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)

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!

“ 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

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.

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.

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.

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

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.).

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.

Find the unknown constants An and Bn.

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:

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

Examples The function Expand as a Fourier series.

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

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!