Fourier Series of $f(x) = 0$ from $(−1,0)$ , $3x$ from $(0,1)$ , f=(x+2)

Question

I need to determine the Fourier series of the following function.

$$ f(x) = \begin{cases} 0 & \text{if } -1 < x < 0 \\ 3x & \text{if } \phantom{{}-{}} 0 \le x < 1 \end{cases}.$$

Answer 1

Define problem

Piecewise function: Resolve $f(x)$ into a left and right piece

$$ \begin{align} % l(x) &= 0, \quad -\lambda \le x < 0 \\ % r(x) &= 3x, \quad \ 0 \le x \le \lambda % \end{align} $$

Find the Fourier expansion $$ f(x) = \frac{1}{2}a_{0} + \sum_{k=1}^{\infty} \left( a_{k} \cos \left( \frac{k \pi} {\lambda}x \right) + b_{k} \sin \left( \frac{k \pi} {\lambda}x \right) \right) $$ where $\lambda = 1$ for this problem.

Given $k=1,2,3\dots$, the amplitudes are given by

$$ \begin{align} % a_{0} &= \frac{1}{2} \int_{-\lambda}^{\lambda} f(x) dx \\ % a_{k} &= \frac{1}{2} \int_{-\lambda}^{\lambda} f(x) \cos \left( \frac{k \pi} {\lambda}x \right) dx \\ % b_{k} &= \frac{1}{2} \int_{-\lambda}^{\lambda} f(x) \sin \left( \frac{k \pi} {\lambda}x \right) dx \\ % \end{align} $$

Basic integrals

Left hand piece

$$ \begin{align} % \int_{-\lambda}^{0} l(x) dx &= 0 \\ % \int_{-\lambda}^{0} l(x) \cos \left( \frac{k \pi} {\lambda}x \right) dx &= 0 \\ % \int_{-\lambda}^{0} l(x) \sin \left( \frac{k \pi} {\lambda}x \right) dx &= 0 \\ \end{align} $$

Right hand piece

$$ \begin{align} % \int_{0}^{\lambda} r(x) dx &= \frac{3}{2} \\ % \int_{0}^{\lambda} r(x) \cos \left( \frac{k \pi} {\lambda}x \right) dx &= \frac{3 \left((-1)^k-1\right)}{\pi ^2 k^2} \\ % \int_{0}^{\lambda} r(x) \sin \left( \frac{k \pi} {\lambda}x \right) dx &= -\frac{3 (-1)^k}{\pi k} % \end{align} $$

Results

$$ \begin{align} % a_{0} & = \frac{3}{2} \\ % a_{k} &= \frac{3 \left((-1)^k-1\right)}{\pi ^2 k^2} \\[2pt] % b_{k} &= -\frac{3 (-1)^k}{\pi k} % % \end{align} $$

The first terms of each series: $$ \begin{align} % \left\{ a_{k} \right\}_{k=1}^{7} &= -\frac{6}{\pi^{2}} \left\{ 1,0,\frac{1}{9},0,\frac{1}{25},0,\frac{1}{49} \right\} \\[5pt] % \left\{ b_{k} \right\}_{k=1}^{7} &= \frac{3}{\pi} \left\{ 1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\frac{1}{7} \right\} % \end{align} $$

Approximation sequence

$$ g_{n}(x) = \frac{3}{4} + \frac{3}{\pi} \sum_{k=1}^{n} \left( \frac{(-1)^k-1}{\pi k^2} \cos \left( \frac{k \pi} {\lambda} x \right) + \frac{(-1)^{k+1}}{k} \sin \left( \frac{k \pi} {\lambda} x \right) \right) $$

Visual verification

Periodicity