Fourier series

Definition of Fourier series

\(\begin{align} f(x) = a_0 + \sum^\infty_{n=1} a_n \cos nx + \sum^\infty_{n=1} b_n \sin nx \end{align}\)

Formula of Fourier coefficients

\(\begin{align} & a_0 = \frac{1}{2\pi}\int^\pi_{-\pi}f(x)dx \\ \\ & a_n = \frac{1}{\pi}\int^\pi_{-\pi}f(x)\cos nxdx \\ \\ & b_n = \frac{1}{\pi}\int^\pi_{-\pi}f(x)\sin nxdx \end{align}\)

Derivation of \(a_0\)

\(\begin{align} \int^\pi_{-\pi}f(x)dx &= \int^\pi_{-\pi} \Bigg[ a_0 + \sum^\infty_{n=1} a_n \cos nx + \sum^\infty_{n=1} b_n \sin nx \Bigg]dx \\ \\ &=\int^\pi_{-\pi} a_0 + \sum^\infty_{n=1}\int^\pi_{-\pi}a_n\cos nxdx + \sum^\infty_{n=1}\int^\pi_{-\pi}b_n\sin nxdx \\ \\ &=2\pi a_0 + 0 + 0 \\ \\ &=2\pi a_0 \\ \end{align}\) \(\begin{align} \\ \\ \Rightarrow a_0 = \frac{1}{2\pi}\int^\pi_{-\pi}f(x)dx \end{align}\)

Derivation of \(a_n\)

\(\begin{align} \int^\pi_{-\pi}f(x)\cos ndx &= \int^\pi_{-\pi} \Bigg[ a_0 + \sum^\infty_{m=1} a_m \cos mx + \sum^\infty_{m=1} b_m \sin mx \Bigg]\cos nxdx \\ \\ & = \int^\pi_{-\pi}a_0\cos nxdx + \sum^\infty_{m=1}\int^\pi_{-\pi}a_m \cos mx\cos nxdx + \sum^\infty_{m=1}\int^\pi_{-\pi}b_m \sin mx\cos nxdx \\ \\ & =0 + \pi a_n + 0 \\ \\ & = \pi a_n \\ \end{align}\) \(\begin{align} \\ \\ \Rightarrow a_n = \frac{1}{\pi}\int^\pi_{-\pi}f(x)\cos nxdx \end{align}\)

Derivation of \(b_n\)

\(\begin{align} \int^\pi_{-\pi}f(x)\sin ndx &= \int^\pi_{-\pi} \Bigg[ a_0 + \sum^\infty_{m=1} a_m \cos mx + \sum^\infty_{m=1} b_m \sin mx \Bigg]\sin nxdx \\ \\ & = \int^\pi_{-\pi}a_0\sin nxdx + \sum^\infty_{m=1}\int^\pi_{-\pi}a_m \cos mx\sin nxdx + \sum^\infty_{m=1}\int^\pi_{-\pi}b_m \sin mx\sin nxdx \\ \\ & =0 + 0 + \pi b_n \\ \\ & = \pi b_n \\ \end{align}\) \(\begin{align} \\ \\ \Rightarrow b_n = \frac{1}{\pi}\int^\pi_{-\pi}f(x)\sin nxdx \end{align}\)