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

y 複數平面化複數平面 (x,y) 挪威人 – Caspar Wessel Polar representation y r x = r cosθ y = r sinθ z = r (cosθ + i sinθ) θ r : the modulus or magnitude of z x z = r eiθ O x 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

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

Q : 試解 A : 1. 2. 3. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

y z θ (x,y) (x,-y) 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 1 mapping 1 x u Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

複變對數函數 z = r eiθ x y z - plane u = lnr v = θ + 2nπ w - plane mapping n = 1 n = 0 n = -1 n = -2 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

複變冪函數其中 a 亦為複數為多值函數亦為多值函數 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

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

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

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

映射轉換(mapping transformation) 放大(enlargement) x y u v 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

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

映射轉換(mapping transformation) two-to-one correspondence 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

映射轉換(mapping transformation) one-to-two correspondence two-to-one correspondence , z = 0的點除外 w-plane z-plane 在z平面上某一點映射到w平面時,可以有兩個值. , , z-plane w-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

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

holomorphic 複變函數的微分—直角座標 f (z) is analytic at z = z0 y regular 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

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

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

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

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

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

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

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

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

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

線積分(Contour Integrals) y Consider the sum : z0’=zn z2 Let n  with for all j z1 ζ2 z0 ζ1 If the sum exists and is independent of the details of choosing the points zj and j . 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

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舉例 : 此處C 是以z = 0為中心,半徑 r 的圓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

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

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 : 對於第一項 Let u = Vx and v = -Vy 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

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 C2 C3 與路徑無關 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

全函數之積分假設 77交大控制計算解為全函數只考慮端點 0 , i Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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

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

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

複變級數收斂(Complex series convergence) 絕對收斂(absolute convergence) : convergence and convergence 條件收斂(conditional convergence) : divergence but convergence 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

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

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

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

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

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

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

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

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

複變勞倫級數(Complex Laurent series) C2 r2 f(z)在區域中為解析 C1 z0 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

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

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

複變勞倫級數(Complex Laurent series) C : 區域複變泰勒級數(Complex Taylor series) 泰勒級數必定是以解析點為展開中心,而勞倫級數的展開中心可以是解析點,也可以是不解析點. 泰勒級數與勞倫級數主要的差別在於: 泰勒級數之有效範圍必是一個圓的內部,而勞倫級數的有效範圍必定是一個環狀區域. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

複變勞倫級數(Complex Laurent series) 試求函數在原點展開之所有的勞倫級數 (89清大動機) 對f(z)而言,不解析點: z = 0, z = 1 在在 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

複變勞倫級數(Complex Laurent series) 試求函數在原點展開之所有的勞倫級數 (89清大動機) C : 區域 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

複變勞倫級數(Complex Laurent series) 試求函數在z = -1展開之勞倫級數在-1處是可解析的, 故可用泰勒展開式 for 已是在-1處的勞倫展開式 for for Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

奇異點(Singularities) 當f(z)在區域中為解析函數,但在z0處不為解析函數時,我們稱f(z)有一孤立奇異點(isolated singularity) z0.. 我們將f(z)對z0做勞倫級數展開 1. 當展開式中沒有(z-z0)的負冪次項,則稱z0為可移除奇異點(removable singularity). 2. 當展開式中有無窮多個(z-z0)的負冪次項,則稱z0為本質奇異點(essential singularity). 3. 當展開式中有(z-z0)的負冪次項一個以上,則稱z0為極點(pole). 4. 當展開式中 (z-z0)的負冪次項部分稱為主要部份(principal part). 5. 當展開式中 (z-z0)的負冪次項部分只到第k項,則稱z0為k階極點(kth order pole). 6. 當展開式中沒有主要部分,且則稱z0為k階零點(kth order zero). 7. b1稱為f(z)在z0的殘(留)數(residue). Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

8. 當z0為f(z)的一個k階零點,則z0為1/ f(z)的一個k階極點,反之亦然. 9. 當展開式中(z-z0)的負冪次項只一個,則稱z0為簡單極點(simple pole). 當z0為f(z)的k階極點,則f(z)在z0的殘數記為R(z0): 若f(z)為有理函數: ,而z0是一個簡單極點: Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

找出以及在z = 0之極點階數與殘數 h(z)在z = 0處解析,且h(0)  0, 故f(z)在z = 0處為一個二階極點為本質奇異點 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

當f(z)在區域中為解析函數,但在z0處不為解析函數時,我們稱f(z)有一孤立奇異點(isolated singularity) z0. 我們將f(z)對z0做勞倫級數展開 C z0 r 由歌西積分公式: 殘數定理 f(z)在區域中為解析函數,但除了z0, z1,…, zn點為不為解析,當C為包含上述奇異點的單連封閉曲線時,則 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

計算之值,C為包含z = 1之單連封閉曲線. (89清華動機) z = 1處為三階極點 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

z = 0處為簡單極點, z = 2i, -2i處為簡單極點 z = 2i, -2i處為簡單極點 z = 0處為簡單極點 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

殘數定理之應用 -- 三角函數的積分 Where f is both finite and single-valued for all values of  Let The path of integration is the unit circle By the residue theorem Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

計算很重要! 分母有兩個根單位圓之外單位圓之內 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

分母有兩個根單位圓之內單位圓之外 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

殘數定理之應用 --有理函數的積分 CR 在CR上取當n-m-1 > 0時,上式之極限值將趨近於零當q(z)之次方數  p(z)之次方數+2時 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

計算之值 q(z)之次方數  p(z)之次方數+2 分母有兩個根下半面上半面 q(z)沒有實根,以避免函數p(z)/q(z)在實軸上出現極點 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

歌西積分主值(The Cauchy Principal Value : PV) 當線積分路徑上出現極點時 counterclockwise clockwise CR Infinite radius semicircle 當走下面箭頭之路徑時,小半球Cx0包含x0點 net Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

歌西積分主值(The Cauchy Principal Value : PV) CR Infinite radius semicircle 當走上面箭頭之路徑時,小半球Cx0排除x0點 net 歌西積分主值 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

常出現在傅氏轉換(Fourier Transform) and Consider CR 在CR上收斂至零與否取決於 : 在0到之間sin恆為正實數 a > 0即可 Jordan’s Lemma : q(z)之次方數  p(z)之次方數+1即可 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

計算之值 sinx與x均為奇函數線積分路徑上出現極點 x = 0 利用歌西積分主值公式 1 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

歌西不等式(Cauchy’s Inequality) 當f(z)在區域中為解析函數時,若M為在上之極大值,則恆有 : 圓證明取 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

理奧維爾定理(Liouville Theorem) 任何有界(bounded)的全函數,必然是一個常數函數. 證明當f(z)為全函數時,f(z)在區域當然為解析函數,假設m為在圓上的極大值,則若f(z)為有界,則必定存在一正實數M為在整個複數平面上的絕對極大值: 此不等式與r的大小無關,因此當r趨近於無窮大時 f(z)是一個常數函數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

最大模數定理(Maximum Modulus Theorem) 若f(z)在單連封閉曲線C上及內部為解析函數,則在C上及內部的極大值,必發生在解析區域的邊界C上,而不在C內部. C 單連封閉曲線C的長度為L, z0距離C之最短距離為d 假設M為在曲線C上的極大值. d Z0 當時 z0為C內之任一點,因此得證! Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

最小模數定理(Minimum Modulus Theorem) 若f(z)在單連封閉曲線C上及內部為解析函數,且 ,則在C上及內部的極小值,必發生在解析區域的邊界C上,而不在C內部. Example 試求在上之極大與極小模數 Sol. 在為解析且恆不為零的極大值與極小值均發生在上的極值發生在若n為偶數若n為奇數 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

幅角原理(The Argument Principle) 在區域D中,除了幾個極點外,f(z)是可解析的.假設單連封閉曲線C在區域D中沒有通過任何極點或零點時,則曲線C中f(z)之零點數目-曲線C中f(z)之極點數目 i 在曲線C中,sinz包含五個零點,且沒有極點 -3 -2 -  2 3 -i Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

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Chapter 12 Complex Numbers and Functions

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