From this answer:
The Fourier series:
\$ V_t = \dfrac{a_0}{2} + \displaystyle \sum_{i=1}^{\infty}[a_i sin(i \omega_0 t) + b_i cos(i \omega_0 t) ] \$
Why is the DC level written as \$\dfrac{a_0}{2}\$, and not \$a_0\$? stevenvh says it's convention, but there has to be an explanation. Where does it come from?
Best Answer
This stumped me for a while as well but it is actually quite simple.
The general Fourier series:
\$ g(x) = a_0 + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x) ] \$
I am not going to do all the math but if you use the orthogonal signal space \$ \left\{{1, cos(nx),sin(nx)}\right\} n\in N \$ you can derive the fourier series terms as:
\$ a_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)cos(n x)\,\mathrm{d}x\$
\$ b_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)sin(n x)\,\mathrm{d}x\$
And:
\$ a_0 = \dfrac{1}{2\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$
The "regular" Fourier series
\$ g(x) = \dfrac{a_0}{2} + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x)]\$
For the sake of symmetry the \$a_0\$ term was redefined as:
\$ a_0 = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$
to be similar to \$a_n\$ and \$b_n\$. Therefore the \$\dfrac{1}{2}\$ term was added to the expression of the Fourier series.
Reference:
Generalized Fourier Series
Fourier Series