Complex plane-wave fields: real parts of complex exponentials

Fix a vector $\mathbf{k}\in {\mathbb{R}}^3$, a parameter $\omega >0$ and amplitudes $\mathbf{e},\mathbf{b}\in {\mathbb{C}}^3$. Here ${\mathbb{C}}^3$ is the set of functions $\{0,1,2\}\to \mathbb{C}$ under componentwise operations, the complex analogue of the convention for ${\mathbb{R}}^n$ in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; we relabel the function values at $0,1,2$ by $1,2,3$ and write the coordinates of $\mathbf{e}$ as $(e_1,e_2,e_3)$ and those of $\mathbf{b}$ as $(b_1,b_2,b_3)$. For $\mathbf{x}\in {\mathbb{R}}^3$, $⟨\mathbf{k},\mathbf{x}⟩$ is the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, the real number ${\sum }_{j=1}^3{k}_j{x}_j$; for $t\in \mathbb{R}$ put

$\theta (\mathbf{x},t):=⟨\mathbf{k},\mathbf{x}⟩-\omega t\in \mathbb{R}\mathrm{.}$

Let $\exp$ be the complex exponential of The complex exponential by its power series (the power series ${\sum }_{n\ge 0}{z}^n\mathrm{/}n!$, convergent for every $z\in \mathbb{C}$), let $\mathrm{Re}$ be the real part of Real and imaginary parts, complex conjugation, and modulus, and let $i$ be the imaginary unit of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$.

For each $t\in \mathbb{R}$ define maps $\mathbf{E}(\cdot ,t),\mathbf{B}(\cdot ,t):{\mathbb{R}}^3\to {\mathbb{R}}^3$ componentwise by

$E_j(\mathbf{x},t):=\mathrm{Re}(e_j\exp (i\theta (\mathbf{x},t)\left)\right),B_j(\mathbf{x},t):=\mathrm{Re}(b_j\exp (i\theta (\mathbf{x},t)\left)\right),j\in \{1,2,3\}\mathrm{.}$

Equivalently, $\mathbf{E}(\cdot ,t)$ is the real part, taken componentwise, of the complex vector $\mathbf{e}\exp (i\theta (\mathbf{x},t))$, and likewise for $\mathbf{B}(\cdot ,t)$ with $\mathbf{b}$ in place of $\mathbf{e}$.

Well-definedness and conventions

For every $(\mathbf{x},t)\in {\mathbb{R}}^3\times \mathbb{R}$ the number $\theta (\mathbf{x},t)$ is real, so $i\theta (\mathbf{x},t)$ is a complex number and $\exp (i\theta (\mathbf{x},t))$ is defined by the everywhere-convergent exponential series; multiplying by the complex number $e_j$ or $b_j$ and taking real parts is likewise defined. Both amplitudes may be zero, in which case the corresponding field vanishes identically. The joint map $(\mathbf{x},t)\mapsto (\mathbf{E}(\mathbf{x},t),\mathbf{B}(\mathbf{x},t))$ has ${\mathbb{R}}^3\times \mathbb{R}$, an open subset of ${\mathbb{R}}^4$, as its domain, so the $C^k$ conventions of $C^k$ Euclidean maps and diffeomorphisms apply to any regularity statement made about it; this definition asserts no regularity of its own beyond well-definedness. The exponential is entire with ${\exp }^{\mathrm{\prime }}=\exp$ (The complex exponential is entire and its complex derivative is itself), which is what makes the encoded expressions differentiable in the plane-wave form; derivations that differentiate them use Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ and the chain rule for holomorphic maps where they are applied.

Complex notation here is an encoding of real fields and of real linear equations, not a complex physical field. The symbols $\mathbf{e}$ and $\mathbf{b}$ are constant complex amplitudes, not fields; the complex quantity $\mathbf{e}\exp (i\theta (\mathbf{x},t))$ is an auxiliary object whose real part is the field. Real parts, imaginary parts and complex conjugation are the explicit operations of Real and imaginary parts, complex conjugation, and modulus. Since $\mathrm{Re}$ is additive and commutes with real scalar multiples, any identity obtained from the encoded expressions by sums, real scalar multiples, partial differentiation or integration may be projected by $\mathrm{Re}$ term by term; identities that multiply two encoded real fields may not be read off from a product of complex expressions without conjugation, and the correct second-order average rule is proved separately. No electromagnetic interpretation, equation of motion or physical identification is part of this definition.