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Question 3. 1. Let f: C\ {0, 1,2} → C be given by f(z) expansion of f in each of the following regions: (a) C₁ = {z € C: 0 < |z| < 1} (b) C₂ = {z € C: 1 < |z| <2} (c) C3 = {2 € C : |2| > 2}. = 1 z(z − 1)(z-2)* Give the Laurent

Question 3. 1. Let f: C\ {0, 1,2} → C be given by f(z) expansion of f in each of the following regions: (a) C₁ = {z € C: 0 < |z| < 1} (b) C₂ = {z € C: 1 < |z| <2} (c) C3 = {2 € C : |2| > 2}. = 1 z(z − 1)(z-2)* Give the Laurent

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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Question 1. 1. Find the limits of the following sequences as n→∞ if they exist (Note: n N - {0}): (a) Zn = √√2+ i sin ²/2 (b) Zn = (c) Zn = (d) Zn = 2. Find all the points of discontinuities of the following functions: (a) f(2)=z(z²+1) (b) ƒ(2) = 2+2 3. Let (1+) (cos+ i sin =) in +(-1)n n² vn+im. g(z) = 4i (a) Prove that limzi f(z) exists and determine its value. (b) Determiner whether f(z) continuous at z = i. (c) Determine whether f(z) continuous at z ‡i. if zi if z = i Question 2. 1. Find all the solutions to the equation cosh z = -3. 2 2. Show that if f(z) = u(x, y) + iv(x, y) is analytic, then |Vu| = |Vv|. 3. Find the most general analytic function f(z) of the variable z = x+iy whose real part is u(x, y) = = sin x · sinh . Express your answer explicitly in z. 1 4. Find the most general analytic function f(z) of the variable z = x+iy whose imaginary part is v(x, y) = y Y x² + y² Express your answer explicitly in z, and also give the domain of analyticity of f. Question 3. 1. Let f: C\ {0, 1,2} → C be given by f(z) expansion of f in each of the following regions: (a) C₁ = {z € C: 0 < |z| < 1} (b) C₂ = {z € C: 1 < |z| < 2} (c) C3 = {z € C : |z| > 2}. 1 z(z − 1)(z-2) - Give the Laurent
Transcribed Image Text:Question 1. 1. Find the limits of the following sequences as n→∞ if they exist (Note: n N - {0}): (a) Zn = √√2+ i sin ²/2 (b) Zn = (c) Zn = (d) Zn = 2. Find all the points of discontinuities of the following functions: (a) f(2)=z(z²+1) (b) ƒ(2) = 2+2 3. Let (1+) (cos+ i sin =) in +(-1)n n² vn+im. g(z) = 4i (a) Prove that limzi f(z) exists and determine its value. (b) Determiner whether f(z) continuous at z = i. (c) Determine whether f(z) continuous at z ‡i. if zi if z = i Question 2. 1. Find all the solutions to the equation cosh z = -3. 2 2. Show that if f(z) = u(x, y) + iv(x, y) is analytic, then |Vu| = |Vv|. 3. Find the most general analytic function f(z) of the variable z = x+iy whose real part is u(x, y) = = sin x · sinh . Express your answer explicitly in z. 1 4. Find the most general analytic function f(z) of the variable z = x+iy whose imaginary part is v(x, y) = y Y x² + y² Express your answer explicitly in z, and also give the domain of analyticity of f. Question 3. 1. Let f: C\ {0, 1,2} → C be given by f(z) expansion of f in each of the following regions: (a) C₁ = {z € C: 0 < |z| < 1} (b) C₂ = {z € C: 1 < |z| < 2} (c) C3 = {z € C : |z| > 2}. 1 z(z − 1)(z-2) - Give the Laurent
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