Doubt regarding Fourier series coefficients.

Question

I have some doubts while I was self studying Fourier series. I wanted to ask, there are three types of Fourier series namely; Trigonometric Fourier series, Polar Fourier series, Complex Fourier series, does complex Fourier coefficients can be found out for complex functions only? Do complex functions have non-zero Trigonometric and Polar Fourier coefficients? Do real functions have Complex Fourier coefficients? Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)? Can Complex Fourier coefficients be purely real? If any answer is 'Yes', tell me when does this happen.

Answer 1

Many pages would be written for answering those questions. Briefly:

Given a function $f$ with domain $\Omega$ real or complex valuated, there is only one type of Fourier series, but different perspectives for thinking on it.

Does complex Fourier coefficients can be found out for complex functions only?

Observe $e^{inx}$ is complex valued, thus also $f(x)e^{inx}$. Then, for calculating the $n$th coefficient, we have $\int\limits_{\Omega} f(x)e^{inx} dx$ complex valuated in general. So the answer is: No. Complex coefficients also appear for real-valued functions.

Do complex functions have non-zero Trigonometric and Polar Fourier coefficients?

Trigonometric or polar coefficients is a matter of perspective. Euler's identity $e^{i\theta} = \cos(\theta) + i \sin(\theta)$ translates one into the another. It is easily seen that the modulus corresponds to the amplitude of that harmonic in the function, while the the angle corresonds to its phase.

Complex functions can have some zero coefficients $b_n$ and it can be interpreted as if such function were orthogonal to the $n$th harmonic oscillator.

Can Trigonometric and Polar Fourier coefficients be complex numbers (with non-zero imaginary part)?

Yes.

Can Complex Fourier coefficients be purely real?

Yes, when the phase of that harmonic in the function is exactly zero.