Great Fun: Complex Proof using de Moivre's Theorem

[benjyk]

Homework Statement

Show that
$$\int_{0}^{\pi} {\cos}^{2n}\theta d\theta = \frac{(2n)!\pi}{{2}^{2n}{(n!)}^{2}}$$

Homework Equations

cos n*theta = (z^n + z^-n)/2

The Attempt at a Solution

Ok, this one has kept me busy for a while. I started using de Moivre's theorem to express cos^2n(theta) in terms of multiple angles but got stuck on the integral.
After a while I tried to prove it inductively; showing it true for n=1 but then failed to show it for n=k+1.
If someone could suggest a novel way to start this problem I would be very grateful.

Benjy

 

[Avodyne]

1) First notice that the cosine is an even function, so if you integrate over -pi to +pi, the answer is double what you want.

2) cos theta = (z + z^-1)/2, z=e^(i theta)

3) What is the integral from -pi to +pi of z^n where n is an integer (possibly negative or zero)?

 

[benjyk]

Thanks for your reply Avodyne.

To solve it, I expressed cos^2n(theta) as (z + z^-1)^2n. This expanded to yield:
(cos(2n) + 2nC1*cos(2n-2) + ... + (2n)!/(n1)^2) / 2^2n

When integrated, all the cos's become sin's. The sine of zero or any integer multiple of 2pi will cancell, therefore the only term remaining will be (2n)!/(n1)^2) / 2^2n.

Retrospectively it seems rather easy, although it did keep me busy for a while.

 

[Avodyne]

You could also notice that $\int_{-\pi}^{+\pi} e^{in\theta} d\theta$ is zero unless n=0, in which case it is 2pi. This skips the step of assembling the z's into cosines.

Problems are always easier in retrospect ...

 

[benjyk]

Yes they do :)
Thanks again for the help.