Chapter 6 Integrals and Complex variable function series 6 5 The complex variable function series 6 6 Classification of anomalies The complex variable function series • A series is the sum of the terms of a sequence • For any sequence
Chapter 6: Integrals and Complex variable function series 6.5 The complex variable function series 6.6 Classification of anomalies The complex variable function series • A series is the sum of the terms of a sequence • For any sequence 𝑧𝑚 of real numbers, complex numbers, functions, etc., the associated series is defined as the ordered formal sum ∞ 𝑧𝑚 = 𝑧1 + 𝑧2 + … 𝑚=1 • The sum of the first n terms 𝑠𝑛 = 𝑧1 + 𝑧2 + ⋯ + 𝑧𝑛 is called the nth partial sum of the series • A convergent series is one whose sequence of partial sums converges ∞ lim 𝑠𝑛 = 𝑠 𝑛→∞ Then we write 𝑠= 𝑧𝑚 = 𝑧1 + 𝑧2 + ⋯ 𝑚=1 and call s the sum or value of the series • A series is not convergent is called a divergent series The complex variable function series • Given a sequence of complex variable function: 𝑢1 𝑧 , 𝑢2 𝑧 , … , 𝑢𝑛 𝑧 , … , in a region G, the series of complex variable functions is ∞ 𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯ (1) 𝑛=1 • The sum of the first n terms 𝑆𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 (𝑧) is called the nth partial sum of the series • If the series is convergent with 𝒛 = 𝒛𝟎 then 𝒛𝟎 is called the point of convergence • If the series is divergent with 𝒛 = 𝒛𝟎 then 𝒛𝟎 is called the point of divergence Uniform Convergence ∞ 𝑓 𝑧 = 𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯ (1) 𝑛=1 • Convergence test: The series of functions (1) is convergent if and only if for every given 𝜺 > 𝟎 (no matter how small) we can find a real number 𝑵(𝜺, 𝒛) such that: If 𝒏 > 𝑵 𝜺, 𝒛 then 𝒇 𝒛 − 𝑺𝒏 (𝒛) < 𝜺 • Uniform Convergence: The series of functions (1) is called uniformly convergent in a region G if for every given 𝜺 > 𝟎 we can find an 𝑵 = 𝑵(𝜺), not depending on 𝑧, such that 𝑓 𝑧 − 𝑆𝑛 (𝑧) < 𝜀 for all 𝑛 > 𝑁 𝜀 and all 𝑧 in G • Test for Uniform Convergent (Weierstrass M-Test): if 𝒖𝒏 𝒛 ∞ 𝒏=𝟏 𝒂𝒏 < 𝒂𝒏 for all 𝑧 in G and the series converges then the series of functions (1) is uniformly convergent in G Uniform Convergence Properties ∞ 𝑓 𝑧 = 𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯ (1) 𝑛=1 Continuity of the Sum Let the series (1) be uniformly convergent in a region G Let 𝑓(𝑧) be its sum Then if each term 𝑢𝑛 𝑧 is continuous at a point 𝑧1 in G, the function 𝑓(𝑧) is continuous at 𝑧1 (See the proof in Page 700) Termwise Integration Let the series (1) be a uniformly convergent series of continuous functions in a region G Let C be any path in G Then the series ∞ 𝑓 𝑧 𝑑𝑧 = 𝐶 is convergent and has the sum 𝑢𝑛 𝑧 𝑑𝑧 = 𝑛=1 𝐶 𝐶 𝑢1 𝑧 𝑑𝑧 + 𝐶 𝑓 𝑧 𝑑𝑧 ( See the proof in Page 702) 𝑢2 𝑧 𝑑𝑧 + ⋯ 𝐶 Power Series • A power series in powers of 𝑧 − 𝑧0 is a series of the form ∞ 𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯ (2) 𝑛=0 where 𝑧 is a complex variable, 𝑎0 , 𝑎1 , … are complex (or real) constants, called the coefficients of the series, 𝑧0 is a complex (or real) constant, called the center of the series • If 𝑧0 = 0, we obtain as a particular case a power series in powers of 𝑧: ∞ 𝑎𝑛 𝑧 𝑛 = 𝑎0 + 𝑎1 𝑧 + 𝑎2 𝑧 + ⋯ 𝑛=0 (3) Convergence of a Power Series ∞ 𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯ (2) 𝑛=0 Every power series (2) converges at the center 𝑧0 If (2) converges at a point 𝑧 = 𝑧1 ≠ 𝑧0 , it converges absolutely for every 𝑧 closer to 𝑧0 than 𝑧1 , that is, 𝑧 − 𝑧0 < 𝑧1 − 𝑧0 See the Figure If (2) diverges at 𝑧 = 𝑧2 , it diverges for every 𝑧 farther away from 𝑧0 than 𝑧2 See the Figure Radius of Convergence of a Power Series ∞ 𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯ (2) 𝑛=0 • We consider the smallest circle with center 𝑧0 that includes all the points at which a given power series (2) converges Let 𝑅 denote its radius • The circle 𝑧 − 𝑧0 = 𝑅 is called the circle of convergence and its radius 𝑅 is the radius of convergence of (2) • Determination of the Radius of Convergence from the Coefficients: 𝑎𝑛 𝑅 = lim 𝑜𝑟 𝑅 = lim 𝑛 𝑛→∞ 𝑎𝑛+1 𝑛→∞ 𝑎𝑛 Taylor Series The Taylor Series of a function 𝑓(𝑧), the complex analog of the real Taylor series is ∞ 𝑓 𝑧 = 𝑎𝑛 (𝑧 − 𝑧0 ) 𝑛=1 𝑛 𝑤ℎ𝑒𝑟𝑒 𝑎𝑛 = 𝑓 𝑛! 𝑛 𝑧0 = 2𝜋𝑖 𝑓(𝑧 ∗ ) 𝑑𝑧 ∗ ∗ 𝑛+1 (𝑧 − 𝑧0 ) (4) 𝑧 ∗ is the variable of integration We can write 𝑓 𝑧 = 𝑓 𝑧0 𝑧 − 𝑧0 ′ (𝑧 − 𝑧0 )2 ′′ 𝑧 − 𝑧0 + 𝑓 𝑧0 + 𝑓 𝑧0 + ⋯ + 1! 2! 𝑛! Where 𝑅𝑛 (𝑧) is called the remainder of the Taylor series after the term 𝑎𝑛 (𝑧 − 𝑧0 )𝑛 (𝑧 − 𝑧0 )𝑛+1 𝑅𝑛 𝑧 = 2𝜋𝑖 𝑓(𝑧 ∗ ) 𝑑𝑧 ∗ ∗ 𝑛+1 ∗ 𝑧 − 𝑧0 (𝑧 − 𝑧) Taylor series are Power series A Maclaurin series is a Taylor series with center 𝑧0 = 𝑛 𝑓 𝑛 𝑧0 + 𝑅𝑛 (𝑧) Important Special Taylor Series ∞ 𝑧 𝑒 = 𝑛=0 𝑧𝑛 𝑧 𝑧 =1+ + +⋯ 𝑛! 1! 2! ∞ sin 𝑧 = 2𝑛+1 𝑧 𝑧 𝑧 𝑧 (−1)𝑛 =𝑧− + − +⋯ 2𝑛 + ! 3! 5! 7! 𝑛=0 ∞ cos 𝑧 = (−1)𝑛 𝑛=0 𝑧 2𝑛 𝑧2 𝑧4 𝑧6 = 1− + − +⋯ 2𝑛 ! 2! 4! 6! 𝑧2 𝑧3 𝑧4 𝐿𝑛 + 𝑧 = z − + − + ⋯ 𝑧3 𝑧5 𝑎𝑟𝑐𝑡𝑔𝑧 = 𝑧 − + − ⋯ Laurent Series Let 𝑓 𝑧 be analytic in a domain containing two concentric circles 𝐶1 and 𝐶2 with center 𝑧0 and the annulus between them Then 𝑓(𝑧) can be represented by the Laurent series ∞ ∞ 𝑛 𝑓 𝑧 = 𝑎𝑛 (𝑧 − 𝑧0 ) + 𝑛=0 𝑛=1 𝑏1 𝑏2 𝑏𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 ) + ⋯ + + 𝑛 𝑧 − 𝑧 𝑧 − 𝑧0 (𝑧 − 𝑧0 ) +⋯ consisting of nonnegative and negative powers The coefficient of this Laurent series are given by the integrals 𝑎𝑛 = 2𝜋𝑖 𝑓(𝑧 ∗ ) ∗ 𝑑𝑧 ∗ 𝑛+1 𝐶 (𝑧 − 𝑧0 ) 𝑏𝑛 = 2𝜋𝑖 𝑧 ∗ − 𝑧0 𝐶 taken counterclockwise around any simple closed path 𝐶 that lies in the annulus and encircles the inner circle ∞ 𝑛=1 𝑏𝑛 is called the principal part of 𝑓(𝑧) (𝑧 − 𝑧0 )𝑛 𝑛−1 𝑓 𝑧 ∗ 𝑑𝑧 ∗ (5) Classifying singularities • A singular point of an analytic function 𝑓(𝑧) is a 𝑧0 at which 𝑓 𝑧 is not analytic, but every neighborhood of 𝑧 = 𝑧0 contains points at which 𝑓 𝑧 is analytic • A zero is a 𝑧 at which 𝑓 𝑧 = • A isolated singularity of 𝑓(𝑧) is 𝑧 = 𝑧0 if it has a neighborhood without further singularities of 𝑓 𝑧 𝜋 Ex: tan 𝑧 has isolated singularities at ± , ± 3𝜋 ,… 𝑏 • In Laurent series, if the principal part has only finitely many terms 𝑧−𝑧1 + ⋯ + 𝑏𝑚 𝑧−𝑧0 𝑚 (𝑏𝑚 ≠ 0) then the singularity of 𝑓(𝑧) at 𝑧 = 𝑧0 is called a pole, and 𝑚 is called is order • If the principal part has infinitely many terms, we say that 𝑓(𝑧) has at 𝑧 = 𝑧0 an isolated essential singularity • We say that a function 𝑓(𝑧) has a removable singularity at 𝑧 = 𝑧0 if 𝑓(𝑧) is not analytic at 𝑧 = 𝑧0 but can be made analytic there by assigning a suitable value 𝑓 𝑧0 Ex: 𝑓 𝑧 = sin 𝑧 𝑧 become analytic at 𝑧 = if we define 𝑓 =