There is a simple identity used in the study of complex numbers which is given by
$ |t^z|= t^{Re(z)} $ I'm just curious, how does one actually go about showing this? I had used this in a proof regarding the Gamma function, but it would be better if I understood how to prove this seemingly simple identity.
Verification of a simple identity regarding $z \in \mathbb{C}$ and $t \in \mathbb{R}^{\geq 0}$
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complex-analysis
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0@RobertIsrael: I adjusted that in my title, but thanks for the tip! – 2012-03-10
1 Answers
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If $t > 0$ and $z = a+bi$ where $a$ and $b$ are real, $\log t$ is real, and $t^z = e^{z \log t} = e^{a \log t} e^{i b \log t}$. Now $e^{a \log t} > 0$ while $|e^{i b \log t}| = 1$.