Are the Elementary Binary Operations Analytic?

Question

Are the elementary binary operations of addition, multiplication, and exponentiation -- taken as multivariate functions over the real numbers -- analytic? That is, $f(a, b) = a + b$ and so forth. Does it change if $a, b$ are any complex numbers?

I don't know how to get started assessing multivariate analyticity. Any references on how to determine this would be appreciated.

Answer 1

Yes, all "elementary" operations \begin{align*} (z,w) \mapsto z+w \\ (z,w) \mapsto z-w \\ (z,w) \mapsto zw \\ (z,w) \mapsto \frac{z}{w} \end{align*} are analytic (holomorphic) as functions of two complex variables (and therefore also real-analytic as functions of two real variables), where defined. (Of course, for $z/w$ we have to exclude $w=0$).

For exponentiation, things are a little more tricky, $$ (z,w) \mapsto z^w = \exp(w\log z) $$ is not a well-defined operation. If we fix a particular branch of $\log$, then $z^w$ is holomorphic as a function of two complex variables where defined, but this is more subtle. We can for example view $z^w$ as a holomorphic function on $\mathbb{C}^2$ minus the set where $z$ is ($0$ or) negative real.

Answer 2

Elementry binary operations are analytic. Binary operations, like. Addition, multiplication. For example :a+b=c,a/b{b should not equal to zero to }