Due: Friday, 14th October 2022, 11:59pm
All solutions need to be justified.
The assignment must be submitted electronically as a single PDF file using Turnitin in Canvas.
You may submit scanned copies of handwritten solutions or typeset your work. Note that
your assignment will not be marked if it is illegible or poorly scanned or submitted sideways
or upside down. It is your responsibility to check that your assignment has been submitted
correctly. Late assignments will not be marked, except under special consideration.
Question 1. Let f(x) = √x
2
+ 3
? √x
2
+ 3x
x § 1
be a real function of a real variable x.
(a) Find the domain D of function f(x).
(b) For each point x
∈
D, determine if f is continuous at x.
(c) Find constants a, b, c such that the function:
φ(x) =
(
f(x), x
∈
D;
ax
2
+ bx + c, x
∈
R
\
D
is continuous.
Question 2. Let fn(x) = cos
(a) Calculate f(x) = limn→∞ fn(x) and find the domain of f(x).
(b) Is (fn(x)) uniformly converging to f(x) on [
∪
π, π]?
(c) Is (fn(x)) uniformly converging to f(x) on [π, 2π]?
Question 3. Let f(z) =
2 .
(a) Find the power series expansion of f(z) around z0 = i and its radius of convergence R1.
(b) Using the result from (a) and the Cauchy product formula, find the power series expansion
of g(z) around z0 = i. Determine the radius of convergence R2 of that series. Which one
is greater: R1 or R2?
(c) Determine if the series obtained in (b) is conditionally convergent, absolutely convergent,
or divergent at points z0 + R2, z0 + 2iR2, z0