Page 225 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

f(t) f(t) f(t)
2
t + k
t 2 k
t t t
0 0 0

Figure 6.7. Examples of even functions

A function ft() is an odd function of time if the following relation holds.

f – –() = ft() (6.16)
t

t
that is, if in an odd function we replace with t , we obtain the negative of the function ft() .
–
Thus, polynomials with odd exponents only, and no constants are odd functions. For instance,
the sine function is an odd function because it can be written as the power series
3 5 7
t
t
t
sin t = t – ---- + ----- – ----- + …
-
3! 5! 7!
Other examples of odd functions are shown in Figure 6.8.
f(t) f(t) f(t)

mt t 3

t t t
0 0 0

Figure 6.8. Examples of odd functions

We observe that for odd functions, f0() = 0 . However, the reverse is not always true; that is, if
f0() = 0 , we should not conclude that ft() is an odd function. An example of this is the function

ft() = t 2 in Figure 6.7.
The product of two even or two odd functions is an even function, and the product of an even
function times an odd function, is an odd function.

Henceforth, we will denote an even function with the subscript , and an odd function with the
e
subscript . Thus, f t() and f t() will be used to represent even and odd functions of time
o
e
o
respectively.
Also,
T T
∫ f t() td = 2 ∫ f t() t (6.17)
d
e
e
– T 0

6−8 Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications