} du = \frac{1}{2} \sqrt{\pi}.$$
Combining this result with $(*)$, we have $$\int_0^\infty e^{iz^2} dz = e^{i\pi/4}\frac{\sqrt{\pi}}{2} = \frac{\sqrt{2\pi}}{4}+ i \frac{\sqrt{2\pi}}{4}.$$
• Not sure if you'll see this, but a brief followup -- how does this proof show that $\int_0^{\infty}\sin{x^2}dx = \frac{\sqrt{2\pi}}{4}$? Doesn't it only show that $\int_0^{\infty}\cos{x^2}dx = \frac{\sqrt{2\pi}}{4}$? (i.e. $\cos{x^2}$ is exactly the real part of $e^{iz^2}$?) – Ryan Yu Sep 16 '14 at 7:12
• Similarly, $\sin{x^2}$ is exactly the imaginary part of $e^{iz^2}$ on the real line, so taking imaginary parts gives us the $\sin{x^2}$ integral. – Josh Keneda Sep 16 '14 at 7:21