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(a) = √2+isin 2 (b) = (1+1)(cos+ i sin ) (c) zn = "+(-1)"

(a) = √2+isin 2 (b) = (1+1)(cos+ i sin ) (c) zn = "+(-1)"

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1. 1. Find the limits of the following sequences as n →∞ if they exist (Note: n € N- {0}): (a) = √2+ i sin 2 (b) = (1+) (cos+ i sin ) (c) 2 = "+(-1)" (d) n = √√n+i. 2. Find all the points of discontinuities of the following functions: (a) f(²)=(2²+1) (b) f(2)=¹+ 3. Let 1+² g(z) = (a) Prove that lim,-i f(z) exists and determine its value. (b) Determiner whether f(z) continuous at z = i. (c) Determine whether f(z) continuous at zi. Question 2. 1. Find all the solutions to the equation cosh z = -3. Express your answer explicitly in z. if : #i if z = i 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(2,y) = sin 2 sinhy. 2 1 4. Find the most general analytic function f(z) of the variable z = x+iy whose imaginary part is v(x, y) = y- x² + y² Express your answer explicitly in z, and also give the domain of analyticity of f. Question 3. 1. Let : C\ {0, 1,2} C be given by f(2)= 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 2(2-1)(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) = √2+ i sin 2 (b) = (1+) (cos+ i sin ) (c) 2 = "+(-1)" (d) n = √√n+i. 2. Find all the points of discontinuities of the following functions: (a) f(²)=(2²+1) (b) f(2)=¹+ 3. Let 1+² g(z) = (a) Prove that lim,-i f(z) exists and determine its value. (b) Determiner whether f(z) continuous at z = i. (c) Determine whether f(z) continuous at zi. Question 2. 1. Find all the solutions to the equation cosh z = -3. Express your answer explicitly in z. if : #i if z = i 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(2,y) = sin 2 sinhy. 2 1 4. Find the most general analytic function f(z) of the variable z = x+iy whose imaginary part is v(x, y) = y- x² + y² Express your answer explicitly in z, and also give the domain of analyticity of f. Question 3. 1. Let : C\ {0, 1,2} C be given by f(2)= 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 2(2-1)(z-2) Give the Laurent
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