Question

In Fourier series \(f\left( x \right) = {a_0} + \mathop \sum \limits_{n - 1}^\infty \left\{ {{a_n} + \cos \left( {nx} \right) + {b_n}\sin \left( {nx} \right)} \right\}\)
1. \({a_0} = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\)
2. \({a_0} = \mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\)
3. \({a_0} = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)\sin \left( x \right)dx\)
4. None of these

Answer 1

Correct Answer - Option 1 : \({a_0} = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\)

The Fourier series coefficients are:

\({{\rm{a}}_0} = \frac{1}{{2{\rm{\pi }}}}\mathop \smallint \limits_{ - {\rm{\pi }}}^{\rm{\pi }} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\)

\({{\rm{a}}_{\rm{n}}} = \frac{1}{{2{\rm{\pi }}}}\mathop \smallint \limits_{ - {\rm{\pi }}}^{\rm{\pi }} {\rm{f}}\left( {\rm{x}} \right)\cos {\rm{nxdx}}\)

\({{\rm{b}}_{\rm{n}}} = \frac{1}{{2{\rm{\pi }}}}\mathop \smallint \limits_{ - {\rm{\pi }}}^{\rm{\pi }} {\rm{f}}\left( {\rm{x}} \right)\sin {\rm{nxdx}}\)