I need to know the Fourier series of exponential of general function, represented as


($c_n$ and $\psi_n$ are what I want to know, and the others are given.)

This question gives a specific solution of this problem; the Fourier series of $e^{a\cos x}$:

$\int^{\pi}_{-\pi}e^{a\cos \theta}\cos(n\theta)d\theta=2\pi I_n(a)$,

where $I_n$ is modified Bessel function of the first kind.

This comes from gererating function of $I_n$:


But I have failed to apply this relation to my generalized question because of the complexity.

Could anyone help me solve this question?

(I apologize my poor English.)


Basically you are asking for the Fourier transform ${\cal F}[e^f]$ of the exponential of a function $f(\theta)$, in terms of the Fourier transform ${\cal F}[f]$ of $f$ itself; the formal answer is $${\cal F}[e^f]=1+\sum_{n=1}^\infty \frac{1}{n!}(\underset{\underbrace{\text{n times}}}{{\cal F}[f]*\cdots *{\cal F}[f]})$$ where $\ast$ is a convolution. There is no further simplification for the general case.


Thank you for your answer.

Well, I have already known that relation, but I was motivated by your post.

$$\exp\left(\sum_k a_k\cos(k\theta+\phi_k)\right)=\sum_{n=0}^\infty\frac{1}{n!}\left(\sum_{k=0}^\infty a_k\cos(k\theta+\phi_k)\right)^n$$

Although this is difficult to calculate integral as it is, after some deformations (which maybe contain some mistakes), this will be:

$$=\sum_{n=0}^\infty\frac{1}{2^{n+1}n!}\sum_{p_0=0}^\infty\sum_{p_1=0}^\infty\cdots\sum_{p_n=0}^\infty\left(\prod_{l=0}^n a_{p_l}\right)\sum_{q_0=\pm1}\sum_{q_1=\pm1}\cdots\sum_{q_n=\pm1}e^{i\sum_{m=0}^nq_m\phi_{p_m}}\cdot e^{i\sum_{m=0}^nq_m p_m\theta}$$

This (some complicated & formal) notation can be integrated,

$$c_j:=\int_{-\pi}^{\pi}d\theta\exp\left(\sum_k a_k\cos(k\theta+\phi_k)\right)\cos j\theta$$ $$=\sum_{n=0}^\infty\frac{1}{2^{n+1}n!}\sum_{p_0=0}^\infty\sum_{p_1=0}^\infty\cdots\sum_{p_n=0}^\infty\left(\prod_{l=0}^n a_{p_l}\right)\sum_{q_0=\pm1}\sum_{q_1=\pm1}\cdots\sum_{q_n=\pm1}e^{i\sum_{m=0}^nq_m\phi_{p_m}}\cdot(\delta_{\sum_{m=0}^n q_m p_m+j}+\delta_{\sum_{m=0}^n q_m p_m-j})$$ where $\delta_r$ represents Kronecker delta.

It is difficult to make this more simple for me, but the interior of the summation seems to decrease rapidly with "well-behaved" $a_k$.

So this Fourier integral may be dominated by the smaller $n$ terms and I have managed to write down the cases $n=0$ and $n=1$.

Then the approximated Fourier cosine series for small $k$ : $$c_0\sim\frac{1}{2}a_0+\frac{1}{4}\left(a_0^2+\sum_{p=0}^\infty a_p^2\right)$$

$$c_1\sim\frac{1}{2}a_1\cos\phi_1+\frac{1}{2}\left(a_0a_1\cos(\phi_1)+\sum_{p=0}^{\infty}a_p a_{p+1}\cos(\phi_p-\phi_{p+1})\right)$$

As I am originally interested in lower-order terms, possibly this caluclation will be sufficient (not tested yet).


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