Proof of the Week for 20120827

Theorem: The natural logarithm of a complex variable: \ln z = \ln r + i\theta + i 2\pi k, k \in \mathbb{Z}

Proof: Let z = r e^{i\theta} = r e^{i\left(\theta + 2 \pi k \right)} , then take natural logs of both sides to yield, \ln z = \ln \left(r e^{i\left(\theta + 2 \pi k \right)} \right) = \ln r + \ln e^{i\left(\theta + 2 \pi k \right)} . Then we have \ln z = \ln r + i\theta + i 2\pi k, k \in \mathbb{Z} .

Notes:

  1. This is an infinitely valued function. To recover the single valued function familiar for real numbers, we designate z = 0 as a branch point and the negative real axis as a branch cut. This allows us to define the principal branch of this multivalued function as Ln z = \ln r + i Arg z, -\pi < Arg z \le \pi. If z is real, then Arg z = 0 and Ln z = \ln r .
  2. We have used the fact that the complex exponential function has a period of 2 \pi . Thus e^{i\theta} = e^{i\left(\theta + 2 \pi k \right)}, k \in \mathbb{Z}
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