Chapter 12 Complex Numbers and Functions

Chapter 12 Complex Numbers and Functions 複數(complex numbers)與複變數(complex variables) 複數(complex numbers) : z = a + i b , 其中 a 與 b 均為實數, . 複變數(complex variables) : z = x + i y , 其中 x 與 y 均為實變數, . 複數運算規則 : 相等(equality) : z1 = z2 x1 = x2 , y1 = y2 加法(addition) : z1 + z2 = (x1 , y2) + (x2 , y2) = (x1+ x2 , y1+y2) 相乘(multiplication) : z1 z2 = (x1 , y2) · (x2 , y2) = (x1 x2 - y1y2 , x1 y2 + x2y1) 相除(division) : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions y 複數平面化複數平面 (x,y) 挪威人 – Caspar Wessel Polar representation y r z = r (cosθ + i sinθ) x = r cosθ y = r sinθ θ r : the modulus or magnitude of z x z = r eiθ x O θ : the argument or phase of z z = x + i y Euler’s Formula : 相乘(multiplication) : 相除(division) : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions De Moivre’s (隸美弗) Formula Q : 試證明 A : 二項式定理展開 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions Q : 試解 1. 2. 3. A : 1. 2. 3. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

y (x,y) z θ x θ (x,-y) Chapter 12 Complex Numbers and Functions 共軛複數(Complex Conjugation) 複變函數(complex functions) Complex function w(z) = u(x,y) + iv(x,y) where u(x,y) and v(x,y) are pure real y v For example : z - plane w - plane w(z) = z2 = (x + iy)2 = (x2 - y2) + i 2xy 2 2 mapping 1 1 x u Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 100oC Q : 某一楔形金屬板,其兩面之溫度固定為恆溫(如圖所示),試求其中之溫度分佈. π/3 A : 0oC v y π/3 100oC π/3 x u 0oC 在u-v 平面上的解為 : ,又在x-y 平面上的解為 : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

y v = θ + 2nπ mapping n = 1 n = 0 x n = -1 u = lnr n = -2 z - plane w - plane Chapter 12 Complex Numbers and Functions 複變對數函數 z = r eiθ 主值(Principle value) θ : 主幅角(the principle argument) w : 多值函數 Q : 試計算之值 A : 假設通解主值 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變冪函數其中 a 亦為複數亦為多值函數為多值函數 Q : 試計算下列之值 A : 通解主值 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變指數函數複變指數函數具有虛週期複變三角函數複變三角函數仍具有週期 , 但為無界函數複變雙曲函數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions Q : 1.試解 2.求之值 A : 1. 2. or  恆為正 n 為偶數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

y v v y x u u x Chapter 12 Complex Numbers and Functions 映射轉換(mapping transformation) 平移(translation) z-plane w-plane y0 x0 旋轉(rotation) z-plane w-plane θ0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

y v x u Chapter 12 Complex Numbers and Functions 映射轉換(mapping transformation) 放大(enlargement) z-plane w-plane 反轉(inversion) v y w-plane z-plane x u Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 映射轉換(mapping transformation) 反轉(inversion) line  circle y z-plane y = c1 1 2 3 4 x v w-plane 4 u 3 2 1 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions two-to-one correspondence 映射轉換(mapping transformation) 非線性轉換: 係數平方,幅角變兩倍 Upper half-plane of z, 0  θ < π  whole plane of w, 0  φ < 2π Cover by two times Lower half-plane of z, π θ < 2π  whole plane of w, 0  φ < 2π For example : two-to-one correspondence , w(z) = z2 = (x + iy)2 = (x2 - y2) + i 2xy v y w-plane z-plane u x Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions one-to-two correspondence 映射轉換(mapping transformation) two-to-one correspondence , z = 0的點除外 z-plane w-plane 在z平面上某一點映射到w平面時,可以有兩個值. , , w-plane z-plane How to make the function of w a singled-values function ? one-to-one correspondence y z-plane branch point singularities 限制z的幅角 x cut line Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions many-to-one correspondence 映射轉換(mapping transformation) w-plane z-plane the same point any points also If y y cut line x x The Riemann surface for ln z Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions holomorphic 複變函數的微分—直角座標 f (z) is analytic at z = z0 regular y δx0 z0 δy = 0 δx = 0 δy0 x First approach δy = 0 δx0 δx = 0 Second approach δy0 Cauchy-Riemann conditions : if exists, then , if does not exist at z = z0, then z0 is labeled a singular point . Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變函數的微分—極座標 Cauchy-Riemann conditions 實部對實部,虛部對虛部極座標直角座標 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變函數的微分性質複變函數的微分性質與實變函數相同 1. 2. For any points 3. 4. Except for branch points and cut lines Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變解析(analytic)函數定義: 若複變函數f (z)在z0處以及包圍z0點的小封閉曲線範圍內均可微分, 則稱f (z)為複變解析函數. 有理函數除分母為零的位置外,均為解析函數. 對數函數與冪函數除了在分支點與分支切割外,均為解析函數. 複變全(entire)函數定義: 若複變函數f (z)在整個複數平面均可微分, 則稱f (z)為複變全(entire)函數. 複變多項式函數以及 , , , , ,等均為全函數奇異點(singular point) 若複變函數f (z)在z0處不可微分,則稱z0處為奇異點. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions Q : 試問下列函數在何處解析? A : Cauchy-Riemann conditions : 1. 均不成立任何地方皆不解析 1. 僅原點成立原點處解析 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions A : 3. 均成立任何地方皆解析 4. 均不成立任何地方皆不解析 5. 除原點不成立除原點外均解析 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 共軛座標系(conjugate coordinates) 複數z = x + iy 為以實數 x 與 y 為變數的函數使用 z 及 z* 為變數的座標系 (z , z*)稱為共軛座標系複變函數 f (z) = u + iv 在共軛座標系的表示方式時此時Cauchy-Riemann conditions為 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 共軛座標系與直角座標系在偏微分的關係 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions Q : 試將拉式運算子以共軛座標表示之 A: Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 諧和函數(harmonic functions) 對於任何實變函數u (x , y), 若其滿足Laplace’s equation : 則函數u (x , y)稱之為諧和函數. 若複變函數f (z) = u (x , y) + i v (x , y)在某區域內為解析函數,則實變函數u (x , y) 以及v (x , y)在此區域內必為諧和函數,但反之未必然. 此證明利用下列定理: 若複變函數f (z) = u (x , y) + i v (x , y)在某區域內為解析函數,則複變函數f (z)的各階導數均存在且仍為解析函數. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 線積分(Contour Integrals) y Consider the sum : z0’=zn z2 Let n  with for all j z1 ζ2 z0 If the sum exists and is independent of the details of choosing the points zj and j . ζ1 x 積分路徑 then f (z)沿著特定路徑C (由z = z0到 z = z0’)的線積分 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 線積分(Contour Integrals) 線積分定義將複變積分簡化為實變積分的複數和在極座標下另一種做法此處C 是以z = 0為中心,半徑 r 的圓舉例 : 我們以極座標來處理當 n  -1 當 n = -1 與r無關! Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分定理(Cauchy’s Integral Theorem) 當一複變函數f (z)在某封閉區域中為可解析(analytic),且其微分仍為連續的, 則對於在此封閉區域的任一封閉路徑C, f (z)的線積分為零. 記得上一例子中 : 此乃因f (z) = 1/z 在z = 0 處為不解析 ? 含原點不含原點 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分定理(Cauchy’s Integral Theorem)證明利用Stokes’s theorem與Cauchy-Riemann condition可證明Cauchy’s Integral Theorem Stokes’s theorem : Let u = Vx and v = -Vy 對於第一項對於第二項 Let v = Vx and u = Vy If f (z) is analytic Cauchy-Riemann condition Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分定理(Cauchy’s Integral Theorem) 當一複變函數f (z)在某封閉區域中為可解析(analytic),且其微分仍為連續的, 則對於在此封閉區域的任一封閉路徑C, f (z)的線積分為零. C1 C1 C2 C3 C2 與路徑無關 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 全函數之積分假設 77交大控制計算解只考慮端點 0 , 2 + i 為全函數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分公式(Cauchy’s Integral Formula) 當一複變函數f (z)在某封閉區域中為可解析(analytic),且其微分仍為連續的, 則對於在此封閉區域的任一封閉路徑C,則其中z0位於封閉路徑C內部因為f (z)可解析, 在z = z0處不是解析的,除非f (z0) = 0 C1 contour line z0 C2 As r  0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分公式(Cauchy’s Integral Formula) z0 interior z0 exterior f(z)的微分可利用歌西積分公式來表示同理 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分公式(Cauchy’s Integral Formula) 計算之值 z0 interior z0 exterior 取 z0 = i , f (z) = sinz 取 z0 = -i , f (z) = sinz Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 歌西積分公式(Cauchy’s Integral Formula) 計算之值, 其中 C 為單位圓 z0 interior z0 exterior 取 z0 = 0 , f (z) = ez Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變序列(Complex sequences) A complex sequence : z1, z2, z3 , z4,…. zn,… A complex sequence z1, z2,… is said to converge to the number L if ,given ε > 0, there is some positive integer N such that whenever n  N. zN+3 ε L zN zN+2 zN+1 Cauchy sequence Theorem : Let zn = xn +iyn. Then, znA + iB if and only if xn  A and yn  B Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變級數(Complex series) Given a complex sequence : z1, z2, z3 , z4,…. zn,… The complex series : The sum : Theorem : Let zn = xn +iyn. Then, if and only if and Theorem : If converges, then Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變級數收斂(Complex series convergence) 絕對收斂(absolute convergence) : convergence and convergence 條件收斂(conditional convergence) : convergence divergence but 要判斷收斂,可以利用比例測試法(ratio test) : 取 1. 對滿足的 z 而言, 此級數為絕對收斂 2. 對滿足的 z 而言, 此級數為發散 3. 對滿足的 z 而言, 無法判定收斂性 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變冪級數(Complex power series) 複變冪級數利用比例測試法(ratio test)判斷收斂 : where 對滿足的 z 而言, 此冪級數為絕對收斂 ρ z0 收斂區域收斂半徑展開中心 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變冪級數(Complex power series) 複變冪級數在其收斂範圍內可以逐項微分及積分,且其收斂範圍不會因積分或微分而改變. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變冪級數(Complex power series) 定理: 若f(z)為連續,且在區域D中存在  R+, 而有 ,則在區域D中證明 : 對於複變冪級數取因為f(z)在區域D中為收斂且小於1 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變冪級數(Complex power series) 試求複變冪級數之收斂區域及其和區域內為絕對收斂利用 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變泰勒級數(Complex Taylor series) 複變函數在解析點的無限級數展開形式考慮單連封閉曲線C為圓 ,函數f(z)在C上及其內部均為解析,z為C內部的一點. 歌西積分公式(Cauchy’s Integral Formula) ρ z0 z C z為C內部的一點, 而s在圓上 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變泰勒級數(Complex Taylor series) f(z)在z0點處的複變泰勒級數(Complex Taylor series)  冪級數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變泰勒級數(Complex Taylor series) Let and 得到驗證 For all z 試求在0 之泰勒展開級數展開中心在z = 0, f(z)的奇異點在z = -1,因此收斂半徑ρ = 1 if Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變泰勒級數(Complex Taylor series) 試求在展開之泰勒級數,並求收斂半徑 (74台大化工) 展開中心在z = 1, f(z)的奇異點在z = 2,因此收斂半徑ρ = 1 試求在之泰勒展開級數展開中心在z = -2i, f(z)的奇異點在z = -1,因此收斂半徑ρ = Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變勞倫級數(Complex Laurent series) C2 r2 f(z)在區域中為解析 z0 C1 r1 z 取0 C f(z)在C2中並非都解析 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變勞倫級數(Complex Laurent series) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions 複變勞倫級數(Complex Laurent series) 此時z0是不解析的 m = -n C : 區域 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions