# Evaluating Fresnel Integrals

1. Feb 1, 2016

### iggyonphysics

1. The problem statement, all variables and given/known data
Evaluate the following integrals C = 0inf∫cos(x2) dx and S = 0inf∫sin(x2) dx

2. Relevant equations

Hint: use Euler formula to write the integral for F = C + iS. Square the integral and evaluate it in polar coordinates. Temporary add a convergence factor.

Answer: C = S = sqrt(pi/8)

3. The attempt at a solution

(Abbreviated form)

F2 = 1/4 0inf∫ eix^2dx 0inf∫eiy^2dy

F2 = pi/2 0inf∫ eir^2 r dr

F2 = pi/4 0inf∫eiudu

Now I think the convergence factor comes in here, but I am not entirely sure how that works.

Thanks!

Last edited: Feb 1, 2016
2. Feb 1, 2016

### Ray Vickson

You can look at
$$J_r = \int_0^{\infty} e^{-ru} e^{iu} \, du,$$
where $r > 0$. Then take the limit as $r \to 0$.

3. Feb 1, 2016

### iggyonphysics

Great, so that just comes to i, correct?

So,

F^2=π/4 eiπ/2 --> F = ei/4sqrt(π/4)

How do I get to S and C from here? (I know if I evaluate the product of sqrt(π/4) and sin(π/4)2 it spits out sqrt(π/8). Why does this work?)