Tagged Questions

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Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
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Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
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Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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function $(-2)^{x}$
What is the real and imaginary parts for the function $f(x)=(-2)^{x}$ ? Is there a unique solution to this question?
Take for example $f(z)=e^z$, so the inverse is $z(f) = \ln(f) + n\pi i$ for an arbitrarily chosen (but fixed) branch $n\in\mathbb N$. Now if $f$ is restricted to e.g. $0<|1-f|<1$ such that the ...