Friday, August 17, 2012

Odd and Even Functions - Oddly Interesting

The idea that functions can be either "odd" or "even" is an initially a surprising idea.

Yes, a number can be odd or even and some people are odd but can a function be odd or even ?

The formal definition of an even function is that f(-t) = f(t) for all of t, while for an odd function f(-t) = - f(t) for all of t.

These formal definitions don't immediately expose the power of the concept. Examining graphs of odd and even functions quickly reveal why the concept is useful and interesting. For example, below is a graph of the function is x^3sin(x), which satisfies the condition of an even function.

 Figure: Plot of x^3 sin (x) (image from Wolfram Alpha)

Please note the following that the function is fully symmetrical around the y-axis (which is is the same as saying f(-x) = f(x)) As a result, iintegrating the function from 0 to any positive value of x will have the identical answer as integrating from 0 to the same negative value. This also means that integrating f(x) from x= -b to x= b is the same as doubling the definite integral from x= 0 to x= b.

Some common even functions include, f(x) = x^2,  f(x) = cos(x), f(x) = SQRT(1 + x^2), f(x) = 1/(4-x^2) and many more.

Odd functions also have an underlying symmetry but it is 180 degrees around the origin. For example, plotting  f(x) = x^2 sin(x) quickly reveals a quite different symmetry from the x^3 sin(x).

Figure: Plot of x^2 sin(x) (image from Wolfram Alpha)

For odd functions, the y value for a particular positive value of x (say x= b) will by -y for x = -b. As a result, the definite integrals from 0 to b will all be the same absolute value but the opposite sign for 0 to -b. Thus, integrating any odd function from -b to b will always results in "0" as the answer (a nice relationship to exploit when simplifying definite integrals in Fourier Analysis). This is all obvious from looking at the graph above. As often the case in mathematics, symmetry is a wonderful thing !

The relationships that follow on from these simple definitions are even also satisfying, namely:

a) An Odd function multiplied by an Odd function results in an Even function ( O x O = E)
b) An Odd function multiplied by an Even function results in an  Odd function (O x E = O)
c) An Even function multiplied by an Even function results in an Even function (E x E = E)
d) The reciprocal of a Odd function is an Odd function ( 1/O = O)
e) The reciprocal of an Even function is an Even function (1/E = E)

Somehow, these relationships seem intuitive and are in keeping with our numerical sense of "odd" and "even". They can also be easily confirmed through plotting and/or simple substitution.

In fact, I strongly encourage you to prove these relationships to yourself. That way, you will really see what is oddly interesting about symmetry.

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