Ch_05 - Fourier Series - Summary

# Ch_05 - Fourier Series - Summary - Chapter 5 The Fourier...

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Chapter 5 The Fourier Series, Integrals and Transforms

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Table of Contents 1. Periodic Functions, Trigonometric Series 1.1 Periodic Functions 1.2 Trigonometric Series 2. Fourier Series 2.1 Euler Formulas for the Fourier Series 3. Functions of Period 22 L 4. Half-Range Expansions 4.1 Even and Odd Functions 4.2 Half-Range Expansions 5. Forced Oscillation 6. Approximation by Trigonometric Polynomials 7. Fourier Integrals 8. Fourier Transforms 8.1 Integral Transforms 8.2 Fourier Transforms and Its Inverses 8.3 Linearity. Fourier Transform of Derivatives 9. Fourier Cosine and Sine Transforms 9.1 Finite Fourier Cosine and Sine Transforms 9.2 Fourier Cosine and Sine Transforms Pair 9.3 Linearity. Transforms of Derivatives Appendix
The Fourier Series, Integrals and Transforms Some functions can be represented by power series (which would have to be their Taylor’s series). Power series 0 () n n n f xa x Not all functions have to be Taylor’s series, if they do necessary equal to the Taylor’s series 1. To have a Taylor’s series, () f x must have all derivative at x a . 2. If () f x has a Taylor’s series, it only equals it if lim 0 n x R  in Taylor’s formula. Taylor’s Series: f x be a continuous function with (1 ) n derivative defined through out an interval containing the number a , then expansion of () f x about some point x a in power of x a 0 2 ! = ( ) ( ) '( ) "( ) ( ) 2! ! n n n n n n fx f a n f ax a f a f a f a R n   where the remainder n R is given by 1 ) ) ! n n n Rf c n with c some of acx EXAMPLE: 23 24 35 1 3! cos 1 4! sin 3! 5! x n n n xx ex R xR R     What about all the rest of the functions? Instead of using series of power functions, may be we can use series of trigonometric functions, like 11 2 2 3 3 ( ) ( cos sin ) ( cos2 sin 2 ) ( cos3 sin3 ) a x b x a x a x 

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The Fourier Series, Integrals, and Transforms 1. Periodic Functions, Trigonometric Series 1. Periodic Functions, Trigonometric Series 1.1 Periodic Functions Definition: A function () f x is “ periodic ”, if () f x exists a positive number 2 L such that for every x in the domain () fx , (2 ) ( ) f xLf x . The number 2 L is called a period of 2 L . Definition: Let () be defined on [,] ab . The periodic extension of () with respect to [,] is normalization of the function 0 ( ) if () ( ( )) otherwise f sa x b fx b a  Ex: periodic extension of () | | x  on [, ]  Definition: Continuous f x is said to be “ continuous ” at x c on [,] f x is defined at x c and lim ( ) ( ) xc f xf c . Definition: Piecewise Continuous f x is said to be “ piecewise continuous ” on [,] if and only if there exists a finite subdivision 012 n ax x x x b   , such that f x is continuous on each subinterval 1 (,) ii x x and the left and right hand limits of () f x exists at each i x .
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Ch_05 - Fourier Series - Summary - Chapter 5 The Fourier...

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