Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

394 Approximate Calculations \Ch 10
The Fourier coefficients an b , n (n = 0, 1, 2, 3) of the function y = f(x)
may be determined approximately from the formulas:
where 0.866 =
We have
a 2 = s s, + 0.5(s, s 2),
.
10 30 '
6 t = 0.50! + 0.866a 2 + a 8 ,
&, = 0.866 (1, + *,),
& = 8 a, a8 ,
f(x) zz + (an cos nx + b n sin nx).
Other schemes are also used. Calculations are simplified by the use of
patterns.
Example. Find the Fourier polynomial for the function y = f(x)
represented by the table
From formulas (1) we have
= 9.7; a, = 24.9; 2 =i0.3; a a = 3.8;
Consequently,
6, = 13.9; 6 2 = 8.4; 6, = 0.8.
/ (x) ^ 4.8 + (24.9 cos x + 13.9 sin x) + (10.3 cos2x 8.4 sin 2x) +
+ (3. 8 cos 3* + 0.8 sin 3x).
Using the 12-ordinate scheme, find the Fourier polynomials
for the following functions defined in the interval (0,2:rc) by the
(1)