Chap4 Lec8 Fourier Series (Student)

# Chap4 Lec8 Fourier Series (Student) - H UT DEPARTM ENT OF M...

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HUT – DEPARTMENT OF MATH. APPLIED -------------------------------------------------------------------------------------------------------- -- IDE – ADVANCED PROGRAM CHAPTER 4: FOURIER SERIES PhD. NGUYEÃN QUOÁC LAÂN (11/2007)

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CONTENTS ------------------------------------------------------------------------------------------------------------------------------- 8.1 EXPRESSING A FUNCTION IN A TRIGONOMETRIC SERIES 8.2 FOURIER COEFFICIENTS 8.3 FOURIER SERIES OF EVEN AND ODD FUNCTION 8.4 FOURIER SERIES ON HALF – RANGE. EXTENSION 8.5 ORTHOGONAL EXPANSION
BEGINNING PROBLEM ------------------------------------------------------------------------------------------------------------------------------------ Express a given function f(x) in the trigonometric serie: { } { } [ ] [ ] π π , , sin cos 2 ) ( : , Find 1 0 - + + = = x nx b nx a a x f b a n n n n n The trigono. system {sinnx, cosmx}, m, n 0 is orthogonal with respect to the integral scalar product: u v (u,v) = 0 m n dx mx nx , 0 sin cos 2200 = - π π = = = = - 0 , 2 0 , , 0 cos cos m n m n m n dx mx nx π π π π = = - m n m n dx mx nx , , 0 sin sin π π π

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FOURIER COEFFICIENTS ------------------------------------------------------------------------------------------------------------------------------------ ( 29 0 , cos 1 = - n nxdx x f a n π π π ( 29 1 , sin 1 = - n nxdx x f b n π π π Fourier coefficients of the periodic function f(x) over (– π , π ) [ ] ( 29 ( 29 e.! almost sin cos 2 1 0 x f nx b nx a a n n n + + = Fourier serie:
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