Message Boards Message Boards

GROUPS:
Mathematics Wavelets and Fourier Series
afreeka b

Cosine coefficient of the Fourier serie of $\sqrt{1 - k^2 \sin{t}^2}$

afreeka b
Posted 9 years ago
5488 Views
|
2 Replies
|
1 Total Likes
|

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.
POSTED BY: afreeka b
2 Replies
Sort By:

You could use directly

In[25]:= FourierCosSeries[Sqrt[1 - k^2 Sin[t]^2], t, 8]
Out[25]= (2 EllipticE[k^2])/\[Pi] + (
 2 Cos[2 t] (-2 (-2 + k^2) EllipticE[k^2] + 
    4 (-1 + k^2) EllipticK[k^2]))/(3 k^2 \[Pi]) + (
 2 Cos[4 t] (-2 (16 - 16 k^2 + k^4) EllipticE[k^2] + 
    16 (2 - 3 k^2 + k^4) EllipticK[k^2]))/(15 k^4 \[Pi]) + (
 2 Cos[6 t] ((512 - 768 k^2 + 268 k^4 - 6 k^6) EllipticE[k^2] + 
    4 (-128 + 256 k^2 - 155 k^4 + 27 k^6) EllipticK[k^2]))/(
 105 k^6 \[Pi]) + (
 2 Cos[8 t] (-2 (2048 - 4096 k^2 + 2496 k^4 - 448 k^6 + 
       5 k^8) EllipticE[k^2] + 
    64 (64 - 160 k^2 + 138 k^4 - 47 k^6 + 5 k^8) EllipticK[k^2]))/(
 315 k^8 \[Pi])

In[27]:= FourierCosCoefficient[Sqrt[1 - k^2 Sin[t]^2], t, 8]
Out[27]= (2 (-2 (2048 - 4096 k^2 + 2496 k^4 - 448 k^6 + 
      5 k^8) EllipticE[k^2] + 
   64 (64 - 160 k^2 + 138 k^4 - 47 k^6 + 5 k^8) EllipticK[
     k^2]))/(315 k^8 \[Pi])

Now you have to reconstruct the denominator and the polynomials in front of EllipticE as well as in front of EllipticK in dependence from n.

Possibly you can do so by understanding how the elliptic integrals do appear in the formulae.
POSTED BY: Udo Krause
afreeka b
afreeka b
Posted 9 years ago

Thanks a lot. it was very helpful. Your comment gave me some hint and I found the analytical form of the Fourier transform which is a sum of Gauss's hypergeometric fonctions.:)
POSTED BY: afreeka b
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Tag limit exceeded
Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract