A Fourier Series is an expansion of a periodic function f (x) in terms of an infinite sum of sines and cosines. Fourier series makes use of the orthogonality relationships of the sine and cosine functions. Since infinite cosine functions and infinite sine functions are mutually orthogonal/exclusive. Summary of Fourier Series Suppose f is a piecewise continuous periodic function of period 2 L, then f has a Fourier series representation ∑ ∞ = = + + 1 0 cos sin 2 ( ) n n n L n x b L n x a a f x π π. Where the coefficients a's and b's are given by the Euler-Fourier formulas: ∫ − = L L m dx L m x f x L a π ( )cos 1, m = 0, 1, 2 These notes introduce Fourier series and discuss some applications. 1Introduction Joseph Fourier (1768-1830) who gave his name to Fourier series, was not the first to use Fourier series neither did he answer all the questions about them. These series had already been studied by Euler, d'Alembert, Bernoulli and others be-fore him. Lecture Notes 2 August 24, 2015 Fourier Series We have come across the term Fourier Series in the last chapter. This is a term so dear to Signal Processing, a panacea for many problems there. A natural question (often forgotten) here is why Fourier Series/Analysis?". This is particularly signi cant to this class, where students are from di Fourier Series! The solution of the original problem of heat conduction in a bar would then be solved analytically by the infinite series u(x,t)= ∞ n=1 b n sin nπx L e−n 2π2 L2 Kt, where the b n are called the Fourier coefficientsof f on the interval [0,L]. Fourieractuallygaveaproofofthe convergenceofthe serieshe developed(in his FOURIER SERIES We write L2([ ˇ;ˇ]) for the set of functions f: [ ˇ;ˇ] !R which are square-integrable, i.e. for which Z ˇ ˇ f(x)2 dxconverges. Given an element of this set, you can think of it as giving a function on all of R with period 2ˇ{ just repeat the same function from [ˇ;3ˇ] that you used on [ ˇ;ˇ], etc. (2) Fourier Series is a different type of infinite series where a periodic function is represented in terms of a suitable series of sines and/or cosines. One such Fourier series is, g(t) = 1 6 − X∞ n=1 cos2πnt n2π2 = 1 6 − 1 π2 cos2πt + cos4πt 22 + cos6πt 32 + cos8πt 42 +··· , which represents a function with a unit period. This is called a Fourier Series. fT(x) = a0 2 + ∞X n=1 [an cos (n ω0x) + bn sin (n ω0x)] where ω0 = 2π T is the fundamental frequency. The frequencies of the Sine and Cosine terms are integer multiples of the frequency of the original function fT (x) which is 1 T corresponding to a radian frequency of ω0 = 2π T. Advantages of Fourier Series-d If a function f(x) is not periodic it evidently cannot be expanded in a Fourier series for all values of x. Neverthelless we can find a Fourier series to represent it over any range of width 2π, say from -π to π or from 0 to 2π. For consider a new function, say g(x), obtained by taking the Fourier Series Fourier series started life as a method to solve problems about the ow of heat through ordinary materials. It has grown so far that if you search our library's catalog for the keyword Fourier" you will nd 618 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication Fourier Series pdf. This note covers the following topics: Computing Fourier Series, Computing an Example, Notation, Extending the function, Fundamental Theorem, Musical Notes, Parseval's Identity, Periodically Forced ODE's, General