物數(二) 第十章 傅立葉分析 (Fourier Analysis) 第十一章 偏微分方程式 (Partial Differential Equations) 第十二章 複數與函數 (Complex Numbers and Functions) 第十三章 複變積分 (Complex Integrations) 第十四章 羃級數與泰勒級數 (Power Series, Taylor Series) 第十五章 勞倫級數與餘數積分 (Laurent Series, Residue Integration) 第十六章 應用在位勢理論的複變分析 (Complex Analysis Applied to Potential Theory) 評分標準: 平時成績 : 30% 期中考 : 35% 期末考 : 35% Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 對於週期性函數而言,包含餘弦與正弦項的傅立葉級數是一個重要的代表級數.它在求解常微分與偏微分方程式方面是一個重要的工具. 傅立葉級數的理論是較為複雜,但是它的應用卻是簡單方便的.對於非連續性週期函數的表示上,它強過泰勒級數. 學習完傅立葉級數後,我們將傅立葉級數的觀念與技巧將以推廣到非週期性函數上,這將利用到傅立葉積分與傅立葉轉換. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 週期性函數(Periodic Functions) 假如對於所有的實數 x 並存在有任一正數 p 使得一函數 f(x)具有以下特性: 則此函數 f(x)稱為週期性函數. p 稱之為 f(x)的基本週期(fundamental period). 2p , 3p, 4p, … np 亦為 f(x)的週期. Q : 此時 f (ax) 的週期為何? Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 週期性函數(Periodic Functions) * 假如 f(x) 與 g(x) 的基本週期均為 p 時, 則 a, b 為常數 h(x)函數的基本週期亦為 p. * 函數 f(x) = c = 常數 : 亦為週期性函數,但是它沒有基本週期. * 正弦與餘弦函數為常見的週期性函數 * 常見的非週期性函數包括有 : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 三角的級數(Trigonometric Series) 這些正弦與餘弦函數均為具有 2π週期的週期性函數 三角的級數(trigonometric series) 其中 a0, a1, a2, b1, b2, b3, .. 為實數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉級數(Fourier Series) 一個實數週期性函數(週期為 2)可以三角的級數來展開,此級數亦稱之為傅立葉級數(Fourier series) where n = 1,2,3,….. n = 1,2,3,….. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉係數之尤拉公式(Euler Formulas for the Fourier Coefficients) 常數項係數 a0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) The Orthogonality Condition for all integral m and n Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉係數之尤拉公式(Euler Formulas for the Fourier Coefficients) 餘弦項係數 an Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉係數之尤拉公式(Euler Formulas for the Fourier Coefficients) 正弦項係數 bn Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 矩形波(Rectangular Wave) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 矩形波(Rectangular Wave) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 矩形波(Rectangular Wave) Theorem 1 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Homework Problem Set 10.2 8 16 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 任何週期 p = 2L的函數(Functions of Any Period p = 2L) 週期為 2 函數的傅立葉級數 週期為 2L 函數的傅立葉級數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 任何週期 p = 2L的函數(Functions of Any Period p = 2L) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 任何週期 p = 2L的函數(Functions of Any Period p = 2L) n = 1 n = 2, 3, …. n 為奇數時 n = 2, 4, 6, …. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 任何週期 p = 2L的函數(Functions of Any Period p = 2L) n = 1 n = 2, 3, …. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 偶函數與奇函數(Even and Odd Functions) The Odd Function f(x) is an odd function, e.g. For all integer m and n Fourier sine series Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 偶函數與奇函數(Even and Odd Functions) The Even Function f(x) is an Even function, e.g. For all integer m and n Fourier cosine series Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) The General Function E(-x) = E(x) O(-x) = -O(x) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) f(x) Example 1– Square wave 2k x -4π -3π -2π -π π 2π 3π 4π For n = odd For n = even f(x)-k is an odd function Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Example 2– Sawtooth wave Theorem 2 --- Sum of functions f1 + f2 的傅立葉係數為 f1 與 f2 個別傅立葉係數的加總 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Half-Range Expansions f1 is an even periodic extension of f f1  Fourier cosine series f2 is an odd periodic extension of f f2  Fourier sine series Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Half-Range Expansions Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Even periodic extension Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 複數傅立葉級數(Complex Fourier Series) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 複數傅立葉級數(Complex Fourier Series) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Example 1 f(x)=ex, -<x<, f(x+2 )=f(x),將此函數以複數傅立葉級數來表示 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 驅動振盪(Forced Oscillations) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) m = 1g, c = 0.02 g/s, k = 25 g/s2 n = 1,3,5,… 振幅 振幅最大 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 三角多項式的近似(Approximation by Trigonometric Polynomials) 當一週期(2)函數 f(x)以傅立葉級數展開時: 當我們計算前N項總和時,此近似值將會是誤差最小的! Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 當 當 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 定理一 : 最小平方誤差 在區間- ≤ x ≤ 中,若且唯若F(x)的係數為 f 的傅立葉係數,則F(N為固定值)相對於 f 的總平方誤差為最小,此最小值 E* 可由下式求得: 貝索不等式(Bessel inequality) 傅立葉係數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉積分(Fourier Integrals) 週期性函數 非週期性函數 傅立葉級數 傅立葉積分 2L>2 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Amplitude spectrum 2L=8 當2L = 2k 時,每半波有 2k-1 -1個振幅 2L=16 當L   , wn  0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 現考慮週期 2L 的週期性函數 fL(x) 令 令 L  且假設非週期性函數 為絕對可積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉積分(Fourier Integrals) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) example Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 狄里西雷不連續因數(Dirichlet’s discontinuous factor) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) x = 0 正弦積分(sine integral) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 當L   , oscillation x =  1 吉布斯現象(Gibb’s Phenomenon) 令 w+wx=t 令 w-wx=-t Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉餘弦積分(Fourier Cosine Integrals) 若 f(x)為偶函數 B(w) = 0 傅立葉正弦積分(Fourier Sine Integrals) 若 f(x)為奇函數 A(w) = 0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) example 求此函數的傅立葉餘弦積分和正弦積分: 傅立葉餘弦積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉正弦積分 Laplace integral Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉餘弦轉換(Fourier Cosine Transforms) 傅立葉餘弦積分 現令 為 f(x)的傅立葉餘弦轉換 為 的反傅立葉餘弦轉換 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉正弦轉換(Fourier Sine Transforms) 傅立葉正弦積分 現令 為 f(x)的傅立葉正弦轉換 為 的反傅立葉正弦轉換 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) example 當 f(x) 在 0<x< 時均等於一常數k時,則上兩式轉換均不存在 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉正(餘)弦轉換之線性運算 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 導數之傅立葉餘弦轉換 假設 f(x)在 x 軸上為連續且絕對可積分,且 f’(x)在各有限區間中為片段連續, 而且當x時f(x)0,則 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 導數之傅立葉餘弦轉換 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 導數之傅立葉正弦轉換 假設 f(x)在 x 軸上為連續且絕對可積分,且 f’(x)在各有限區間中為片段連續, 而且當x時f(x)0,則 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 導數之傅立葉正弦轉換 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 複數型式之傅立葉積分 實數之傅立葉積分 與w無關 W的偶函數 所以[…]為w的偶函數, 稱之為F(w) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 複數型式之傅立葉積分 If Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉轉換(Fourier Transforms) 為 f 的傅立葉轉換 為 的反傅立葉轉換 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Physical Interpretation : Spectrum 想像成所有可能頻率之正弦振盪的重疊 : 頻譜密度, 代表 f(x)在頻率區間w與w+Δw之間的強度 為系統之總能量 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 單一頻率 若系統較為複雜,且 y = f(x)為具有傅立葉級數表示式的週期函數時,則 不再只是一個能量項 ,而是各係數 cn 之 的級數,此為離散頻 譜(discrete spectrum)或點頻譜(point spectrum),包含不同的各種頻率 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉轉換之線性運算 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 導數之傅立葉轉換 假設 f(x)在 x 軸上為連續且當 時 f(x)0,假設f’(x)在x軸上為絕對可積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 傅立葉轉換之摺積定理(convolution theorem) 函數 f 與 g 的摺積定義 : 傅立葉轉換之摺積定理(convolution theorem) 假設 f(x)和 g(x)在 x 軸上為片段連續,有界且絕對可積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) 令 x – p = q, 則 x = p + q Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 10 傅立葉分析 (Fourier Analysis) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung