How are complex and real Fourier coefficients related for periodic functions?

[nickmai123]

I have a quick question about the relationship between the complex Fourier coefficient,$$\alpha_n$$ and the real Fourier coefficients, $$a_n$$ and $$b_n$$.

Given a real-valued function, I could just find the real coefficients and plug them into the relation below, right?Fourier Coefficients for periodic functions of period 2a.
Complex Form:
$$\alpha_n = \frac{1}{2a}\int_{-a}^{a} f\left(t\right)e^{\frac{-jn\pi t}{a}dt$$

Real Form:
$$a_0 = \frac{1}{a}\int_{-a}^{a} f\left(t\right)dt$$

$$a_n = \frac{1}{a}\int_{-a}^{a} f\left(t\right) cos\left(\frac{n\pi t}{a}\right)dt$$

$$b_n = \frac{1}{a}\int_{-a}^{a} f\left(t\right) sin\left(\frac{n\pi t}{a}\right)dt$$

Relation
$$\alpha_n = \left\{ \begin{array}{lr} \frac{1}{2}\left(a_n + jb_n\right) & : n < 0\\ \\ \frac{1}{2}a_0 & : n = 0\\ \\ \frac{1}{2}\left(a_n - jb_n\right) & : n > 0 \end{array} \right.$$

 

[vela]

Yup, except that $\alpha_0 = a_0$. The factor of 1/2 for that coefficient isn't needed with the formulas you're using.

 

[nickmai123]

Thanks.