Math 303 - Complex Variables
Homework due February 23
Recall that in class we showed that the function Log zis analytic on C− {non-pos R-axis}. That is,
Log zis analytic everywhere on the complex plane except the non-positive real axis. On this domain
of analyticity, we have that
d
dz Log z=1
z.
Question 1. Use the above to determine the domain of analyticity for f(z) = Log(4 + i−z); that is,
determine the complex numbers zsuch that Log(4 + i−z) is analytic. On your domain of analyticity,
what is the derivative?
Question 2. Find the solutions to the following equations. Be sure to list al l solutions (if multiple
solutions exist).
(a) ez=ie
(b) Log(z−1) = iπ
2
(c) ez+1 =i
(d) Log(z2−1) = iπ
2
Question 3. [Note: This is Question 3.3.5c in our text.] In this question, we will find the solutions to
the equation
e2z+ez+ 1 = 0.
(a) Use the quadratic equation to factor the polynomial
ω2+ω+ 1 = 0.
(b) Use (a) to factor e2z+ez+ 1 into two factors. [Hint: let ω=ezand thus ω2=e2z.]
(c) In (b), you factored e2z+ez+ 1 into two factors. Thus, e2z+ez+ 1 will equal 0 when either of
those factors equals 0. Use the complex logarithm to solve for the zfor which each of these two
factors equal zero. [You should get two sets of answers.]
(d) Verify that your solutions from (c) do indeed solve the equation e2z+ez+ 1 = 0 by plugging your
solutions into zand verifying that the equation holds.
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