《高等数学》课程电子教案（PPT课件）Chapter 12 Infinite series Sec 7 Fourier Series

11Chapter中InfiniteSeriesSec.7FourierSeries

Chapter 11 Infinite Series Sec.7 Fourier Series

$7 Fourier SeriesI. Introduction-1.when -π≤t<0u(t)Periodic function1.when 0≤t<元uQ......i...........................................R.........................................1元........................元0................C

§7 Fourier Series I. Introduction o t u −  1 −1 Periodic function      − −   =   t t u t 1, when 0 1, when 0 ( )

$7 FourierSeriesI. Introductionu=-sint元u10.5t-212-13C

§7 Fourier Series u sint 4  = I. Introduction

$7 Fourier SeriesI. Introductionsin3t)(sint +u=3元3341

§7 Fourier Series sin3 ) 3 1 (sin 4 u t + t  = I. Introduction

$7 FourierSeriesI. Introduction(sint +=sin3t+sin5t)U=3元110.521132O

§7 Fourier Series sin5 ) 5 1 sin3 3 1 (sin 4 u t + t + t  = I. Introduction

$7 FourierSeriesI. Introductionsin3t +sin5t +sin7t)(sint +U=一I53元u0.5-212-13-0

§7 Fourier Series sin7 ) 7 1 sin5 5 1 sin3 3 1 (sin 4 u t + t + t + t  = I. Introduction

$7 Fourier SeriesI.Introductionsin3t +sin5t +sin 7t +--sin9t)(sint+.u=-93元5n5-2123-14sin3t +=sin5t +=sin7t +..)u(t) = -(sint +=s35元(一元<t<元,t±0)

§7 Fourier Series sin7 ) 7 1 sin5 5 1 sin3 3 1 (sin 4 ( ) + + + +  u t = t t t t (−  t  ,t  0) sin9 ) 9 1 sin7 7 1 sin5 5 1 sin3 3 1 (sin 4 u t + t + t + t + t  = I. Introduction

$7 Fourier SeriesI, Orthogonality of Trigonometric functions1.Trigonometric series80aoZ(a, cos nx + b, sin nx)2n=l2.Trigonometric functions1, cos x,sin x,cos 2x, sin 2x, ...cos nx, sinnx,

§7 Fourier Series II、Orthogonality of Trigonometric functions 1. Trigonometric series   = + + 1 0 ( cos sin ) 2 n an nx bn nx a 2. Trigonometric functions 1,cos x,sin x,cos 2x,sin2x, cos nx,sinnx, 

$7 FourierSeriesII, Orthogonality of Trigonometric functions3.OrthogonalityAcos nxdx = 0,sinnxdx = 0, (n =1,2,3,...)元一元0,m+n元sinmx sin nxdx :元，元m=n[0,m≠n元cos mx cos nxdx :一元[元,m=n元sin mx cos nxdx = 0.元

§7 Fourier Series II、Orthogonality of Trigonometric functions 3. Orthogonality cos = 0, −   nxdx sin = 0, −   nxdx (n = 1,2,3, ) −   sinmx sinnxdx , , 0, cos cos     =  =   − m n m n mx nxdx −   sinmx cosnxdx , , 0,     =  = m n m n  = 0

S7FourierSeriesIll Fourier Series Expansion of FunctionsIf f(x)is periodic function with the period 2πXaoZ(ax cos kx +be sin kx)f(x)2k=1Questions:1. If a function can be expanded by a Fourier series.what is a, and bk:2. Does the Fourier series converge to f (x)?

§7 Fourier Series III Fourier Series Expansion of Functions Questions:   = = + + 1 0 ( cos sin ) 2 ( ) If ( )is periodic function with the period 2 k ak kx bk kx a f x f x  2. Does the Fourier series converge to f (x)? 1. If a function can be expanded by a Fourier series, what is and . ak bk