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Em Causal Response And Radiation Prerequisites

1 · Prerequisites

2 · Summary

The mathematical layer beneath causal response theory and compact-source radiation. The first item, Causal convolution response, transfer function and dispersion relations, develops the convolution response $P (t) = \int_{0}^{\infty} K (s) E (t - s) d s$ of a real matrix kernel on bounded continuous histories, proves that it is bounded, continuous, linear and causal, and analyzes its transfer function $χ (z) = \int_{0}^{\infty} K (s) e^{i z s} d s$ in the compactly supported and integrable branches: the principal-value dispersion relations hold entrywise under exact decay hypotheses, and the monochromatic period-averaged work is identified with the quadratic form of the Hermitian dissipative part $(χ - χ^{*}) / (2 i)$ , whose positivity is a passivity condition on the model rather than a consequence of causality. The second item, Uniform far-zone asymptotics and retarded-time flux limit for compact conserved smooth sources, treats smooth sources of compact spatial support with compact time support satisfying the continuity equation; it proves the retarded field integrals on the exterior of the source, identifies the retarded potentials with the zero-data wave Cauchy solution, establishes the uniform far-zone expansion with explicit error control by finitely many source suprema, and derives the outward flux limit at fixed retarded time for the sphere at infinity.

Both items are mathematics: they carry domain: mathematics, depend only on mathematical suppliers, and assert no physical conclusion. Empirical qualifications of any physical use of these estimates belong to the consuming physical item. The companion examples page records the two hypothesis boundaries that these proofs depend on, namely that causality alone does not force a nonnegative work form and that the far-zone flux limit is taken at fixed retarded time rather than fixed observation time.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Causal convolution response, transfer function and dispersion relations

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the smooth Riemann/Lebesgue comparison route below; in clause (c), also assume Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) inherited from the AC supplier. Let $d \geq 1$ . All kernels below are real $d \times d$ matrix kernels and all histories are $R^{d}$ -valued; conditions on matrices are read entrywise, every matrix- or vector-valued integral below is read entrywise as a scalar Lebesgue integral, and $L^{1} (0, \infty; R^{d \times d})$ means entrywise Lebesgue integrability on $(0, \infty)$ with Lebesgue measure.

For a real-variable scalar function $F$ and real $ω$ , the principal value is defined by $PV ⁣ \int_{R} \frac{F (ξ)}{ξ - ω} d ξ : = \lim_{R \to \infty} \lim_{η ↓ 0} (\int_{ω - R}^{ω - η} + \int_{ω + η}^{ω + R}) \frac{F (ξ)}{ξ - ω} d ξ,$ read entrywise when applied to matrix entries; under the bounds proved in (b) and (c) the tails converge absolutely, so this limit exists and the centering at infinity does not affect it.

(a) (Causal convolution.) Let $K \in L^{1} (0, \infty; R^{d \times d})$ and let $E : R \to R^{d}$ be bounded and continuous. Then for every $t \in R$ the integral $P (t) : = \int_{0}^{\infty} K (s) E (t - s) d s$ converges absolutely, read componentwise as $P_{j} (t) = \int_{0}^{\infty} (\sum_{k = 1}^{d} K_{j k} (s) E_{k} (t - s)) d s (j = 1, \dots, d),$ and satisfies $∣ P_{j} (t) ∣ \leq \sum_{k = 1}^{d} (\sup_{s \in R} ∣ E_{k} (s) ∣) \int_{0}^{\infty} ∣ K_{j k} (s) ∣ d s (j = 1, \dots, d);$ the response $P$ is bounded and continuous, the map $E \mapsto P$ is linear, and it is causal: if $E_{1}, E_{2}$ are bounded continuous and coincide on $(- \infty, t_{0}]$ , then their responses coincide on $(- \infty, t_{0}]$ .

(b) (Compact branch.) The support of a scalar, vector or matrix map is the closure, in its domain, of the set where it is nonzero. If $K \in C_{c}^{\infty} ((0, \infty); R^{d \times d})$ (smooth, with compact support contained in the open half-line), then the transfer function $χ (z) : = \int_{0}^{\infty} K (s) e^{i z s} d s$ is entire, satisfies $χ (- ω) = \overline{χ (ω)}$ for every real $ω$ , obeys $∣ χ_{j k} (z) ∣ \leq ∥ K_{j k}^{''} ∥_{L^{1} (0, \infty)} ∣ z ∣^{- 2}$ for every $z \neq 0$ with $Im z \geq 0$ , and for any $b > 0$ with $supp K \subseteq (0, b)$ the full entire-plane bound is $∣ χ_{j k} (z) ∣ \leq e^{b \max (0, - Im z)} ∥ K_{j k}^{''} ∥_{1} ∣ z ∣^{- 2}$ for every $z \neq 0$ . Its entries satisfy the principal-value dispersion relations $Re χ_{j k} (ω) = \frac{1}{π} PV ⁣ \int_{R} \frac{Im χ_{j k} (ξ)}{ξ - ω} d ξ, Im χ_{j k} (ω) = - \frac{1}{π} PV ⁣ \int_{R} \frac{Re χ_{j k} (ξ)}{ξ - ω} d ξ$ for every real $ω$ .

(c) (Integrable branch.) If instead $K : [0, \infty) \to R^{d \times d}$ is locally absolutely continuous entrywise in the sense of Absolute continuity on a compact interval on every compact subinterval of $[0, \infty)$ , with $K^{'}$ its almost-everywhere derivative with $K, K^{'} \in L^{1} (0, \infty; R^{d \times d})$ and $s \mapsto s K (s)$ in $L^{1} (0, \infty; R^{d \times d})$ , then $χ (z) : = \int_{0}^{\infty} K (s) e^{i z s} d s$ is well defined and holomorphic on the open upper half-plane ${Im z > 0}$ , continuous on the closed upper half-plane with continuous first derivative on the real boundary, satisfies the same pair of dispersion relations for every real $ω$ , and is bounded on ${Im z \geq 0}$ by $∣ χ_{j k} (z) ∣ \leq ∣ K_{j k} (0) ∣ + ∥ K_{j k}^{'} ∥_{L^{1} (0, \infty)}$ for $∣ z ∣ \geq 1$ ; for an instantaneous real response matrix $χ_{\infty} \in R^{d \times d}$ define $χ_{t o t a l} : = χ_{\infty} + χ$ and apply the relations to $χ_{t o t a l} - χ_{\infty} = χ$ .

(d) (Monochromatic work and passivity.) Let $ω > 0$ , $e \in C^{d}$ , and let $ε_{0} > 0$ be a real parameter (a response normalization; the equivalence below is independent of its value). Put $E (t) : = Re (e e^{- i ω t})$ and $P (t) : = Re (ε_{0} χ (ω) e e^{- i ω t})$ , real parts taken componentwise and $χ (ω)$ acting by matrix multiplication. Then, with $T : = 2 π / ω$ and the Euclidean dot product on $R^{d}$ , $\frac{1}{T} \int_{0}^{T} E (t) \cdot P^{'} (t) d t = \frac{ε_{0} ω}{2} Im (e^{*} χ (ω) e), e^{*} χ (ω) e : = \sum_{j, k = 1}^{d} \overline{e_{j}} χ_{j k} (ω) e_{k},$ where $χ (ω)^{*}$ denotes the conjugate transpose, $(χ (ω)^{*})_{j k} : = \overline{χ_{k j} (ω)}$ , a complex matrix $H$ is Hermitian when $H^{*} = H$ , and a Hermitian $H$ is positive semidefinite when $e^{*} H e \geq 0$ for every $e \in C^{d}$ . This period-averaged work is nonnegative for every $e \in C^{d}$ if and only if the Hermitian dissipative part $M (ω) : = (χ (ω) - χ (ω)^{*}) / (2 i)$ satisfies $e^{*} M (ω) e \geq 0$ for every $e \in C^{d}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), and in clause (c) additionally Dependent Choice $D C$ (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain); a fixed dimension $d \geq 1$ ; a real $d \times d$ matrix kernel $K = (K_{j k})$ and an $R^{d}$ -valued history $E = (E_{k})$ ; all matrix- and vector-valued integrals are read entrywise as scalar Lebesgue integrals on $(0, \infty)$ . In clause (b), $K \in C_{c}^{\infty} ((0, \infty); R^{d \times d})$ with $supp K \subseteq (0, b)$ for some real $b > 0$ ; in clause (c) the kernel satisfies the integrable-branch hypotheses of [F14]; in clause (d), $ω > 0$ , $T = 2 π / ω$ , $ε_{0} > 0$ , $e \in C^{d}$ , and $a : = ε_{0} χ (ω) e \in C^{d}$ .

[F1]

A measurable $h = u + i v : Ω \to C$ is integrable exactly when $\int ∣ h ∣ d μ < + \infty$ , and then $\int h d μ = \int u d μ + i \int v d μ$ ; $L^{1} (μ)$ is the class of integrable complex functions. (Integrable real and complex functions, and their integrals, The class $L^{1} (μ)$ of integrable functions)

[F2]

For integrable functions $∣ \int f d μ ∣ \leq \int ∣ f ∣ d μ$ ; for measurable $0 \leq f \leq g$ one has $\int f \leq \int g$ and $\int c f = c \int f$ for $c \geq 0$ ; the integral over a measurable set is $\int_{E} f = \int f χ_{E}$ ; and the integral is complex-linear, $\int (α f + β g) d μ = α \int f d μ + β \int g d μ$ . (The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset, The Lebesgue integral is linear on $L^{1} (μ)$ )

[F3]

Every continuous map $R^{n} \to R^{m}$ is Borel measurable (under ${A C}_{ω}$ ), and sums, scalar multiples, absolute values and pointwise products of measurable functions are measurable. (Continuous functions on Euclidean spaces are Borel measurable, Arithmetic and lattice operations preserve measurability whenever they are defined)

[F4]

Dominated convergence: if $f_{n} \to f$ almost everywhere and $∣ f_{n} ∣ \leq g$ almost everywhere for a single integrable $g$ , then $\int f_{n} \to \int f$ . (Dominated convergence)

