1. Fourier Series Introduction Fourier sine series (square wave)
Some physics!
Other types of Fourier series
Symmetry
Series expressions of constants
Application to the wave equation
Examples
Dr Mervyn Roy (S6)

2. Fourier Series Lectures www.le.ac.uk/Members/mr6 Notes on Blackboard
Notes on symmetry
Notes on trigonometric identities
Computing exercises
Exam tips
Mock paper 2008
Mock paper 2004
Books
214 course text
Mathematical Methods in the Physical Sciences (M. L. Boas)
Plus, many more!

3. “ One point … cannot be emphasized too strongly
“ One point … cannot be emphasized too strongly. To use mathematics effectively in problems, you need not just knowledge, but skill. Skill can be obtained only through practice. “
- Mary L. Boas

4. Lecture 1, revision…. Expand any function f(x) as a Fourier series.
Use Fourier Sine series, Fourier Cosine series, or Fourier full range series depending on the symmetry.

5. Lecture 1, revision…. Expand any function f(x) as a Fourier series.
Use Fourier Sine series, Fourier Cosine series or Fourier full range series depending on the symmetry.

6. Lecture 1, revision…. Expand any function f(x) as a Fourier series.
Use Fourier Sine series, Fourier Cosine series or Fourier full range series depending on the symmetry.

7. Obtaining series expressions for constants
True for all
Try some values and see what happens (e.g )

8. Obtaining series expressions for constants
True for all
Try some values and see what happens (e.g )
Expansion for  first discovered around 1400 by Indian mathematician, Madhava (found  correct to 11 d.p.).

9. Application to the wave equation
General solution for a wave fixed at x=0 and x=L,
Apply boundary conditions
to find the unknown constants An and Bn.

10. Find the unknown constants An and Bn.

13. Examples
The function is expanded in (a) a Fourier sine series, and (b) a Fourier cosine series. Sketch the form of the function represented by these series in the range
(a) is an odd sine Fourier series.
The sketch must follow:
(b) is an even cosine Fourier series. The sketch must follow:

14. Examples
A function defined between is known to have a Fourier series of the form.
Show that
If between , what function does the Fourier series represent in the range

15. Examples
The function
Expand as a Fourier series.

16. Examples The function Expand as a Fourier series.
Can always use full range Fourier series, apply the formulae and do the algebra.
But, can greatly simplify the algebra if understand the symmetry.
Find,
which is very similar to the Fourier half range sine series for

17. Summary Fourier series are important. used in many areas of physics.
There are 3 types of Fourier series.
which one we use depends on symmetry.
understanding symmetry is important!
see Blackboard: Notes on symmetry
Revise integration by parts!