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Domain of a logarithm

The domain of a logarithm can be expanded into negative and complex numbers.

 

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Explanation

The logarithm of a complex number z in the interval [0, 2π) is called the principle value. The interval (−π, π] can also be taken. The logarithm is a multi-valued function

ln z = ln r + i (φ + 2kπ)

Negative numbers are a special case of complex numbers.

 

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Example 1

We start with Euler's identity and find

e^iπ + 1 = 0      ⇒      −1 = e^iπ

and take the logarithm

ln (−1) = ln (e^iπ)

so that

ln (−1) = iπ

 

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