MA 222 Using symmetries to simplify Fourier series K. Rotz
Recall the theorem in your textbook about Fourier series, which says the following:
Fourier’s Theorem: If f(t) is a periodic function with one period defined on the interval (−p, p), then the Fourier series of f is given by ∞ a0 X nπt nπt f(t) = + a cos + b sin 2 n p n p n=1 a πt 2πt 3πt = 0 + a cos + a cos + a cos + ··· 2 1 p 2 p 3 p πt 2πt 3πt + b sin + b sin + b sin + ··· 1 p 2 p 3 p where the coefficients a0, an, and bn are given by 1 Z p a0 = f(t) dt p −p 1 Z p nπt an = f(t) cos dt p −p p 1 Z p nπt bn = f(t) sin dt. p −p p To find the Fourier series of a given function f, all that’s really necessary is to find the coefficients a0, an, and bn, which amounts to computing the three integrals in the statement of the theorem. This unfortunately can be quite a bit of work. To top it off, many times on your homework the function you’re given is a piece-wise defined function, so each of these integrals involves finding two or more integrals on sub-intervals of (−p, p). These problems can get very long and tedious very quickly, so this document is meant to simplify Fourier series for certain types of functions.
1 Even and odd functions
In your high-school algebra class, you probably learned about even and odd functions. If you haven’t (or if you forget – it’s okay if you do!) a function f(t) is called even if f(−t) = f(t) for every value of t, and odd if f(−t) = −f(t) for every value of t. Graphically, even functions have symmetry around the y-axis, i.e. when you reflect the graph of an even function around the y-axis, you get back the same graph. Examples of even functions include even 0 2 16 nπx powers of x (e.g. 1 = x , x , x ), cos(kx) for any constant k (e.g. cos(3x), cos(πx), and cos p ), and the absolute value function |x|.
1 MA 222 Using symmetries to simplify Fourier series K. Rotz
Even Odd Neither
On the other hand, odd functions have symmetry about the origin, i.e. if you reflect the graph of an odd function around both the x-axis and the y-axis, you get back the original graph. Typical examples of odd functions are odd powers of x, sin(kx) for any constant k, and the signum function −1 if x < 0 sgn(x) = 0 if x = 0 1 if x > 0.
2 Algebra with even and odd functions
Here are some useful properties of even and odd functions. All of them are easy to check from the definition of even and odd functions.
1. A product of two even functions is even. Therefore, the function x2 cos(2x) is even since both of its factors are even.
2. A product of two odd functions is even. Thus x3 sin x is even.
3. A product of an odd function with an even function is odd. So the function x cos x is odd since x is odd and cos x is even.
You can relate these to how even and odd numbers behave with respect to sums: the sum of two even numbers is even, the sum of two odd numbers is even, and the sum of an odd number and an even number is odd.
3 Calculus with even and odd functions
The real reason to study even and odd functions is when one is dealing with an integral over an interval which is symmetric around zero, e.g. the interval (−p, p).
2 MA 222 Using symmetries to simplify Fourier series K. Rotz
Principle 1: The integral of an odd function from −p to p is zero. If one looks at the graph of an odd function, the area under the curve and to the left of x = 0 is exactly the same as the area A under the curve and to the right of x = 0, but opposite in sign. Therefore, area under f and area under f and −A + = 0, to the left of x = 0 to the right of x = 0 or, in terms of integrals,
Z p f(t) dt = 0 if f is an odd function. −p
Principle 2: The integral of an even function from −p to p is twice the integral from 0 to p. Again looking at the graph of a typical even function, the area under the curve and to the left of x = 0 is again exactly the same as the area under the curve and to the right of x = 0, but unlike the case for odd functions, it has the same sign. Thus we have area under f and area under f and + to the left of x = 0 to the right of x = 0 A A area under f and = 2 . to the right of x = 0 Translating this into integrals, we have the mathematical state- ment Z p Z p f(t) dt = 2 f(t) dt if f is an even function. −p 0
4 Application to Fourier Series
This section is where the payoff is. We can use the two bolded properties of the previous section to simplify computing Fourier series for functions which are either even or odd. There are two extra
3 MA 222 Using symmetries to simplify Fourier series K. Rotz examples on my web page which use these symmetries to simplify a lot of the computations. The url is http://www.math.purdue.edu/~krotz/teaching .
Case 1: (Odd functions) Suppose that you’re given a function f(t) defined on (−p, p) which is odd, and you’re asked to find the Fourier series of that function. Using the theorem on the first page, and principle 1 from section 3, we have
1 Z p a0 = f(t) dt = 0 if f is odd. p −p
nπt nπt Since f(t) is odd and cos( p ) is even, it follows from property 3 in section 2 that f(t) cos p is also odd, and so again applying the first principle from section 3 we have the following:
1 Z p nπt an = f(t) cos dt = 0 if f is odd. p −p p
nπt Finally, to compute the bn’s, note that f(t) sin p is even since both of its factors are odd. Therefore applying the second principle from section 3, 1 Z p nπt 2 Z p nπt bn = f(t) sin dt = f(t) sin dt. p −p p p 0 p While this isn’t necessarily equal to zero, it’s usually easier to compute this integral. This is especially true for piece-wise defined functions, because instead of doing two separate integrals, you can do a single integral and just double its value.
Case 2: (Even functions) Now let’s suppose that the function f(t) defined on (−p, p) is even. Starting nπt out with the bn coefficients, note that f(t) sin( p ) is odd by property 3 of section 2. Therefore by principle 1 in section 3,
1 Z p nπt bn = f(t) sin dt = 0 if f is even. p −p p
nπt Since f is even, it follows that f(t) cos p is also even. Therefore by principle 2, 1 Z p 2 Z p a0 = f(t) dt = f(t) dt p −p p 0 1 Z p nπt 2 Z p nπt an = f(t) cos dt = f(t) cos dt. p −p p p 0 p
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