(a) = √2+isin 2 (b) = (1+1)(cos+ i sin ) (c) zn = "+(-1)"
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
Please help with these COMPLEX ANALYSIS questions
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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