Comment by bheadmaster
17 days ago
My biggest pet peeve in complex analysis is the concept of multi-value functions.
Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something.
Madness.
You can think of it as returning an equivalence class if you like. Then it's single-valued.
More explicitly, it returns an equivalence class whose members are complex numbers that differ by integer multiples of 2*pi*i.
When it's important to distinguish members of the class, we speak of branches of the logarithm.
Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
> You can think of it as returning an equivalence class if you like. Then it's single-valued.
I can, but I still have to be very mindful of how I use the "=" sign. Sometimes it's an (in)equality, sometimes it's... the equivalence class thing. The ambiguity doesn't seem very elegant.
> Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
I'll check that out, thanks.
Idk, to me it feels much much better than just picking one root when defining the inverse function.
This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).
Well, yeah, the alternative is also bad.
But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.
Just think of it as a function that returns an array or a set: it still one value in a sense