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Definition of Fourier Series and Typical Examples
 Baron Jean Baptiste Joseph Fourier ($$1768-1830$$) introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related. To consider this idea in more detail, we should introduce some definitions and common terms.
Basic Definitions
A function $$f\left( x \right)$$ is said to have period $$P$$ if $$f\left( {x + P} \right) = f\left( x \right)$$ for all $$x.$$ Let the function $$f\left( x \right)$$ has period $$2\pi.$$ In this case, it is enough to consider behavior of the function on the interval $$\left[ { - \pi ,\pi } \right].$$
1. Suppose that the function $$f\left( x \right)$$ with period $$2\pi$$ is absolutely integrable on on $$\left[ { - \pi ,\pi } \right]$$ so that the following so-called Dirichlet integral is finite: $\int\limits_{ - \pi }^\pi {\left| {f\left( x \right)} \right|dx} < \infty ;$

2. Suppose also that the function $$f\left( x \right)$$ is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).

If the conditions $$1$$ and $$2$$ are satisfied, the Fourier series for the function $$f\left( x \right)$$ exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions.)

At a discontinuity $${x_0}$$, the Fouries Series converges to $\lim\limits_{\varepsilon \to 0} \frac{1}{2}\left[ {f\left( {{x_0} - \varepsilon } \right) - f\left( {{x_0} + \varepsilon } \right)} \right].$ The Fourier series of the function $$f\left( x \right)$$ is given by $f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}} ,$ where the Fourier coefficients $${{a_0}},$$ $${{a_n}},$$ and $${{b_n}}$$ are defined by the integrals ${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} ,}\;\; {{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nx dx} ,}\;\; {{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nx dx} .}$ Sometimes alternative forms of the Fourier series are used. Replacing $${{a_n}}$$ and $${{b_n}}$$ by the new variables $${{d_n}}$$ and $${{\varphi_n}}$$ or $${{d_n}}$$ and $${{\theta_n}},$$ where ${{d_n} = \sqrt {a_n^2 + b_n^2} ,}\;\; {\tan {\varphi _n} = \frac{{{a_n}}}{{{b_n}}},}\;\; {\tan {\theta _n} = \frac{{{b_n}}}{{{a_n}}},}$ we can write: ${f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {{d_n}\sin \left( {nx + {\varphi _n}} \right)} \;\;\text{or}\;\;} {f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {{d_n}\cos\left( {nx + {\theta _n}} \right)} .}$
Fourier Series of Even and Odd Functions
The Fourier series expansion of an even function $$f\left( x \right)$$ with the period of $$2\pi$$ does not involve the terms with sines and has the form: $f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,$ where the Fourier coefficients are given by the formulas ${{a_0} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)dx} ,}\;\; {{a_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} .}$ Accordingly, the Fourier series expansion of an odd $$2\pi$$-periodic function $$f\left( x \right)$$ consists of sine terms only and has the form: $f\left( x \right) = \sum\limits_{n = 1}^\infty {{b_n}\sin nx} ,$ where the coefficients $${{b_n}}$$ are ${b_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\sin nxdx} .$ Below we consider expansions of $$2\pi$$-periodic functions into their Fourier series, supposing that these expansions exist and are convergent.

Example 1
Let the function $$f\left( x \right)$$ be $$2\pi$$-periodic and suppose that it is presented by the Fourier series: $f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}} .$ Calculate the coefficients $${{a_0}},$$ $${{a_n}},$$ and $${{b_n}}.$$

