Fourier Sine and Cosine Series

Date: 02/23/2001 at 10:50:06
From: Bong Yong Chuen
Subject: Fourier Series

Dear sir,

I'm a student from National University of Singapore.

1. Why do we need to separate Fourier series into "Fourier Sine 
Series" and "Fourier Cosine Series"? What is the use of them?

2. Each series has its own formulae. How do we know when to use 
which series formalae to solve questions?

3. The Fourier Sine/Cosine Series are both applicable to Half-range 
Expansions. Is the use the same as in the Fourier series?

Thank you.

Date: 02/25/2001 at 14:13:50
From: Doctor Fenton
Subject: Re: Fourier Series

Hi Bong,

Thanks for writing to Dr. Math.  

>1. Why do we need to separate Fourier series into "Fourier Sine >Series" and "Fourier Cosine Series"? What is the use of them?

This is a reflection of the symmetry of the trigonometric functions:
sin(nx) is an odd function, and cos(nx) is an even function. For the
basic period [-pi,pi], if we are expanding a function f periodic on
[-pi,pi] into a Fourier Series, the coefficients are given by 
                  
                    pi
                1  / 
        a(n) = --- | f(x)cos(nx)dx
                pi /
                   -pi

and a similar integral for b(n) with integrand f(x)sin(nx). If f(x)
is even, then the integrand for the a(n) is even, while the integrand
for b(n) is odd. An odd function integrated over a symmetric interval
always yields 0, so the Fourier series for an even function will 
contain no sin(nx) terms, and f(x) has a cosine series.  

If f(x) is odd, then f(x)sin(nx) is even and f(x)cos(nx) is odd, 
so all the a(n)'s will be 0, and f(x) will have a sine series.


>2. Each series has its own formulae. How do we know when to use 
>which series formalae to solve questions?

I think the answer to 1 explains this: if f(x) is even, you will 
obtain a cosine series; if f(x) is odd, you will obtain a sine series.  
There is no need to compute coefficients that you know will be 0 by 
symmetry.


>3. The Fourier Sine/Cosine Series are both applicable to Half-range 
>Expansions. Is the use the same as in the Fourier series?

There are times when you want sine series or cosine series, perhaps to
satisfy boundary conditions when solving a partial differential 
equation. If f(x) is given on [0,pi], you can obtain a representation 
for f(x) on [0,pi] by first extending f(x) to [-pi,pi], and two ways 
to do this are

(1) extend f(x) to be even on [-pi,pi], i.e. define f(-x)=f(x) for x 
    in (0,pi];

or 

(2) extend f(x) to be odd on [-pi,pi] by defining f(-x)=-f(x) for
    x in (0,pi] .

The extended f(x) on [-pi,pi] is either even, and therefore has a 
cosine series on [-pi,pi], which in particular applies on [0,pi], 
or it is odd, so that the extended function has a sine series on 
[-pi,pi], which in particular applies on the subinterval [0,pi].

Does this answer your questions adequately?  If not, or if you have 
further questions, please write again.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/