[F5]

Continuity under the integral sign: if $x \mapsto f (x, t)$ is integrable for every $t$ in an interval $I$ , if $t \mapsto f (x, t)$ is continuous for almost every $x$ , and if $∣ f (x, t) ∣ \leq g (x)$ for a single integrable $g$ , then $t \mapsto \int f (x, t) d μ (x)$ is continuous on $I$ . (Continuity under the integral sign)

[F6]

Differentiation under the integral sign in a real parameter: under the hypotheses stated there (integrability for every parameter, almost-everywhere differentiability, measurability of the derivative and a single integrable dominating function for the derivative), $F (t) = \int f (x, t) d μ (x)$ is differentiable with $F^{'} (t) = \int \partial_{t} f (x, t) d μ (x)$ . (Differentiation under the integral sign)

[F7]

The complex exponential is defined by $\exp z = \sum_{n \geq 0} z^{n} / n!$ , that series converges absolutely for every $z$ , $\exp$ is entire with $\exp^{'} = \exp$ and hence continuous, and $∣ \exp (x + i y) ∣ = e^{x}$ for real $x, y$ ; conjugation is a real-field automorphism with $z \overline{z} = ∣ z ∣^{2}$ , hence $∣ \overline{z} ∣ = ∣ z ∣$ and conjugation is continuous, while $Re$ , $Im$ and $\overline{}$ are the coordinate maps of Real and imaginary parts, complex conjugation, and modulus. In particular $Re (e^{i θ}) = \cos θ$ and $Im (e^{i θ}) = \sin θ$ for real $θ$ , and $Re (\overline{w}) = Re w$ . (The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument, The complex exponential is entire and its complex derivative is itself, Complex differentiability at a point implies continuity there, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, Real and imaginary parts, complex conjugation, and modulus)

[F8]

If the four first partial derivatives of $u, v$ exist on a neighbourhood of $a$ and are continuous at $a$ with $u_{x} (a) = v_{y} (a)$ and $u_{y} (a) = - v_{x} (a)$ , then $f = u + i v$ is complex differentiable at $a$ ; holomorphic means complex differentiable on an open set and entire means holomorphic on $C$ . (Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set, Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions)

[F9]

If $f$ is holomorphic on an open $Ω \subseteq C$ and the closed rectifiable contour $γ$ is path-homotopic relative to endpoints to a constant loop in $Ω$ , then $\int_{γ} f d z = 0$ . (A closed contour path-homotopic to a constant loop has zero integral against every holomorphic function)

[F10]

Integration by parts: if $u, v$ are differentiable at every point of $[a, b]$ with integrable derivatives, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ; the bounded Riemann integrable case has the same Lebesgue integral; and integrable functions that agree almost everywhere have the same integral. (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree, $C^{k}$ Euclidean maps and diffeomorphisms)

[F11]

The real sine and cosine are differentiable with $(\sin)^{'} = \cos$ and $(\cos)^{'} = - \sin$ ; the chain rule for total derivatives and the algebra of derivatives give the derivative of every finite combination $t \mapsto c_{0} + c_{1} \cos (ω t) + c_{2} \sin (ω t)$ . (The derivatives of sine and cosine are cosine and minus sine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F12]

On $R^{d}$ the Euclidean inner product is $x \cdot y = \sum_{k = 1}^{d} x_{k} y_{k}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). On $C^{d}$ its complex-bilinear extension is also denoted $x \cdot y = \sum_{k} x_{k} y_{k}$ ; this bilinear form is not used as a positive inner product. The conjugated pairing used below is $x \cdot \overline{y} = \sum_{k} x_{k} \overline{y_{k}}$ , and $e^{*} a = \sum_{k} \overline{e_{k}} a_{k}$ , with conjugation as in [F7].

[F13]

Absolute continuity on a compact interval is the finite disjoint-interval condition of the cited definition; it is the exact hypothesis by which clause (c) reads its kernel, and no differentiability of the kernel beyond the stated almost-everywhere derivative is assumed. (Absolute continuity on a compact interval)

[F14]

Integrable-branch supplier, consumed verbatim in clause (c): assume ${A C}_{ω}$ and $D C$ ; for a locally absolutely continuous real matrix kernel $K$ with $K, K^{'} \in L^{1} (0, \infty; R^{d \times d})$ and $s K (s) \in L^{1} (0, \infty; R^{d \times d})$ , the transform $χ (z) = \int_{0}^{\infty} K (s) e^{i z s} d s$ has holomorphic entries on the upper half-plane, continuous and $C^{1}$ on the real boundary, is bounded on the closed upper half-plane with $∣ χ_{j k} (z) ∣ \leq ∣ K_{j k} (0) ∣ + ∥ K_{j k}^{'} ∥_{1}$ for $∣ z ∣ \geq 1$ (indeed $∣ χ_{j k} (z) ∣ \leq (∣ K_{j k} (0) ∣ + ∥ K_{j k}^{'} ∥_{1}) / ∣ z ∣$ for every nonzero $z$ with $Im z \geq 0$ ), satisfies the two displayed principal-value formulas for every real $ω$ , and carries the instantaneous-response subtraction for a real matrix $χ_{\infty}$ ; causality alone supplies neither further growth hypotheses nor the passivity quadratic-form condition. (Dispersion relations for an absolutely continuous integrable causal kernel)

