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Principle value (of a logarithm)

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.

 


Explanation

The logarithm of a complex number z = r · e is

ln z = ln r + (φ + 2kπ) i

You write a complex number in polar form as z = r · e. The logarithm of this number can be expressed as the complex number x + i y, so

Where

so that

ln z = ln r + (φ + 2kπ) i

The logarithm of a complex number has infinitely many values, that all have the same real part ln r and that differ a multiple of in the imaginary part.

For k = 0 you get the principle value.

 


Example 1

The logarithm of −1 has a principle value of

 


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