Evaluating Fresnel Integrals Using Euler Formula

[iggyonphysics]

Homework Statement

Evaluate the following integrals C = ₀^inf∫cos(x²) dx and S = ₀^inf∫sin(x²) dx

Homework Equations

[/B]
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)

The Attempt at a Solution

(Abbreviated form)

F² = 1/4 ₀^inf∫ e^{ix^2}dx ₀^inf∫e^{iy^2}dy

F² = pi/2 ₀^inf∫ e^{ir^2} r dr

F² = pi/4 ₀^inf∫e^iudu

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

Thanks!

 

[Ray Vickson]

Homework Statement

Evaluate the following integrals C = ₀^inf∫cos(x²) dx and S = ₀^inf∫sin(x²) dx

Homework Equations

[/B]
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)

The Attempt at a Solution

(Abbreviated form)

F² = 1/4 ₀^inf∫ e^{ix^2}dx ₀^inf∫e^{iy^2}dy

F² = pi/2 ₀^inf∫ e^{ir^2} r dr

F² = pi/4 ₀^inf∫e^iudu

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

I imagine I wind up with something like F²= i*pi/4 and F = sqrt(i*pi/4) and since S and C are equal, they must each equal sqrt(pi/8).

Thanks!

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

 

[iggyonphysics]

Great, so that just comes to i, correct?

So,

F^2=π/4 e^iπ/2 --> F = e^i/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?)