Fourier expansion od cos( a sin x)

husseine57 · Feb 17, 2013

What is the Fourier series of the function f(x)=cos (a sin(x))?

_Eduardo_ · Feb 17, 2013

\[\cos(a \sin{x}) = J_0(a) + 2 \displaystyle\sum_{k=1}^{\infty} J_k(a)\cos(kx)\]

\[J_k(x)\] is the Bessel function

husseine57 · Feb 17, 2013

_Eduardo_ said:
\[\cos(a \sin{x}) = J_0(a) + 2 \displaystyle\sum_{k=1}^{\infty} J_k(a)\cos(kx)\]

\[J_k(x)\] is the Bessel function

Thanks.
What about these functions:
sin(a cos(x))
sin(a sin(x))
cos(a cos(x))

_Eduardo_ · Feb 17, 2013

husseine57 said:
Thanks.
What about these functions:
sin(a cos(x))
sin(a sin(x))
cos(a cos(x))

See the Jacobi expansion

I wrote the above formula wrong. k must be even

It must be : \[\cos(a \sin{x}) = J_0(a) + 2 \displaystyle\sum_{k=1}^{\infty} J_{2k}(a)\cos(2kx)\]