Imaginary unit
You can express the imaginary unit with logarithms as
Calculation 1
Euler's identity can be written as
We take the logarithm on both sides
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so
Calculation 2
The logarithm of a complex number z = r · eiφ is the multi-valued function
The complexe number z = −1 is on the circle group with radius r = 1 and is turned a half circle φ = π. So the logarihtm of −1 has a principle value of
so that