So I have a function $f(x,y)$ where $x \in \mathbb{R}^n$, $y$ is a real variable, and $f(x,y)$ is a complex function of these variables (where for a given vector $x$ and given $y$, the function $f$ is a complex number). For example, if $x = [x_1,x_2]$, then $$f(x,y) = x_1 + x_2 \frac{1}{iy + 1}$$ Would it be correct to state the following? $$f(x,y):\mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{C}$$
Thanks!
No, you are implying that the function is called $f(x,y)$, while the function is called $f$. $f(x,y)$ is what the function does to the couple $(x,y)$
It would be correct to write:
$$f: \mathbb{R^n} \times \mathbb{R} \to \mathbb{C}: (x,y) \mapsto f(x,y)$$
In fact, the last part is redundant (as you didn't specify what $f$ is) and we can really just write:
$$f: \mathbb{R^n} \times \mathbb{R} \to \mathbb{C}$$