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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\psi_n)$.

($c_n$ 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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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.)

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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\psi_n)d\theta$.

($c_n$ 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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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.)

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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\phi_n)$.

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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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.)

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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\psi_n)$.

($c_n$ 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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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.)

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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\phi_n)$.

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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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

Could anyone help me solve this question?

(I approgize my poor English.)

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

$c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\phi_n)$.

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 $I_n$:

$\exp(\frac{a}{2}(z+\frac{1}{z}))=\sum_{k=1}^{\infty}I_k(a)z^k$.

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.)