The Complex Plane as ℝ²: Standing Notation

(i) Base. This setting extends Complex Analysis: Standing Notation and Multivariable Calculus: Notation, the latter with $n=m=2$: their conventions and notation are in force, except that $U$ denotes the subset of $\mathbb{C}$ fixed in (iii).

(ii) The identification. $\Phi:\mathbb{R}^{2}\to\mathbb{C}$ is the map of The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection, $\Phi(x,y)=x+iy$.

(iii) Domains. Unless stated otherwise, $U$ denotes an open subset of $(\mathbb{C},d_{\mathbb{C}})$, and

$$V=\{p\in\mathbb{R}^{2}:\Phi(p)\in U\}$$

is the corresponding subset of $\mathbb{R}^{2}$; $V$ plays the role of $U$ in Multivariable Calculus: Notation (ii).

(iv) A function read in real coordinates. For a function $f:U\to\mathbb{C}$, $F$ denotes the function

$$F:V\to\mathbb{R}^{2},\qquad F(p)=\Phi^{-1}\bigl(f(\Phi(p))\bigr)=\bigl(\operatorname{Re}f(\Phi(p)),\operatorname{Im}f(\Phi(p))\bigr)$$

(Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), and $u=F_{1}$ and $v=F_{2}$ are its component maps.

(v) Partial derivatives. For a function $g:V\to\mathbb{R}$ and $p\in V$, $\dfrac{\partial g}{\partial x}(p)$ and $\dfrac{\partial g}{\partial y}(p)$ denote the partial derivatives $\partial_{1}g(p)$ and $\partial_{2}g(p)$, when they exist.

(vi) Background. The following are used without further citation. $\Phi$ is a real-linear isometric bijection with inverse $\Phi^{-1}(z)=(\operatorname{Re}z,\operatorname{Im}z)$, and a subset of $\mathbb{C}$ is open in $(\mathbb{C},d_{\mathbb{C}})$ exactly when its preimage under $\Phi$ is open in $(\mathbb{R}^{2},d_{E})$, by The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (i), The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (ii), The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (iii) and The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection (iv); in particular $V$ is open in $(\mathbb{R}^{2},d_{E})$. Since both the earlier and the current versions of the definitions of metric spaces, open sets and continuity are in force here, they are used interchangeably: they agree, by The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (i), The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (v) and The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree (vii).