- Prove that
for all
.
- Find all
such that
.
- Proof. Let
for real numbers
. Then from the definition of the complex exponential, and the fact that
for complex numbers
and
from Theorem 9.3 (page 367 of Apostol) we have,
But from the properties of the real exponential function we know
for any
. Furthermore, we know
for all
since
which contradicts the Pythagorean identity (
). Hence,
for any
- We compute
Then, setting real and imaginary parts equal this implies
This implies
and
. Thus,