Solution.
To define $${{a_0}},$$ we integrate the Fourier series on the interval $$\left[ { - \pi ,\pi } \right]:$$ ${\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\pi {a_0} + \sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ - \pi }^\pi {\cos nxdx} + {b_n}\int\limits_{ - \pi }^\pi {\sin nxdx} } \right]} .}$ For all $$n > 0$$, ${\int\limits_{ - \pi }^\pi {\cos nxdx} = \left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{ - \pi }^\pi = 0\;\;\text{and}\;\;} {\int\limits_{ - \pi }^\pi {\sin nxdx} = \left. {\left( { - \frac{{\cos nx}}{n}} \right)} \right|_{ - \pi }^\pi = 0.}$ Therefore, all the terms on the right of the summation sign are zero, so we obtain ${\int\limits_{ - \pi }^\pi {f\left( x \right)dx} = \pi {a_0}\;\;\text{or}\;\;} {{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} .}$ In order to find the coefficients $${{a_n}},$$ we multiply both sides of the Fourier series by $$\cos mx$$ and integrate term by term: ${\int\limits_{ - \pi }^\pi {f\left( x \right)\cos mxdx} } = {\frac{{{a_0}}}{2}\int\limits_{ - \pi }^\pi {\cos mxdx} } + {\sum\limits_{n = 1}^\infty {\left[ {{a_n}\int\limits_{ - \pi }^\pi {\cos nx\cos mxdx} + {b_n}\int\limits_{ - \pi }^\pi {\sin nx\cos mxdx} } \right]} .}$ The first term on the right side is zero. Then, using the well-known trigonometric identities, we have ${\int\limits_{ - \pi }^\pi {\sin nx\cos mxdx} } = {\frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\sin\left( {n + m} \right)x + \sin \left( {n - m} \right)x} \right]dx} = 0,}$ ${\int\limits_{ - \pi }^\pi {\cos nx\cos mxdx} } = {\frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\cos \left( {n + m} \right)x + \cos \left( {n - m} \right)x} \right]dx} = 0,}$ if $$m \ne n.$$

In case when $$m = n$$, we can write: $\require{cancel} {\int\limits_{ - \pi }^\pi {\sin nx\cos mxdx} } = {\frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\sin 2mx + \sin 0} \right]dx} ,}\;\; {\Rightarrow \int\limits_{ - \pi }^\pi {{\sin^2}mxdx} = \frac{1}{2}\left[ {\left. {\left( { - \frac{{\cos 2mx}}{{2m}}} \right)} \right|_{ - \pi }^\pi } \right] } = {\frac{1}{{4m}}\left[ { - \cancel{\cos \left( {2m\pi } \right)} + \cancel{\cos \left( {2m\left( { - \pi } \right)} \right)}} \right] = 0;}$ ${\int\limits_{ - \pi }^\pi {\cos nx\cos mxdx} } = {\frac{1}{2}\int\limits_{ - \pi }^\pi {\left[ {\cos 2mx + \cos 0} \right]dx} ,}\;\; {\Rightarrow \int\limits_{ - \pi }^\pi {{\cos^2}mxdx} } = {\frac{1}{2}\left[ {\left. {\left( {\frac{{\sin 2mx}}{{2m}}} \right)} \right|_{ - \pi }^\pi + 2\pi } \right] } = {\frac{1}{{4m}}\left[ {\sin \left( {2m\pi } \right) - \sin \left( {2m\left( { - \pi } \right)} \right)} \right] + \pi = \pi .}$ Thus, ${\int\limits_{ - \pi }^\pi {f\left( x \right)\cos mxdx} = {a_m}\pi ,}\;\; {\Rightarrow {a_m} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos mxdx} ,}\;\; {m = 1,2,3, \ldots }$ Similarly, multiplying the Fourier series by $$\sin mx$$ and integrating term by term, we obtain the expression for $${{b_m}}:$$ ${{b_m} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin mxdx} ,}\;\; {m = 1,2,3, \ldots }$ Rewriting the formulas for $${{a_n}},$$ $${{b_n}},$$ we can write the final expressions for the Fourier coefficients: ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} ,}\;\; {{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nxdx} .}$
Example 2
Find the Fourier series for the square $$2\pi$$-periodic wave defined on the interval $$\left[ { - \pi ,\pi } \right]:$$ $f\left( x \right) = \begin{cases} 0, & \text{if} & - \pi \le x \le 0 \\ 1, & \text{if} & 0 < x \le \pi \end{cases}.$

Solution.
First we calculate the constant $${{a_0}}:$$ ${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\frac{1}{\pi }\int\limits_0^\pi {1dx} } = {\frac{1}{\pi } \cdot \pi = 1.}$ Find now the Fourier coefficients for $$n \ne 0:$$ ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} } = {\frac{1}{\pi }\int\limits_0^\pi {1 \cdot \cos nxdx} } = {\frac{1}{\pi }\left[ {\left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_0^\pi } \right] } = {\frac{1}{{\pi n}} \cdot 0 = 0,}$ ${{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nxdx} } = {\frac{1}{\pi }\int\limits_0^\pi {1 \cdot \sin nxdx} } = {\frac{1}{\pi }\left[ {\left. {\left( { - \frac{{\cos nx}}{n}} \right)} \right|_0^\pi } \right] } = { - \frac{1}{{\pi n}} \cdot \left( {\cos n\pi - \cos 0} \right) } = {\frac{{1 - \cos n\pi }}{{\pi n}}.}$ As $$\cos n\pi = {\left( { - 1} \right)^n},$$ we can write: ${b_n} = \frac{{1 - {{\left( { - 1} \right)}^n}}}{{\pi n}}.$ Thus, the Fourier series for the square wave is $f\left( x \right) = \frac{1}{2} + \sum\limits_{n = 1}^\infty {\frac{{1 - {{\left( { - 1} \right)}^n}}}{{\pi n}}\sin nx} .$ We can easily find the first few terms of the series. By setting, for example, $$n = 5,$$ we get ${f\left( x \right) = \frac{1}{2} + \frac{{1 - \left( { - 1} \right)}}{\pi }\sin x } + {\frac{{1 - {{\left( { - 1} \right)}^2}}}{{2\pi }}\sin 2x } + {\frac{{1 - {{\left( { - 1} \right)}^3}}}{{3\pi }}\sin 3x } + {\frac{{1 - {{\left( { - 1} \right)}^4}}}{{4\pi }}\sin 4x } + {\frac{{1 - {{\left( { - 1} \right)}^5}}}{{5\pi }}\sin 5x + \ldots } = {\frac{1}{2} + \frac{2}{\pi }\sin x } + {\frac{2}{{3\pi }}\sin 3x } + {\frac{2}{{5\pi }}\sin 5x + \ldots }$ The graph of the function and the Fourier series expansion for $$n = 10$$ is shown in Figure $$1.$$
 Fig.1, n = 10 Fig.2, n = 5, n = 10
Example 3
Find the Fourier series for the sawtooth wave defined on the interval $$\left[ { - \pi ,\pi } \right]$$ and having period $$2\pi.$$

Solution.
Calculate the Fourier coefficients for the sawtooth wave. Since this function is odd (Figure $$2$$), then $${a_0} = {a_n} = 0.$$ Find the coefficients $${b_n}:$$ ${{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nxdx} } = {\frac{1}{\pi }\int\limits_{ - \pi }^\pi {x\sin nxdx} .}$ To calculate the latter integral we use integration by parts formula: $\int\limits_{ - \pi }^\pi {udv} = \left. {\left( {uv} \right)} \right|_{ - \pi }^\pi - \int\limits_{ - \pi }^\pi {vdu} .$ Let $$u = x,\;dv = \sin nxdx.$$ Then $$du = dx,\;v = \int {\sin nxdx} = - {\large\frac{{\cos nx}}{n}\normalsize},$$ so the integral becomes ${{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {x\sin nxdx} } = {\frac{1}{\pi }\left[ {\left. {\left( { - \frac{{\cos nx}}{n}} \right)} \right|_{ - \pi }^\pi - \int\limits_{ - \pi }^\pi {\left( { - \frac{{\cos nx}}{n}} \right)dx} } \right] } = {\frac{1}{\pi }\left[ {\left( { - \frac{{\pi \cos n\pi }}{n} + \frac{{\left( { - \pi } \right)\cos \left( { - n\pi } \right)}}{n}} \right) + \frac{1}{n}\int\limits_{ - \pi }^\pi {\cos nxdx} } \right] } = {\frac{1}{{n\pi }}\left[ { - 2\pi \cos n\pi + \left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{ - \pi }^\pi } \right] } = {\frac{1}{{n\pi }}\left[ { - 2\pi \cos n\pi + \frac{1}{n}\left( {\sin n\pi - \sin \left( { - n\pi } \right)} \right)} \right] } = {\frac{1}{{n\pi }}\left[ { - 2\pi \cos n\pi + \frac{{2\sin n\pi }}{n}} \right] } = {\frac{2}{{n\pi }}\left[ {\frac{{\sin n\pi }}{n} - \pi \cos n\pi } \right].}$ Substituting $$\sin n\pi = 0$$ and $$\cos n\pi = {\left( { - 1} \right)^n}$$ for all integer values of $$n,$$ we obtain ${{b_n} = \frac{2}{{n\pi }}\left( { - \pi {{\left( { - 1} \right)}^n}} \right) } = { - \frac{2}{n}{\left( { - 1} \right)^n} } = {\frac{2}{n}{\left( { - 1} \right)^{n + 1}}.}$ Thus, the Fourier series expansion of the sawtooth wave is (Figure $$2$$ above) $x = \sum\limits_{n = 1}^\infty {\frac{2}{n}{{\left( { - 1} \right)}^{n + 1}}\sin nx} .$
Example 4
Let $$f\left( x \right)$$ be a $$2\pi$$-periodic function such that $$f\left( x \right) = {x^2}$$ for $$x \in \left[ { - \pi ,\pi } \right].$$ Find the Fourier series for the parabolic wave.

Solution.
Since this function is even, the coefficients $${b_n} = 0.$$ Then ${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\frac{1}{\pi }\int\limits_{ - \pi }^\pi {{x^2}dx} = \frac{2}{\pi }\int\limits_0^\pi {{x^2}dx} } = {\frac{2}{\pi } \cdot \left[ {\left. {\left( {\frac{{{x^3}}}{3}} \right)} \right|_0^\pi } \right] } = {\frac{2}{\pi } \cdot \frac{{{\pi ^3}}}{3} } = {\frac{{2{\pi ^2}}}{3},}$ ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} } = {\frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} } = {\frac{2}{\pi }\int\limits_0^\pi {{x^2}\cos nxdx} .}$ Apply integration by parts twice to find: ${{a_n} = \frac{2}{\pi }\int\limits_0^\pi {{x^2}\cos nxdx} } = {\left[ {\begin{array}{*{20}{l}} {u = {x^2}}\\ {dv = \cos nxdx}\\ {du = 2xdx}\\ {v = \int {\cos nxdx} = \frac{{\sin nx}}{n}} \end{array}} \right] } = {\frac{2}{\pi }\left[ {\left. {\left( {\frac{{{x^2}\sin nx}}{n}} \right)} \right|_0^\pi - \int\limits_0^\pi {2x\frac{{\sin nx}}{n}dx} } \right] } = {\frac{2}{{\pi n}}\left[ {{\pi ^2}\sin n\pi - {{\left( { - \pi } \right)}^2}\sin \left( { - n\pi } \right) - 2\int\limits_0^\pi {x\sin nxdx} } \right] } = {\frac{2}{{\pi n}}\left[ {2{\pi ^2}\sin n\pi - 2\int\limits_0^\pi {x\sin nxdx} } \right] } = { - \frac{4}{{\pi n}}\int\limits_0^\pi {x\sin nxdx} } = {\left[ {\begin{array}{*{20}{l}} {u = x}\\ {dv = \sin nxdx}\\ {du = dx}\\ {v = \int {\sin nxdx} = - \frac{{\cos nx}}{n}} \end{array}} \right] } = { - \frac{4}{{\pi n}}\left[ {\left. {\left( { - \frac{{x\cos nx}}{n}} \right)} \right|_0^\pi - \int\limits_0^\pi {\left( { - \frac{{\cos nx}}{n}} \right)dx} } \right] } = {\frac{4}{{\pi {n^2}}}\left[ {\pi \cos n\pi - \int\limits_0^\pi {\cos nxdx} } \right] } = {\frac{4}{{\pi {n^2}}}\left[ {\pi \cos n\pi - \left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_0^\pi } \right] } = {\frac{4}{{\pi {n^2}}}\left[ {\pi \cos n\pi - \frac{{\sin n\pi }}{n}} \right].}$ As $$\sin n\pi = 0$$ and $$\cos n\pi = {\left( { - 1} \right)^n}$$ for integer $$n,$$ we have ${{a_n} = \frac{4}{{\pi {n^2}}} \cdot \pi {\left( { - 1} \right)^n} } = {\frac{4}{{{n^2}}}{\left( { - 1} \right)^n}.}$ Then the Fourier series expansion for the parabolic wave is (Figure $$3$$) ${x^2} = \frac{{{\pi ^2}}}{3} + \sum\limits_{n = 1}^\infty {\frac{4}{{{n^2}}}{{\left( { - 1} \right)}^n}\cos nx} .$
 Fig.3, n = 2, n = 5 Fig.4, n = 1, n = 2
Example 5
Find the Fourier series for the triangle wave $f\left( x \right) = \begin{cases} \frac{\pi }{2} + x, & \text{if} & - \pi \le x \le 0 \\ \frac{\pi }{2} - x, & \text{if} & 0 < x \le \pi \end{cases},$ defined on the interval $$\left[ { - \pi ,\pi } \right].$$

Solution.
The constant $${a_0}$$ is ${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^0 {\left( {\frac{\pi }{2} + x} \right)dx} + \int\limits_0^\pi {\left( {\frac{\pi }{2} - x} \right)dx} } \right] } = {\frac{1}{\pi }\left[ {\left. {\left( {\frac{\pi }{2}x + \frac{{{x^2}}}{2}} \right)} \right|_{ - \pi }^0 + \left. {\left( {\frac{\pi }{2}x - \frac{{{x^2}}}{2}} \right)} \right|_0^\pi } \right] } = {\frac{1}{\pi }\left[ {0 - \left( { - \cancel{\frac{{{\pi ^2}}}{2}} + \cancel{\frac{{{{\left( { - \pi } \right)}^2}}}{2}}} \right) + \left( {\cancel{\frac{{{\pi ^2}}}{2}} - \cancel{\frac{{{\pi ^2}}}{2}}} \right) - 0} \right] = 0.}$ Determine the coefficients $${a_n}:$$ ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^0 {\left( {\frac{\pi }{2} + x} \right)\cos nxdx} + \int\limits_0^\pi {\left( {\frac{\pi }{2} - x} \right)\cos nxdx} } \right] } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^0 {\frac{\pi }{2}\cos nxdx} + \int\limits_{ - \pi }^0 {x\cos nxdx} } \right. } + {\left. {\int\limits_0^\pi {\frac{\pi }{2}\cos nxdx} - \int\limits_0^\pi {x\cos nxdx} } \right].}$ Integrating by parts, we can write ${\int {x\cos nxdx} = \frac{{x\sin nx}}{n} - \int {\frac{{x\sin nx}}{n}dx} } = {\frac{{x\sin nx}}{n} + \frac{{\cos nx}}{{{n^2}}}.}$ Then ${{a_n} = \frac{1}{\pi }\left[ {\frac{\pi }{2}\left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{ - \pi }^0 + \left. {\left( {\frac{{x\sin nx}}{n} + \frac{{\cos nx}}{{{n^2}}}} \right)} \right|_{ - \pi }^0} \right. } + {\;\left. {\frac{\pi }{2}\left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_0^\pi - \left. {\left( {\frac{{x\sin nx}}{n} + \frac{{\cos nx}}{{{n^2}}}} \right)} \right|_0^\pi } \right].}$ The values of $$\sin nx$$ at $$x = 0$$ or $$x = \pm \pi$$ are zero. Therefore, ${{a_n} = \frac{1}{{\pi {n^2}}}\left[ {\left. {\left( {\cos nx} \right)} \right|_{ - \pi }^0 - \left. {\left( {\cos nx} \right)} \right|_0^\pi } \right] } = {\frac{1}{{\pi {n^2}}}\left[ {\cos 0 - \cos \left( { - \pi n} \right) - \cos \pi n + \cos 0} \right] } = {\frac{2}{{\pi {n^2}}}\left[ {1 - \cos \pi n} \right] } = {\frac{2}{{\pi {n^2}}}\left[ {1 - {{\left( { - 1} \right)}^n}} \right].}$ When $$n = 2k,$$ then $${a_{2k}} = 0.$$ When $$n = 2k + 1,$$ then $${a_{2k + 1}} = {\large\frac{4}{{\pi {n^2}}}\normalsize},\;k = 0,1,2,3, \ldots$$
Since the function $$f\left( x \right)$$ is even, the Fourier coefficients $${b_n}$$ are zero. Therefore, the complete Fourier expansion for the triangle wave (see Figure $$4$$ above) is $f\left( x \right) = \frac{4}{\pi }\sum\limits_{k = 0}^\infty {\frac{{\cos \left( {2k + 1} \right)x}}{{{{\left( {2k + 1} \right)}^2}}}} .$
Example 6
Find the Fourier series for the function $f\left( x \right) = \begin{cases} 0, & \text{if} & - \pi \le x \le 0 \\ \sin x, & \text{if} & 0 < x \le \pi \end{cases},$ defined on the interval $$\left[ { - \pi ,\pi } \right].$$

