Page 267 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

• In general, for ω ≠ , 1

1
1
)
C = --- ∫ T ft()e – jnωt ( d ωt = ------ ∫ 2π ft()e – jnωt ( d ωt )
n
T
0 2π 0
• We can derive the trigonometric Fourier series from the exponential series from the relations
a = C + C – n
n
n
and
(
b = jC – C ) – n
n
n

• For even functions, all coefficients C i are real
• For odd functions, all coefficients C i are imaginary

• If there is half−wave symmetry, C = 0 for n = even
n
• C – n = C ∗ always
n
• A line spectrum is a plot that shows the amplitudes of the harmonics on a frequency scale.

⁄
• The frequency components of a recurrent rectangular pulse follow a sin x x form.
• We can evaluate the Fourier coefficients of a function based on observed values instead of an
analytic expression using numerical evaluations with the aid of a spreadsheet.

• A power series has the form
∞
2
k
∑ a x = a + a x + a x + …
k
0
1
2
k = 0
• A function fx() that possesses all derivatives up to order at a point x = x 0 can be expanded
n
in a Taylor series as
f'' x ( ) f n () x ( )
0
fx() = fx ( 0 ) f' x ) ( + 0 ( x – x ) 0 -------------- xx ) ( + – 0 2 + … + ------------------- x –( n! 0 x ) 0 n
2!
If x = 0 , the series above reduces to
0
f'' 0() 2 f n () 0 () n
fx() = f0() + f' 0()x + ------------x + … + -----------------x
2! n!
and this relation is known as Maclaurin series

• We can also obtain a Taylor series expansion with the MATLAB taylor(f,n,a) function where
f is a symbolic expression, n produces the first terms in the series, and a defines the Taylor
n
approximation about point . a

6−50 Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications