Math 412 Group Work #10 Spring 2009
SOLUTIONS
Problem 1. Compute the sets below, and then find the principal values:
(a): (i2)i
SOLUTION: We have
(i2)i= (−1)i=eilog(−1) =ei(i(π+2πn)) =e−(π+2πn).
The principal value is e−π.
(b): i2i
SOLUTION: We have
i2i=e2ilog i=e2i(i(π
2+2πn)) =e−(π+4πn).
The principal value is e−π.
What do your calculations above show about what can happen when you
raise a complex number to a complex power?
SOLUTION: When you raise a complex number to a complex power, you can get
a real value!
Problem 2. Prove that e−z=1
ezfor all z∈C.
SOLUTION: Since
1 = e0=ez+(−z)=eze−z,
we conclude that
e−z=1
ez.
Problem 3. Use Problem 2 to show that if c∈Cis fixed and z∈C∗, then
z−c=1
zc.
SOLUTION: We have
z−c= exp[−c·log(z)] = 1
exp[c·log(z)] =1
zc,
where we used Problem 2 in the middle step.
