Fourier Series Coefficient Question

Question

In calculating the Fourier Coefficients a0, an, bn: Why are the an and bn coefficients integrated over 2 times the inverse of the period, 2(1/T) while the a0 coefficient is integrated only over one over the period, (1/T)?

Answer 1

Because the usual orthonomal basis (on the interval $\;[-\pi\,,\,\pi]\;$) used for Fourier series is

$$\left\{\;\frac1{\sqrt{2\pi}}\,,\;\frac1{\sqrt\pi}\cos nx\,,\;\frac1{\sqrt\pi}\sin nx\;\;:\;\;n\in\Bbb N\right\}$$