Range of the exponential function[Bearbeiten]
You are supposed to show that holds for all .
How to get to the proof?
We have already shown that holds for all . Let be an arbitrary complex number with , .
Question: How can we express using and ?
Using the exponential functional equation, we get
As , holds if and only if . A good trick to show that a number is not equal to is to show that its absolute value (or the square of its absolute value) is unequal to .
Question: What is ?
Using the computation rules for complex numbers, we get
Hence, we have and the claim follows.
We have already shown that holds for all . Furthermore, for all , holds that . Hence, for all follows that .
Let be an arbitrary complex number. Then there exist such that . It follows that .