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The solutions to quiz #2 for math 412, a college-level complex analysis course. It includes three problems: finding the limit of a complex function as z approaches a specific point, determining the continuity of a complex function, and defining a linear transformation that rotates points in the complex plane. Students are expected to understand complex numbers, limits, continuity, and linear transformations.
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February 27, 2009 Quiz #2 Name: Math 412
Problem 1. Let f (z) = (12 + 5i)z + 2 − 3 i.
(a): (5 points) Find lim z→2+i f (z).
(b): (5 points) Let g(z) = f
z^2 − 2 zi z^2 + 4
. Find all points z ∈ C such that g is
continuous. Write a very brief justification. (Don’t give an (, δ) proof.)
Problem 2. (4 points) Let α = π/2. Define fα(z) = z^1 /^4 = 4
|z| eiθ/^4 , where π/ 2 < θ ≤ 5 π/2. Find the range of the function w = fα(z).
Problem 3. (6 points) Write the formula for a linear transformation f : C → C that rotates points in the complex plane 90◦^ counter-clockwise about the point z = 2 + i.
[Hint: Do a translation to the origin, rotate about the origin, and then translate back.]
