- 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,