HMS215 Fourier series(Final form Feb 07)

HMS215 Fourier series(Final form Feb 07) - Fourier Series...

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Unformatted text preview: Fourier Series Nian Li, David Lucy and Manmohan Singh Faculty of Engineering and Industrial Sciences, Engineering Mathematics Discipline, Swinburne University of Technology, Hawthorn, Victoria, Australia, 3122. Contents 1 Fourier Series 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Integrals of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . 5 1.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Fourier series for functions of period 2 . . . . . . . . . . . . . . . . . . 8 1.2.3 Functions of Period l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.4 Odd and Even Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Full Range and Half Range Expansions . . . . . . . . . . . . . . . . . . 19 1.3 Dirichlets Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.1 Bonus Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.2 Operations on Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.3 Complex Form of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.4 Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 Use of Fourier Series in Engineering Applications . . . . . . . . . . . . . . . . . 31 1.4.1 Vibrations of a String, (the Wave Equation) . . . . . . . . . . . . . . . . 31 1.5 Problems and Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Problems in Partial differential Equations 48 2.1 Method of Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 Partial Differential Equations: Answers . . . . . . . . . . . . . . . . . . . . . . 52 2.2.1 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2.2 Method of Separation of Variables . . . . . . . . . . . . . . . . . . . . . 52 3 Fourier Transforms 55 3.1 Parsevals Formula for Fourier Transforms . . . . . . . . . . . . . . . . . . . . . 56 3.2 Properties of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Applications to two Port Network Theory . . . . . . . . . . . . . . . . . . . . . 58 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6 Extra Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3....
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This note was uploaded on 10/23/2009 for the course HES 2340 taught by Professor Tomedwards during the Three '09 term at Swinburne.

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HMS215 Fourier series(Final form Feb 07) - Fourier Series...

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