Hi Everyone,
I'm curently working a physical model based on the Kernel Poisson.
I'm struggling with a Fourier serie I need to find the Fourier series of the following function:
$f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$ (eq.1)
The function is even and
$\pi$-periodic.
The Fourier serie should be in this form:
$f(t)=\frac{a_0}2+\sum\limits_{i=0}^\infty a_n\cos[2nt]$ (eq.2)
Then, at
$t\to0$, the Taylor serie is:
$f(t)=\left[\frac{2E[k^2]}\pi+\sum_{i=0}^\infty\frac1{2^{2i-1}}\pmatrix{1/2\\
i}(k)^{2i}\sum_{j=0}^{i-1}(-1)^j\pmatrix{2i\\j}\cos(2(i-j)t)
\right]$ (eq.3)
It's pretty close to the final Fourier serie but I cannot identify the coefficient
$a_n$ from (eq.2) and (eq.3).
Can someone give a help on this? Any support is appreciated.
Thanks in advance. PS:
$k<<1$ is real postive and E is the complete elliptic integral of the 2nd kind.