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Logarithm of a complex number

The logarithm of a complex number is a multi-valued function

ln z = ln r + ( φ+ 2kπ) i

 


Explanation

You write a complex number with polar coordinates as z = r · e. If we assume that the logarithm of this produces the complex number x + i y is, then you get

where

so that

ln z = ln r + (φ + 2kπ) i

The logarithm of a complex number has an infinite number of values, that all have the same real part ln r and the imaginary part differs a multiple of from each other.

For k = 0 you get the principle branch.

 


Example 1

Negative numbers are a special case of complex numbers. Thus z = −1 is a complex number on the unit circle with radius r = 1 and a semicircle rotated φ = π. The logarithm of −1 has a principal value of

ln (−1) = ln (1) + πi = πi

 


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