104 Chapter 1 Fourier Series and Integrals
ixi cosxi cos 2xi cos 3xi sin(xi)/xi
00 1. 01. 01. 01. 0
1 π 6 0. 86603 0 .50 0. 95493
2 π 3 0. 5 − 0. 5 − 1. 00. 82699
3 π 2 0 − 1 .00 0. 63662
4 2 π 3 − 0. 5 − 0. 51. 00. 41350
5 5 π 6 − 0. 86603 0 .50 0. 19099
6 π − 1. 01. 0 − 1. 00. 0
Table 3 Numerical informationn aˆn an Error
00. 58717 0. 58949 0. 00232
10. 45611 0. 45141 0. 00470
2 − 0. 06130 − 0. 05640 0. 00490
30. 02884 0. 02356 0. 00528
Table 4Approximate coefficients
of sin(x)/xThus the graph ofF(x)cuts the graph off(x)at the pointsxi,i= 1 , 2 ,...,r.
Example.
Calculate the approximate Fourier coefficients off(x)=sin(x)/xin−π<
x<π.Sincefis even, it will have a cosine series. We simplify computation by
using the half-range formulas and makingseven. We takes= 6 ,x 0 = 0 ,x 1 =
π/ 6 ,...,x 5 = 5 π/ 6 ,x 6 =π. The numerical information is given in Table 3.
The results of the calculation are given in Table 4. On the left are the approx-
imate coefficients calculated from the table. On the right are the correct values
(to five decimals), obtained with the aid of a table of the sine integral (see Ex-
ercise 2). Figure 12 shows the difference betweenf(x)andF(x)(the sum of the
Fourier series using the approximate coefficients throughaˆ 6 ).
For hand calculation, choosingsto be a multiple of 4 makes many of the
cosines “easy” numbers such as 1 and 0.5. When the calculation is done by
digital computer, this is not a consideration.
EXERCISES
1.Since Table 3 gives sin(x)/xfor seven points, seven cosine coefficients can
be calculated. Findaˆ 6.