7.1: Introduction to Fourier Series
- Page ID
- 106841
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Chapter 3 we learned that a function \(f(x)\) can be expressed as a series in powers of \(x\) as long as \(f(x)\) and all its derivatives are finite at \(x=0\). We then extended this idea to powers of \(x-h\), and called these series “Taylor series”. If \(h=0\), the functions that form the basis set are the powers of \(x: x^0, x^1, x^2...\), and in the more general case of \(h\neq0\), the basis functions are \((x-h)^0, (x-h)^1, (x-h)^2...\)
The powers of \(x\) or \((x-h)\) are not the only choice of basis functions to expand a function in terms of a series. In fact, if we want to produce a series which will converge rapidly, so that we can truncate if after only a few terms, it is a good idea to choose basis functions that have as much as possible in common with the function to be represented. If we want to represent a periodic function, it is useful to use a basis set containing functions that are periodic themselves. For example, consider the following set of functions: \(\sin{(nx)},\;n=1, 2, ..., \infty\):
![basisset.jpg](https://chem.libretexts.org/@api/deki/files/141621/basisset.jpg?revision=1)
We can mix a finite number of these functions to produce a periodic function like the one shown in the left panel of Figure \(\PageIndex{2}\), or an infinite number of functions to produce a periodic function like the one shown on the right. Notice that an infinite number of sine functions creates a function with straight lines! We will see that we can create all kinds of periodic functions by just changing the coefficients (i.e. the numbers multiplying each sine function).
![fourier1.jpg](https://chem.libretexts.org/@api/deki/files/141622/fourier1.jpg?revision=1)
So far everything sounds fine, but we have a problem. The functions \(\sin{nx}\) are all odd, and therefore any linear combination will produce an odd periodic function. We might need to represent an even function, or a function that is neither odd nor even. This tells us that we need to expand our basis set to include even functions, and I hope you will agree the obvious choice are the cosine functions \(\cos{(nx)}\).
Below are two examples of even periodic functions that are produced by mixing a finite (left) or infinite (right) number of cosine functions. Notice that both are even functions.
![Screen Shot 2019-10-25 at 11.18.18 AM.png](https://chem.libretexts.org/@api/deki/files/245106/Screen_Shot_2019-10-25_at_11.18.18_AM.png?revision=1)
Before moving on, we need to review a few concepts. First, since we will be dealing with periodic functions, we need to define the period of a function. As we saw in Section 1.4, a function \(f(x)\) is said to be periodic with period \(P\) if \(f(x)=f(x+P)\). For example, the period of the function of Figure \(\PageIndex{4}\) is \(2\pi\).
![period1.jpg](https://chem.libretexts.org/@api/deki/files/141623/period1.jpg?revision=1)
How do we write the equation for this periodic function? We just need to specify the equation of the function between \(-P/2\) and \(P/2\). This range is shown in a red dotted line in Figure \(\PageIndex{4}\), and as you can see, it has the width of a period, and it is centered around \(x=0\). If we have this information, we just need to extend the function to the left and to the right to create the periodic function:
![period2.jpg](https://chem.libretexts.org/@api/deki/files/141624/period2.jpg?revision=1)