What is the relationship between cos(cosx) and Bessel functions?

[Emmanuel_Euler]

Hi everyone, my friend challenged me to solve this definite integral...integral from -2pi to 2pi ((sin(2sinx)+cos(2cosx))dx, i proved by using definite integral properties that this integral equals to integral from -2pi to 2pi cos(2cosx)dx, can you give me any ideas how to solve this?? I know that i can use the contour integration but how??

 

[scottdave]

Note that cosine oscillates between -1 and 1, but you are taking cosine of 2cos, so you are taking cosine of a function which oscillates between -2 and +2. Since cosine is even, cos(x) is the same as cos(-x) so this new function has a period of pi, rather than 2pi. Analytically how to integrate cosine of cosine, I am not sure. You could do it numerically, for sure.

 

[scottdave]

Since it still has the shape of a cosine wave, just shifted up and scaled down, you should be able to rewrite it as something like A*cos(2*x) + B, where A and B are constants to be determined (by the max and min), then integrate that over 1 period, then do for how many periods in your range.

 

[MAGNIBORO]

$$I = -4 \pi + 8 \int_{0}^{\pi}cos^2(cos(x))\, dx$$
and with Diff Under the Integral Sign we can get the ODE
$$\frac{\partial }{\partial a}\left ( \int_{0}^{\pi}cos^2(cos(ax))\, dx \right )=\frac{1-\pi\, sin^2(cos(a\pi))}{a}\, -\, \frac{1}{a} \int_{0}^{\pi}cos^2(cos(ax))\, dx$$
then you can solve for
$$\int_{0}^{\pi}cos^2(cos(ax))\, dx$$
and set a=1

 

[Emmanuel_Euler]

Ammmmm, I understand what you did, but please if you have the final answer give it to me...and thanks

 

[MAGNIBORO]

Ammmmm, I understand what you did, but please if you have the final answer give it to me...and thanks

wolfram alpha is your friend

 

[DrClaude]

wolfram alpha is your friend

You made an error somewhere, since the answer is not the same as http://www.wolframalpha.com/input/?i=integral+-2pi+to+2pi+cos(2+cos(x))

 

[Emmanuel_Euler]

You made an error somewhere, since the answer is not the same as http://www.wolframalpha.com/input/?i=integral+-2pi+to+2pi+cos(2+cos(x))

No the answer is right, 8 integral from(0>pi) (cos(cosx))^2-4pi=wolframalpha answer.
Ammmm the answer of the previous integral is given in terms of bessel function...interesting!
So there's a relationship between them...

 

[scottdave]

I am not sure exactly where the cos²(cos(ax)) came from either. Cos(2u) is equal to cos²(u) - sin²(u) or 2cos²(u) - 1

 

[Dr Transport]

No the answer is right, 8 integral from(0>pi) (cos(cosx))^2-4pi=wolframalpha answer.
Ammmm the answer of the previous integral is given in terms of bessel function...interesting!
So there's a relationship between them...

\cos(\cos(x)) is the real part of e^{ikr} which is the definition of the Bessel function...