What is the definition of a single valued function

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this is potentially a dumb question but I am a touch confused about some terminology. I'm reading Ahlfor's complex analysis, and I am in a section on integrals of harmonic functions.

I may be being dense, but I am not sure what he is referring to when he calls a harmonic function "single-valued". For context,

A real-vaued function u(z), defined and single valued in $\Omega$, is said to be harmonic...

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1 Answer

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It means that every element maps to exactly one value. Because in complex analysis you can sometimes deal with things like $\sqrt[n]{x}$, which has $n$ solutions if $x$ is non-zero. Some authors call such functions multi-valued functions.

So the text just means it is a "proper" function.

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