Definition:Power (Algebra)/Complex Number/Principal Branch
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Definition
The principal branch of a complex number raised to a complex power is defined as:
- $z^k = e^{k \Ln z}$
where $\Ln z$ is the principal branch of the natural logarithm.
Positive Real Base
Let $t > 0$ be a real number and let $k$ be a complex number.
The principal branch of a positive real number raised to a complex power is defined as:
- $t^k = e^{k \ln t}$
where $\ln$ is the natural logarithm of a positive real number.