By Mark S. Gockenbach (siam, 2010)

Chapter 12: More about Fourier series

Download 2.45 Mb.

Page	20/25
Date	09.06.2018
Size	2.45 Mb.
	#53772

1 ... 17 18 19 20 21 22 23 24 25

Chapter 12: More about Fourier series

Section 12.1: The complex Fourier series

It is no more difficult to compute complex Fourier series than the real Fourier series discussed earlier. You should recall that the imaginary unit sqrt(-1) is denoted by i (or j, but I will use i) in MATLAB. As an example, I will compute the complex Fourier series of on the interval

clear

f=inline('x.^2','x');

syms x pi n

1/2*int(f(x)*exp(-i*pi*n*x),x,-1,1)

ans =

- (exp(pi*n*sqrt(-1))*(pi^2*n^2 + 2*pi*n*sqrt(-1) - 2)*sqrt(-1))/(2*pi^3*n^3) - ((1/exp(pi*n*sqrt(-1)))*(- pi^2*n^2 + 2*pi*n*sqrt(-1) + 2)*sqrt(-1))/(2*pi^3*n^3)

We now have a simplification problem similar to what we first encountered in Chapter 5, and for which I wrote the function mysubs. In this case, since n is an integer, exp(-iπn) equals (-1)ⁿ, as does exp(iπn). For this reason, I wrote another version of mysubs, namely mysubs1, to handle this case:
type mysubs1

function expr=mysubs1(expr,varargin)

%expr=mysubs1(expr,m,n,...)

% This function substitutes (-1)^m for exp(-i*pi*m) and

% for exp(i*pi*m), and similarly for exp(-i*pi*n) and

% exp(i*pi*n), and any other symbols given as inputs.

syms pi i

k=length(varargin);

for jj=1:k

expr=subs(expr,exp(-i*(pi*varargin{jj})),(-1)^varargin{jj});

expr=subs(expr,exp(i*pi*varargin{jj}),(-1)^varargin{jj});

end

expr=simplify(expr);
Now I apply mysubs1:
a=mysubs1(ans,n);

pretty(a)

2 (-1)

-------

2 2

pi n
This formula obviously does not hold for

, and so I need to compute the coefficient a₀ separately:
a0=1/2*int(f(x),x,-1,1)

a0 =

1/3
Now I can define the partial Fourier series. Recall from Section 9.1 of the text that, since f is real-valued, the following partial Fourier series is also real-valued (I choose to use 21 terms in the series):
S=a0+symsum(a*exp(i*n*pi*x),n,-10,-1)+symsum(a*exp(i*n*pi*x),n,1,10);

t=linspace(-1,1,101)';

plot(t,subs(S,x,t))

Warning: Imaginary parts of complex X and/or Y arguments ignored

The approximation is not bad, as the following graph of the error shows:
plot(t,f(t)-subs(S,x,t))

Warning: Imaginary parts of complex X and/or Y arguments ignored

You should notice the above warning messages complaining that the quantity I am trying to graph is complex. This is not important, because I know from the results in the text that the partial Fourier series must evaluate to a real number (since the underlying function f is real-valued). Therefore, any imaginary part must be due to roundoff error, and it is right to ignore it. Here is an example:

subs(S,x,1)

ans =

0.9614 + 0.0000i

imag(ans)

ans =

3.0815e-033
(The imag command extracts the imaginary part of a complex number.)

Download 2.45 Mb.

Share with your friends:

1 ... 17 18 19 20 21 22 23 24 25