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7.1: Introduction to Fourier Series

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  • 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: \(\sin{(nx)},\;n=1, 2, ..., \infty\):

    Figure \(\PageIndex{1}\): Some examples of the family of funtions \(\sin{(nx)}\). From left to right: \(\sin{(x)},\sin{(2x)},\sin{(3x)}\) and \(\sin{(10x)}\) (CC BY-NC-SA; Marcia Levitus)

    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).

    Figure \(\PageIndex{2}\): Examples of periodic functions that are linear combinations of \(sin{(nx)}\) (CC BY-NC-SA; Marcia Levitus)

    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
    Figure \(\PageIndex{3}\): Examples of periodic functions that are linear combinations of \(\cos (nx)\) functions (CC BY-NC-SA; Marcia Levitus)

    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 \(2\pi\).

    Figure \(\PageIndex{4}\): A periodic function with period \(P=2\pi\) (CC BY-NC-SA; Marcia Levitus)

    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:

    Figure \(\PageIndex{4}\) (CC BY-NC-SA; Marcia Levitus)
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