[F15]

Period-average supplier, consumed in clause (d) in its scalar form applied componentwise: for $ω > 0$ , $T = 2 π / ω$ and all $a, d \in C$ , $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \overline{d});$ the vector form follows by summing this over the coordinate pairs of $a, d \in C^{d}$ . (Period averages of products of real parts of complex exponentials)

Proof

technique · direct; the steps are ordered in dependency layers, so the four clauses of the Statement are interleaved, and every step cites only earlier steps

1.1F1F2F3

Fix $t \in R$ , $j \in {1, \dots, d}$ and set $g_{j} (s) : = \sum_{k = 1}^{d} K_{j k} (s) E_{k} (t - s)$ for $s > 0$ . Each summand is a product of the measurable function $K_{j k}$ with the continuous, hence Borel measurable, function $s \mapsto E_{k} (t - s)$ ; so $g_{j}$ is measurable. With $M : = \max_{k} \sup_{R} ∣ E_{k} ∣ < + \infty$ one has $∣ g_{j} (s) ∣ \leq M \sum_{k = 1}^{d} ∣ K_{j k} (s) ∣$ for every $s > 0$ , and $\sum_{k = 1}^{d} ∣ K_{j k} ∣$ lies in $L^{1} (0, \infty)$ ; hence $g_{j} \in L^{1} (0, \infty)$ and $P_{j} (t) : = \int_{0}^{\infty} g_{j} (s) d s$ converges absolutely. The termwise bound $∣ g_{j} (s) ∣ \leq \sum_{k = 1}^{d} (\sup_{R} ∣ E_{k} ∣) ∣ K_{j k} (s) ∣$ gives the exact stated estimate $∣ P_{j} (t) ∣ \leq \sum_{k = 1}^{d} (\sup_{R} ∣ E_{k} ∣) \int_{0}^{\infty} ∣ K_{j k} (s) ∣ d s \leq M \sum_{k = 1}^{d} \int_{0}^{\infty} ∣ K_{j k} (s) ∣ d s$ .

1.2F1F3F7

Fix $z = x + i y \in C$ . The map $s \mapsto e^{i z s}$ is continuous, since it is the composition of the continuous linear map $s \mapsto i z s$ with the continuous exponential, and $K_{j k}$ is measurable; hence $s \mapsto K_{j k} (s) e^{i z s}$ is measurable as a product of measurable functions. On $supp K \subseteq (0, b)$ one has $∣ e^{i z s} ∣ = ∣ e^{i x s} ∣ e^{- y s} = e^{- y s} \leq e^{b ∣ y ∣}$ , so $∣ K_{j k} (s) e^{i z s} ∣ \leq e^{b ∣ y ∣} ∣ K_{j k} (s) ∣$ ; the right-hand side is integrable, so the integrand is integrable and $χ_{j k} (z) : = \int_{0}^{\infty} K_{j k} (s) e^{i z s} d s$ is well defined with $∣ χ_{j k} (z) ∣ \leq e^{b ∣ y ∣} ∥ K_{j k} ∥_{1}$ .

1.3F14F13F8

Clause (c) is exactly the supplied integrable-branch theorem [F14]: its hypotheses on $K$ are those stated in clause (c) (local absolute continuity entrywise in the sense of [F13], $K, K^{'}, s K \in L^{1}$ ), and it concludes the well-definedness and holomorphy of $χ$ on the open upper half-plane, continuity on the closed upper half-plane and continuous first derivative on the real boundary, the entrywise principal-value formulas for every real $ω$ , and the bound $∣ χ_{j k} (z) ∣ \leq ∣ K_{j k} (0) ∣ + ∥ K_{j k}^{'} ∥_{1}$ for $∣ z ∣ \geq 1$ (which follows from its sharper $∣ z ∣^{- 1}$ bound), together with the instantaneous-response subtraction for a real matrix $χ_{\infty}$ and the scalar case $d = 1$ . Continuity on the closed upper half-plane combines the lemma's holomorphy on the open half-plane, hence continuity there, with its continuity at every real boundary point.

1.4F7F12

Write $a_{j} = α_{j} + i β_{j}$ and $e_{j} = p_{j} + i q_{j}$ for the components of $a = ε_{0} χ (ω) e$ and $e$ . By [F7], $e^{i ω t} = \cos (ω t) + i \sin (ω t)$ , and $Re (w) = Re (\overline{w})$ for every $w \in C$ ; therefore $E_{j} (t) = Re (e_{j} e^{- i ω t}) = Re (\overline{e_{j}} e^{i ω t}) = p_{j} \cos (ω t) + q_{j} \sin (ω t)$ and $P_{j} (t) = Re (a_{j} e^{- i ω t}) = Re (\overline{a_{j}} e^{i ω t}) = α_{j} \cos (ω t) + β_{j} \sin (ω t)$ , using the component computation $Re ((u + i v) (\cos + i \sin)) = u \cos - v \sin$ .

