文档介绍：LEIF MEJLBRO
FOURIER SERIES AND SYSTEMS OF
DIFFERENTIAL...
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Leif Mejlbro
Fourier Series and Systems of
Differential Equations and Eigenvalue
Problems
Guidelines for Solutions of Problems
Calculus 4b
Fourier Series and Systems of Differential Equations and
Eigenvalue Problems – Guidelines for Solutions of Problems – Calculus 4b
© 2007 Leif Mejlbro & Ventus Publishing Aps
ISBN 978-87-7681-242-3
Calculus 4b Contents
Contents
Preface 6
1 Review of some important trigonometric formulæ 7
Trigonometric formulæ 7
Integration of trigonometric polynomials 7
2 Fourier series; methods of calculation 11
General 11
Standard procedure 14
Standard Fourier series 18
The square function 18
The identical function in ] - π; π[ 19
The first sawtooth function 19
The second sawtooth function 21
Expansion of cosine in a sinus series over a half period 22
Symmetric parabolic arc in [- π; π] 23
Hyperbolic cosine in [- π; π] 23
Exponential function over ] - π; π] 25
3 A list of problems in the Theory of Fourier series 27
A piecewise constant function 27
4
Calculus 4b Contents
Piecewise linear functions 32
A piecewise polynomial of second degree 37
A piecewise polynomial of third degree 47
A piecewise polynomial of fourth degree 49
Piecewise sinus 50
Piecewise cosine 57
Mixed sinus and cosine 64
A piecewise polynomial times a trigonometric function 66
The exponential function occurs 72
The problem of the assumption of no vertical half tangent 77
4 Systems of linear dierential equations; methods 80
The Existence and Uniqueness Theorem 80
Constant system matrix A 80
Eigenvalue method, real eigenvalues, A a constant (2 x 2)-matrix 82
The eigenvalue method, complex conjugated eigenvalues, comples calculations 83
The eigenvalu