Fourier Series January 26 2009 Outline Fourier Series Review last class Fourier series as expansions in periodic functions Larry Caretto Mechanical Engineering 501B Comparison to eigenfunction expansions Odd and even functions Periodic extensions of non periodic functions Complex Fourier series Seminar in Engineering Analysis January 26 2009 2 Review Last Lecture Review Orthogonal Functions Discussed Sturm Liouville Problem Solutions are a set of orthogonal eigenfunctions ym x that can be used to express other functions f x p x is f x a m y m x weight m 0 b function p x ym x f x dx Have to a ym f a compute m ym ym b p x ym x ym x dx am a 3 Defined in terms of inner product Norm of two like eigenfunctions yi b y y y x y x p x dx i i j j yi ij 2 a Orthonormal beigenfunctions f f f i j i x f j x p x dx ij a Convert orthogonal eigenfunctions to orthonormal eigenfunctions fi yi yi 4 Review Sturm Liouville Review Sturm Liouville Results General equation whose solutions provide orthogonal eigenfunctions Eigenvalues are real Eigenfunctions defined over a region a x b form an orthogonal set over that region Eigenfunctions form a complete set over an infinite dimensional vector space We can expand any function over the region in which the Sturm Liouville problem is defined in terms of the eigenfunctions for that problem Defined for a x b with p x q x and r x continuous and p x 0 Homogenous differential equation and boundary conditions shown below d dy r x dx dx q x p x y 0 k1 y a k 2 l 1 y b l 2 dy dx x a dy dx x b 0 0 5 ME 501B Engineering Analysis 6 1 Fourier Series January 26 2009 Review Eigenfunction Expansions f x a m y m x Eigenfunction expansion formula m 0 Equation for am coefficients in eigenfunction expansion of f x Review Expansion of f x x Start with general equation for am Use ym sin m x over 0 x 1 which is a Sturm Liouville solution Weight function p x 1 1 b am ym f ym ym p x y m x f x dx am a b p x y m ym f y m ym 1 x sin m x dx x sin m x dx 0 1 sin 2 0 1 m x dx 2 2 1 m 1 m 0 x ym x dx f x x a 7 2 sin x sin 2 x sin 3 x sin 4 x L 1 2 3 4 8 Review Partial Sums Large Review Partial Sums Small 1 2 1 1 1 1 1 1 0 9 0 9 0 8 0 8 Exact Series sums Series sum 0 7 1 term 0 6 2 terms 3 terms 0 5 5 terms 0 7 Exact 0 6 10 terms 25 terms 0 5 10 terms 50 terms 100 terms 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 9 Fourier Series 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 10 Fourier Series Have same basic idea as eigenfunction expansions Represent other functions f x as a series of sines and cosines Compute coefficients in a similar way to eigenfunction expansions Fourier series based on periodicity of trigonometric functions 11 ME 501B Engineering Analysis 0 1 x x Equations for series and coefficients defined for L x L n x n x f x a0 an cos bn sin L L n 1 1 1 n x f x dx an f x cos dx 2L L L L L L 1 n x bn f x sin dx L L 12 L L a0 L 2 Fourier Series January 26 2009 Basis for Fourier Series Even and Odd Functions Applies to periodic functions Must be piecewise continuous Derivative must exist at all points in the period At discontinuity both left hand and righthand derivatives exist Fourier series converges to f x At discontinuity the series converges to average of f x and f x Odd function f x f x like sine Even function g x g x like cosine For odd f x For even g x Even and Odd Functions II f x dx 0 L L 0 14 L L For odd functions a 1 f x dx 0 the Fourier series 2L L L coefficients a0 1 n x f x cos and an are zero an dx L L L Can have half range expansions that consider only 0 x L Apply to non periodic functions to get periodic extensions 15 16 Even and Odd Fourier Series We can use relations for even and odd functions to transform Fourier coefficient equations For odd functions f x f x we have only sine terms with coefficients bn Even functions have only cosine and constant terms with coefficients a0 and an Next chart shows series and coefficients 17 ME 501B Engineering Analysis L Even and Odd Functions III The product of an even function g x g x and an odd function f x f x is an odd function Proof define h x f x g x h x f x g x f x g x h x L 1 n x b Fourier coefficient n f x sin dx L L sine g x dx 2 g x dx L 13 Since sine is an odd function bn 0 if f x is even i e f x f x L cosine Odd Function Sine Series f x is an odd function f x f x Sine only series defined for L x L bn computed as integral from 0 to L n x f x bn sin L n 1 bn 1 2 n x n x f x sin dx f x sin dx L L L0 L L L L 18 3 Fourier Series January 26 2009 Even Function Cosine Series Half interval series f x is an even function f x f x Cosine only series defined for L x L an evaluated by integral from 0 to L Based on equations for sine series for odd functions and cosine series for even functions Either the sine or cosine series can apply to any function n x f x a0 an cos L n 1 L L 1 1 a0 f x dx f x dx 2L L L0 Can use sines to expand even functions Can use cosines to expand odd functions Each series defined for 0 x L Behavior outside region 0 x L depends on the function 1 2 n x n x an f x cos dx f x cos dx L L L0 L L L L 19 20 Fourier Sine Series for f x x Half interval Series Example 1 1 …