2.1step 1.1F2F1

The estimate of step 1.1 is uniform in $t$ , so $\sup_{t \in R} ∣ P_{j} (t) ∣ \leq M \sum_{k = 1}^{d} ∥ K_{j k} ∥_{1}$ and $P = (P_{1}, \dots, P_{d})$ is bounded. The map $E \mapsto P$ is linear: for $α, β \in R$ and bounded continuous $E, F$ the integrand defining $P_{α E + β F}$ equals $α$ times that of $P_{E}$ plus $β$ times that of $P_{F}$ pointwise, so the linearity of the scalar integral gives $P_{α E + β F} = α P_{E} + β P_{F}$ .

2.2step 1.1F5F3

Fix $j$ and apply [F5] with $X = (0, \infty)$ carrying Lebesgue measure, $I = R$ and $f (s, t) : = \sum_{k = 1}^{d} K_{j k} (s) E_{k} (t - s)$ : for every $t$ the function $f (\cdot, t)$ is integrable by step 1.1; for every $s$ the map $t \mapsto f (s, t)$ is continuous, being a finite sum of products of constants with the continuous functions $t \mapsto E_{k} (t - s)$ ; and $∣ f (s, t) ∣ \leq M \sum_{k = 1}^{d} ∣ K_{j k} (s) ∣$ for the single integrable function named in step 1.1. Therefore $t \mapsto P_{j} (t)$ is continuous on $R$ , and $P$ is continuous componentwise.

2.3step 1.1F1

Causality: if $E_{1}, E_{2}$ are bounded continuous and agree on $(- \infty, t_{0}]$ , and if $t \leq t_{0}$ , then for every $s > 0$ the argument $t - s$ lies in $(- \infty, t_{0}]$ , so the integrands of step 1.1 agree pointwise and hence $P_{E_{1}} (t) = P_{E_{2}} (t)$ ; thus the response at time $t$ depends only on the history up to time $t$ .

2.4step 1.2F6F4F3

Fix $y$ and apply [F6] in the parameter $x$ to $f (s, x) : = K_{j k} (s) e^{i (x + i y) s}$ on a bounded interval of $x$ -values: for each $x$ the function $f (\cdot, x)$ is integrable by step 1.2, the map $x \mapsto K_{j k} (s) e^{i (x + i y) s}$ is differentiable for every $s$ with derivative $i s K_{j k} (s) e^{i (x + i y) s}$ , and on the bounded parameter set this derivative is dominated by $(1 + b) e^{b ∣ y ∣} ∣ K_{j k} (s) ∣$ , an integrable function. Hence $x \mapsto χ_{j k} (x + i y)$ is differentiable with $\partial_{x} χ_{j k} (z) = \int_{0}^{\infty} i s K_{j k} (s) e^{i z s} d s$ ; exchanging the roles of $x$ and $y$ and repeating the same argument gives $\partial_{y} χ_{j k} (z) = \int_{0}^{\infty} (- s) K_{j k} (s) e^{i z s} d s$ . Both partials are continuous in $(x, y)$ : if $(x_{n}, y_{n}) \to (x, y)$ , the integrands $i s K_{j k} (s) e^{i (x_{n} + i y_{n}) s}$ converge pointwise to $i s K_{j k} (s) e^{i z s}$ and are dominated on a bounded parameter neighbourhood by $(1 + b) e^{b Y} ∣ K_{j k} (s) ∣$ for a suitable $Y$ , so [F4] gives convergence of the integrals.

2.5step 1.2F1F2F7

Reality identity: for real $ω$ , conjugating the partial sums of the absolutely convergent series defining $e^{i ω s}$ term by term gives $\sum_{n \leq m} \overline{(i ω s)^{n} / n!} = \sum_{n \leq m} (- i ω s)^{n} / n!$ , because conjugation is a continuous real-field automorphism with $\overline{i} = - i$ and fixes the real numbers $1 / n!$ ; passing to the limit with continuity of conjugation gives $\overline{e^{i ω s}} = e^{- i ω s}$ . Since $K_{j k}$ is real, $\overline{K_{j k} (s) e^{i ω s}} = K_{j k} (s) e^{- i ω s}$ . For $h : = K_{j k} (\cdot) e^{i ω \cdot} \in L^{1} (0, \infty)$ one has $\overline{\int h} = (\int Re h) - i (\int Im h) = \int (Re h - i Im h) = \int \overline{h}$ by [F1] and the complex linearity of the integral; hence $χ_{j k} (- ω) = \int_{0}^{\infty} K_{j k} (s) e^{- i ω s} d s = \int \overline{h} = \overline{\int h} = \overline{χ_{j k} (ω)}$ .