Solution.
First we find the constant $${a_0}:$$ ${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\frac{1}{\pi }\int\limits_0^\pi {\sin xdx} } = {\frac{1}{\pi }\left[ {\left. {\left( { - \cos x} \right)} \right|_0^\pi } \right] } = {\frac{1}{\pi }\left( { - \cos \pi + \cos 0} \right) } = {\frac{2}{\pi }.}$ Now we calculate the coefficients $${a_n}:$$ ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} } = {\frac{1}{\pi }\int\limits_0^\pi {\sin x\cos nxdx} } = {\frac{1}{{2\pi }}\int\limits_0^\pi {\left[ {\sin \left( {x + nx} \right) + \sin\left( {x - nx} \right)} \right]dx} } = {\frac{1}{{2\pi }}\int\limits_0^\pi {\left[ {\sin \left( {n + 1} \right)x + \sin\left( {n - 1} \right)x} \right]dx} } = {\frac{1}{{2\pi }}\left[ {\left. {\left( { - \frac{{\cos \left( {n + 1} \right)x}}{{n + 1}} + \frac{{\cos \left( {n - 1} \right)x}}{{n - 1}}} \right)} \right|_0^\pi } \right] } = {\frac{1}{{2\pi }}\left[ {\frac{{\cos \left( {n - 1} \right)\pi }}{{n - 1}} - \frac{{\cos \left( {n + 1} \right)\pi }}{{n + 1}} - \frac{1}{{n - 1}} + \frac{1}{{n + 1}}} \right] } = {\frac{1}{{2\pi }}\left[ {\frac{{\cos \left( {n - 1} \right)\pi }}{{n - 1}} - \frac{{\cos \left( {n + 1} \right)\pi }}{{n + 1}} - \frac{2}{{{n^2} - 1}}} \right].}$ Notice that ${\cos \left( {n + 1} \right)\pi } = {\cos \left( {\pi n + \pi } \right) } = {\cos \left( {\pi n - \pi + 2\pi } \right) } = {\cos \left( {\left( {n - 1} \right)\pi + 2\pi } \right) } = {\cos \left( {n - 1} \right)\pi .}$ Since $$\cos \left( {n - 1} \right)\pi = {\left( { - 1} \right)^{n - 1}},$$ we get the following expression for the coefficients $${a_n}:$$ ${a_n} = \frac{1}{{2\pi }} \cdot \frac{2}{{{n^2} - 1}} \cdot \left[ {{{\left( { - 1} \right)}^{n - 1}} - 1} \right].$ It can be seen that $${a_n} = 0$$ for odd $$n.$$ For even $$n,$$ when $$n = 2k\;\left( {k = 1,2,3, \ldots } \right),$$ we have ${a_{2k}} = \frac{1}{{2\pi }} \cdot \left( { - \frac{4}{{{k^2} - 1}}} \right) = - \frac{1}{\pi } \cdot \frac{2}{{{k^2} - 1}}.$ Calculate the coefficients $${b_n}.$$ Start with $${b_1}:$$ ${{b_1} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin xdx} } = {\frac{1}{\pi }\int\limits_0^\pi {\sin x\sin xdx} } = {\frac{1}{\pi }\int\limits_0^\pi {{{\sin }^2}xdx} } = {\frac{1}{{2\pi }}\int\limits_0^\pi {\left( {1 - \cos 2x} \right)dx} } = {\frac{1}{{2\pi }}\left[ {\left. {\left( {x - \frac{{\sin 2x}}{2}} \right)} \right|_0^\pi } \right] } = {\frac{1}{{2\pi }} \cdot \pi = \frac{1}{2}.}$ The other coefficients $${b_n}$$ for $$n > 1$$ are zero. Indeed, ${{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nxdx} } = {\frac{1}{\pi }\int\limits_0^\pi {\sin x\sin nxdx} } = {\frac{1}{{2\pi }}\int\limits_0^\pi {\left[ {\cos \left( {x - nx} \right) - \cos \left( {x + nx} \right)} \right]dx} } = {\frac{1}{{2\pi }}\int\limits_0^\pi {\left[ {\cos \left( {n - 1} \right)x - \cos \left( {n + 1} \right)x} \right]dx} } = {\frac{1}{{2\pi }}\left[ {\left. {\left( {\frac{{\sin \left( {n - 1} \right)x}}{{n - 1}} - \frac{{\sin \left( {n + 1} \right)x}}{{n + 1}}} \right)} \right|_0^\pi } \right] = 0}$ Thus, the Fourier series of the given function is given by ${f\left( x \right) = \frac{1}{\pi } + \frac{1}{2}\sin x } - {\frac{2}{\pi }\sum\limits_{k = 1}^\infty {\frac{1}{{4{k^2} - 1}}\cos \left( {2kx} \right)} .}$ Graphs of the function and its Fourier expansions for $$n = 2$$ and $$n = 8$$ are shown in Figure $$5.$$
 Fig.5, n = 2, n = 8 Fig.6, n = 10
Example 7
Find the Fourier series for the function $f\left( x \right) = \begin{cases} -1, & \text{if} & - \pi \le x \le - \frac{\pi }{2} \\ 0, & \text{if} & - \frac{\pi }{2} \lt x \le \frac{\pi }{2} \\ 1, & \text{if} & \frac{\pi }{2} < x \le \pi \end{cases},$ defined on the interval $$\left[ { - \pi ,\pi } \right].$$

