different formulas for the fourier series.

Question

Quick question, I see both of these

$$f(w) = a_0 + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw) \quad f(w) = \frac{a_0}{2} + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw)$$

Why the difference ( $a_0$ and $\frac{a_0}{2}$)?

Answer 1

If you take the second definition, then the downside is that you have to divide $a_0$ by $2$. The upside is that now all of the coefficients $a_k$ can be concisely described by $$ a_k = \frac 2{T_0}\int_{0}^{T_0} f(t) \cos\left(\frac{2 \pi k t}{T_0} \right)\,dt $$ whereas in the first case, we would have to write the definition something like $$ a_k = \begin{cases} \frac 2{T_0}\int_{0}^{T_0} f(t) \cos\left(\frac{2 \pi k t}{T_0} \right)\,dt & k \geq 1\\ \frac 1{T_0} \int_0^{T_0} f(t)\,dt & k = 0 \end{cases} $$