Integrating with Gamma: cos(theta)^(2k+1)

[Elliptic]

Homework Statement

Solve the integral annd express it through the gamma f

Homework Equations

cos(theta)^(2k+1)

The Attempt at a Solution

 

[Simon Bridge]

You mean:
\int_0^{\frac{\pi}{2}} \cos^{2k+1}(\theta)d\theta... eg: evaluate the definite integral of an arbitrary odd-power of cosine.

The standard approach is to start by integrating by parts.
You'll end up with a reducing formula which you can turn into a ratio of factorials - apply the limits - after which it is a matter of relating that to the factorial form of the gamma function.

eg. http://mathworld.wolfram.com/CosineIntegral.html

 

[Elliptic]

You mean:
\int_0^{\frac{\pi}{2}} \cos^{2k+1}(\theta)d\theta... eg: evaluate the definite integral of an arbitrary odd-power of cosine.

The standard approach is to start by integrating by parts.
You'll end up with a reducing formula which you can turn into a ratio of factorials - apply the limits - after which it is a matter of relating that to the factorial form of the gamma function.

eg. http://mathworld.wolfram.com/CosineIntegral.html

Its difficult do see what is happening here.

 

[Simon Bridge]

If it was easy there'd be no point setting it as a problem.
I'm not going to do it for you ...

Do you know what a gamma function is? You can represent it as a factorial?
Can you identify where you are having trouble seeing what is going on?
Perhaps you should try to do the derivation for yourself?

 

[Elliptic]

If it was easy there'd be no point setting it as a problem.
I'm not going to do it for you ...

Do you know what a gamma function is? You can represent it as a factorial?
Can you identify where you are having trouble seeing what is going on?
Perhaps you should try to do the derivation for yourself?

Thanks.

 

[Simon Bridge]

Really? And I thought I was being mean...

The trig-form of the beta function aye - yep, that's a tad more elegant that the path I was suggesting before (the more usual one)... but relies on a hand-wave: do you know how the beta function is derived?

Also - you have \frac{1}{2}B(\frac{1}{2},k+1) but you've spotted that.

If you look at the cosine formula - you have to evaluate the limits ... at first it looks grim because it gives you a sum of terms like \sin\theta\cos^{2k}\theta which is zero at both limits ... unless k=0 ... which is the first term in the sum, which is 1.

After that it is a matter of subbing in the factorial representation of the gamma function.
Which would be a concrete proof.

Yours is shorter and if you have the beta function in class notes then you should be fine using it.