Fourier Coefficients: A_n & B_n Explained

[dimensionless]

A function $$f(t)$$ can be represented by the expansion

$$f(t) = \frac{1}{2}A_{0} + A_{1}cos(\omega t) + A_{2}cos(2 \omega t) + A_{3}cos(3 \omega t) + ...<br /> B_{1}sin(\omega t) + B_{2}sin(2 \omega t) + B_{3}sin(3 \omega t) + ...$$

Do the constants $$A_{n}$$ and $$B_{n}$$ the same thing as the real and imaginary components of the Fourier transform? If so, why is there no imaginary component in the zeroth term?

 

[mathman]

In computing the Fourier transform, the kernel is of the form e^inwt. For A₀, the kernel is simply 1, so there is no imaginary part.