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\{<br /> \begin{array}{lr}<br /> \frac{1}{2}\left(a_n + jb_n\right) &amp; : n &lt; 0\\ \\<br /> \frac{1}{2}a_0 &amp; : n = 0\\ \\<br /> \frac{1}{2}\left(a_n - jb_n\right) &amp; : n &gt; 0<br /> \end{array}<br /> \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.