2.6step 1.2F10F2F7

By-parts identity: fix $z \neq 0$ . On $[0, b]$ the real and imaginary parts of $s \mapsto K_{j k} (s)$ and of $v (s) : = e^{i z s} / (i z)$ are differentiable with continuous derivatives, and $K_{j k} (0) = K_{j k} (b) = K_{j k}^{'} (0) = K_{j k}^{'} (b) = 0$ because $supp K \subseteq (0, b)$ ; applying [F10] first with $u = K_{j k}$ and then with $u = K_{j k}^{'}$ gives $\int_{0}^{b} K_{j k} (s) e^{i z s} d s = - \frac{1}{i z} \int_{0}^{b} K_{j k}^{'} (s) e^{i z s} d s = \frac{1}{(i z)^{2}} \int_{0}^{b} K_{j k}^{''} (s) e^{i z s} d s = - z^{- 2} \int_{0}^{b} K_{j k}^{''} (s) e^{i z s} d s$ , the boundary terms vanishing at both steps. All three integrands are bounded and Riemann integrable on $[0, b]$ , so [F10] transfers the identity to Lebesgue integrals; and since $K_{j k}$ , $K_{j k}^{'}$ and $K_{j k}^{''}$ all vanish off $(0, b)$ , the a.e.-equality clause of [F10] replaces each integral over $[0, b]$ by the integral over $(0, \infty)$ . Hence $χ_{j k} (z) = - z^{- 2} \int_{0}^{\infty} K_{j k}^{''} (s) e^{i z s} d s$ for every $z \neq 0$ .

2.7step 1.4F11F7F12

Differentiating the expressions of step 1.4 and using [F11] together with the algebra of derivatives gives $P_{j}^{'} (t) = ω (- α_{j} \sin (ω t) + β_{j} \cos (ω t))$ ; on the other hand, expanding $(- i ω) (α_{j} + i β_{j}) = ω β_{j} - i ω α_{j}$ and multiplying by $e^{- i ω t} = \cos (ω t) - i \sin (ω t)$ shows that the real part of $(- i ω) a_{j} e^{- i ω t}$ equals $ω β_{j} \cos (ω t) - ω α_{j} \sin (ω t)$ , the same number. Hence $P^{'} (t) = Re ((- i ω) a e^{- i ω t}) = Re ((- i ω) ε_{0} χ (ω) e e^{- i ω t})$ componentwise, and $P^{'}$ is the real part of a monochromatic complex expression with vector amplitude $d : = (- i ω) ε_{0} χ (ω) e$ .

3.1step 2.4F2F8

By step 2.4 and the linearity of the integral, $\partial_{y} χ_{j k} (z) = \int_{0}^{\infty} (- s) K_{j k} (s) e^{i z s} d s = \int_{0}^{\infty} i (i s K_{j k} (s) e^{i z s}) d s = i \partial_{x} χ_{j k} (z)$ for every $z$ . Writing $χ_{j k} = u + i v$ , this says $u_{y} + i v_{y} = i (u_{x} + i v_{x}) = - v_{x} + i u_{x}$ , that is $u_{y} = - v_{x}$ and $v_{y} = u_{x}$ ; the four partials exist everywhere and are continuous by step 2.4, so [F8] makes $χ_{j k}$ complex differentiable at every point of $C$ , that is, entire.

3.2step 2.6step 1.2F4F2F7

Bounds and boundary continuity: for $z \neq 0$ with $Im z \geq 0$ , step 2.6 and $∣ e^{i z s} ∣ \leq 1$ for $s \geq 0$ give $∣ χ_{j k} (z) ∣ \leq ∣ z ∣^{- 2} ∥ K_{j k}^{''} ∥_{1}$ ; for arbitrary $z \neq 0$ one has $∣ e^{i z s} ∣ = e^{- s Im z} \leq e^{b \max (0, - Im z)}$ on $supp K \subseteq (0, b)$ , whence $∣ χ_{j k} (z) ∣ \leq e^{b \max (0, - Im z)} ∥ K_{j k}^{''} ∥_{1} ∣ z ∣^{- 2}$ . Also $χ_{j k}$ is continuous on the closed upper half-plane: if $z_{n} \to z$ with $Im z_{n} \geq 0$ , the integrands of step 1.2 converge pointwise and are dominated by $e^{b Y} ∣ K_{j k} ∣$ once all $Im z_{n}$ lie below a bound $Y$ , so [F4] gives $χ_{j k} (z_{n}) \to χ_{j k} (z)$ .

3.3step 2.7F15F7F2F12

Applying the period-average identity of [F15] with amplitude vectors $a = e$ and $d = (- i ω) ε_{0} χ (ω) e$ componentwise, and using step 2.7, $\frac{1}{T} \int_{0}^{T} E (t) \cdot P^{'} (t) d t = \sum_{j = 1}^{d} \frac{1}{T} \int_{0}^{T} Re (e_{j} e^{- i ω t}) Re (d_{j} e^{- i ω t}) d t = \sum_{j = 1}^{d} \frac{1}{2} Re (e_{j} \overline{d_{j}}) = \frac{1}{2} Re (e \cdot \overline{d})$ . Since $\overline{d} = i ω ε_{0} \overline{χ (ω) e}$ and $e \cdot \overline{χ (ω) e} = \sum_{j, k} e_{j} \overline{χ_{j k} (ω)} \overline{e_{k}} = \overline{e^{*} χ (ω) e}$ , this equals $\frac{ε_{0} ω}{2} Re (i \overline{e^{*} χ (ω) e}) = \frac{ε_{0} ω}{2} Im (e^{*} χ (ω) e)$ .