Solution.
${{a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)dx} } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^{ - \large\frac{\pi }{2}\normalsize} {\left( { - 1} \right)dx} + \int\limits_{ - \large\frac{\pi }{2}\normalsize}^{\large\frac{\pi }{2}\normalsize} {0dx} + \int\limits_{\large\frac{\pi }{2}\normalsize}^\pi {1dx} } \right] } = {\frac{1}{\pi }\left( { - \cancel{\frac{\pi }{2}} + 0 + \cancel{\frac{\pi }{2}}} \right) = 0.}$ Compute the coefficients $${a_n}:$$ ${{a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\cos nxdx} } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^{ - \large\frac{\pi }{2}\normalsize} {\left( { - \cos nx} \right)dx} + \int\limits_{\large\frac{\pi }{2}\normalsize}^\pi {\cos nxdx} } \right] } = {\frac{1}{\pi }\left[ {\left. {\left( { - \frac{{\sin nx}}{n}} \right)} \right|_{ - \pi }^{ - \large\frac{\pi }{2}\normalsize} + \left. {\left( {\frac{{\sin nx}}{n}} \right)} \right|_{\large\frac{\pi }{2}\normalsize}^\pi } \right] } = {\frac{1}{{\pi n}}\left[ { - \sin \left( { - \frac{{n\pi }}{2}} \right) + \sin \left( { - n\pi } \right) + \sin n\pi - \sin \frac{{n\pi }}{2}} \right] } = {\frac{1}{{\pi n}}\left[ {\cancel{\sin \frac{{n\pi }}{2}} - \cancel{\sin n\pi} + \cancel{\sin n\pi} - \cancel{\sin \frac{{n\pi }}{2}}} \right] = 0.}$ (These results are obvious since this function is odd.)

Calculate the coefficients $${b_n}:$$ ${{b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f\left( x \right)\sin nxdx} } = {\frac{1}{\pi }\left[ {\int\limits_{ - \pi }^{ - \large\frac{\pi }{2}\normalsize} {\left( { - \sin nx} \right)dx} + \int\limits_{\large\frac{\pi }{2}\normalsize}^\pi {\sin nxdx} } \right] } = {\frac{1}{\pi }\left[ {\left. {\left( {\frac{{\cos nx}}{n}} \right)} \right|_{ - \pi }^{ - \large\frac{\pi }{2}\normalsize} - \left. {\left( {\frac{{\cos nx}}{n}} \right)} \right|_{\large\frac{\pi }{2}\normalsize}^\pi } \right] } = {\frac{1}{{\pi n}}\left[ {\cos\left( { - \frac{{n\pi }}{2}} \right) - \cos \left( { - n\pi } \right) - \cos n\pi + \cos \frac{{n\pi }}{2}} \right] } = {\frac{1}{{\pi n}}\left[ {\cos\frac{{n\pi }}{2} - \cos n\pi - \cos n\pi + \cos \frac{{n\pi }}{2}} \right] } = {\frac{2}{{\pi n}}\left( {\cos\frac{{n\pi }}{2} - \cos n\pi } \right).}$ Thus, the Fourier series expansion of the function is given by $f\left( x \right) = \frac{2}{\pi }\sum\limits_{n = 1}^\infty {\frac{1}{n}\left( {\cos\frac{{n\pi }}{2} - \cos n\pi } \right)\sin nx} .$ The graph of the function and the Fourier series expansion for $$n = 10$$ are shown in Figure $$6$$ above.