4.1step 3.1step 3.2F9F4F2

Dispersion relations: fix real $ω$ and $j, k$ , and put $φ (z) : = χ_{j k} (z) / (z - ω)$ for $z \neq ω$ . For $R > ∣ ω ∣ + 1$ and $0 < δ < 1$ let $γ$ be the closed piecewise $C^{1}$ contour running along $[ω - R, ω - δ]$ , then along the clockwise semicircle $z = ω + δ e^{i θ}$ , $π \geq θ \geq 0$ , then along $[ω + δ, ω + R]$ , and then back along the counterclockwise large semicircle $z = ω + R e^{i θ}$ , $0 \leq θ \leq π$ . The contour lies in the slit plane $D : = C ∖ {ω - i v : v \geq 0}$ , where $φ$ is holomorphic because $χ_{j k}$ is entire and $z \mapsto (z - ω)^{- 1}$ is holomorphic off $ω$ . This domain is star-shaped about $ω + i A$ for every $A > 0$ : along a segment from this centre to a point off the removed ray, either its imaginary part stays positive or its real part differs from $ω$ except at the centre. The straight contraction to the centre, with the base-point connector and its reverse, therefore gives a null homotopy relative to the base point. Hence [F9] gives $\int_{γ} φ d z = 0$ . The large arc contributes at most $π ∥ K_{j k}^{''} ∥_{1} / (R - ∣ ω ∣)^{2} \to 0$ by step 3.2; the small arc contributes $- i π χ_{j k} (ω) + o (1)$ as $δ ↓ 0$ because $χ_{j k} (ω + δ e^{i θ}) \to χ_{j k} (ω)$ uniformly in $θ$ ; and the two real segments converge to the principal value of $\int_{R} χ_{j k} (ξ) / (ξ - ω) d ξ$ as first $δ ↓ 0$ and then $R \to \infty$ : near $ω$ the term $χ_{j k} (ω) / (ξ - ω)$ cancels between the symmetric punctured intervals while $(χ_{j k} (ξ) - χ_{j k} (ω)) / (ξ - ω)$ stays bounded, and for $∣ ξ ∣ \geq \max (1, 2 ∣ ω ∣)$ the estimate $∣ χ_{j k} (ξ) / (ξ - ω) ∣ \leq 2 ∥ K_{j k}^{''} ∥_{1} ∣ ξ ∣^{- 2}$ makes the tails absolutely integrable. Passing to the limits in the identity $0 = \int_{γ} φ d z$ therefore yields $PV \int_{R} χ_{j k} (ξ) / (ξ - ω) d ξ = i π χ_{j k} (ω)$ ; taking real and imaginary parts gives the two displayed relations, since $- π Im χ_{j k} (ω)$ and $π Re χ_{j k} (ω)$ are respectively the real and imaginary parts of $i π χ_{j k} (ω)$ .

5.1step 3.3F2F7F12∎

Finally $M (ω)^{*} = (χ (ω)^{*} - χ (ω)) / (- 2 i) = M (ω)$ , so $M (ω)$ is Hermitian, and $M (ω) = (χ (ω) - χ (ω)^{*}) / (2 i)$ has quadratic form $e^{*} M (ω) e = (e^{*} χ (ω) e - e^{*} χ (ω)^{*} e) / (2 i)$ , and $e^{*} χ (ω)^{*} e = \sum_{j, k} \overline{e_{j}} \overline{χ_{k j} (ω)} e_{k} = \overline{e^{*} χ (ω) e}$ because $χ (ω)^{*}$ is the conjugate transpose; hence $e^{*} M (ω) e = Im (e^{*} χ (ω) e)$ for every $e \in C^{d}$ . By step 3.3 the period-averaged work equals $\frac{ε_{0} ω}{2} e^{*} M (ω) e$ with $\frac{ε_{0} ω}{2} > 0$ , so it is nonnegative for every $e$ if and only if $e^{*} M (ω) e \geq 0$ for every $e$ ; clause (a) establishes causality without assuming this positivity condition.

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Uniform far-zone asymptotics and retarded-time flux limit for compact conserved smooth sources

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as used by the declared polar/surface and change-of-variables suppliers. Let $c > 0$ , $μ_{0} > 0$ , $a > 0$ and $t_{0} < T$ be constants, let $ρ \in C^{\infty} (R^{3} \times R)$ and $J \in C^{\infty} (R^{3} \times R; R^{3})$ satisfy (H1) $ρ (y, v) = J (y, v) = 0$ whenever $∣ y ∣ > a$ or $v \notin [t_{0}, T]$ , and (H2) $\partial_{v} ρ + {div}_{y} J = 0$ on $R^{3} \times R$ . Here and below $B_{a} : = {y \in R^{3} : ∣ y ∣ < a}$ and ${\overline{B}}_{a} : = {y \in R^{3} : ∣ y ∣ \leq a}$ are the open and closed Euclidean balls of radius $a$ about the origin, and $S^{2} = {n \in R^{3} : ∣ n ∣ = 1}$ is the unit sphere. On the open exterior $U : = {x \in R^{3} : ∣ x ∣ > a}$ define the retarded scalar and vector potentials $φ (x, t) : = \frac{μ_{0} c^{2}}{4 π} \int_{R^{3}} \frac{ρ (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y, A (x, t) : = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y,$ and the fields $E : = - \nabla φ - \partial_{t} A$ , $B : = \nabla \times A$ , where $\nabla f = (\partial_{1} f, \partial_{2} f, \partial_{3} f)$ , $div F = \sum_{j = 1}^{3} \partial_{j} F_{j}$ and $\nabla \times F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ are the spatial operators of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Divergence and curl of a $C^{1}$ vector field. The fields are defined on $U \times R$ using these ordinary derivatives; the conclusions below use $r : = ∣ x ∣ \geq 2 a$ . Every displayed spatial integrand is evaluated on ${\overline{B}}_{a}$ and extended by zero outside it, including at $y = x$ , so no quotient $0 / 0$ is used. All vector-valued integrals are componentwise scalar Lebesgue integrals.

(a) (Fields as retarded integrals.) On the open set ${∣ x ∣ > 2 a}$ the potentials $φ, A$ are of class $C^{\infty}$ in $(x, t)$ , and there $E (x, t) = \frac{μ_{0}}{4 π} \int_{R^{3}} [\frac{N}{R^{2}} c^{2} ρ (y, t - \frac{R}{c}) + \frac{c N}{R} ρ_{t} (y, t - \frac{R}{c}) - \frac{J_{t} (y, t - \frac{R}{c})}{R}] d y, B (x, t) = - \frac{μ_{0}}{4 π} \int_{R^{3}} N \times [\frac{J (y, t - \frac{R}{c})}{R^{2}} + \frac{J_{t} (y, t - \frac{R}{c})}{c R}] d y,$ where $R : = ∣ x - y ∣$ , $N : = (x - y) / R$ , and $ρ_{t}, J_{t}$ are the partial derivatives in the second variable. Moreover, on ${∣ x ∣ \geq 2 a} \times [t_{0}, T]$ , $φ$ and each component of $A$ coincide with the corresponding solution constructed in the wave Cauchy problem of The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support for zero Cauchy data at $t_{0}$ and scalar source $h = μ_{0} c^{2} ρ$ , respectively $h = μ_{0} J_{k}$ .

(b) (Uniform far-zone expansion.) For $n \in S^{2}$ and $u \in [t_{0}, T]$ put $I (n, u) : = \int_{B_{a}} J (y, u + n \cdot y / c) d y$ and $V (n, u) : = \partial_{u} I (n, u) = \int_{B_{a}} J_{t} (y, u + n \cdot y / c) d y$ . Then $\int_{B_{a}} ρ_{t} (y, u + n \cdot y / c) d y = \frac{1}{c} n \cdot V (n, u)$ , and there are constants $Γ_{E}, Γ_{B}$ , depending only on $a, c, μ_{0}$ and on the suprema of $∣ ρ ∣, ∣ ρ_{t} ∣, ∣ ρ_{t t} ∣, ∣ J ∣, ∣ J_{t} ∣, ∣ J_{t t} ∣$ on ${\overline{B}}_{a} \times [t_{0}, T]$ , such that for every $n \in S^{2}$ , $u \in [t_{0}, T]$ and $r \geq 2 a$ , with $E_{l e a d} (n, u, r) : = \frac{μ_{0}}{4 π r} n \times (n \times V (n, u))$ , $∣ E (r n, u + \frac{r}{c}) - E_{l e a d} (n, u, r) ∣ \leq \frac{Γ_{E}}{r^{2}}, ∣ B (r n, u + \frac{r}{c}) - \frac{n \times E_{l e a d} (n, u, r)}{c} ∣ \leq \frac{Γ_{B}}{r^{2}} .$

(c) (Retarded-time flux limit.) Let $S : = μ_{0}^{- 1} E \times B$ and let $d Ω$ be the surface measure on $S^{2}$ (Surface integration on compact C1 hypersurfaces). Then for every $u \in [t_{0}, T]$ , $\lim_{r \to \infty} \int_{S^{2}} r^{2} n \cdot S (r n, u + \frac{r}{c}) d Ω (n) = \frac{μ_{0}}{16 π^{2} c} \int_{S^{2}} ∣ n \times V (n, u) ∣^{2} d Ω (n) .$ This is a limit at fixed retarded time $u$ , not at fixed observation time $t$ ; no decay or boundary condition at spatial infinity is assumed beyond the compact spatial support in (H1).

Facts & Assumptions

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); constants $c > 0$ , $μ_{0} > 0$ , $a > 0$ , $t_{0} < T$ ; sources $ρ \in C^{\infty} (R^{3} \times R)$ , $J \in C^{\infty} (R^{3} \times R; R^{3})$ satisfying (H1) and (H2) of the Statement; the retarded potentials and fields of the Statement, with all vector integrals read componentwise. For $n \in S^{2}$ , $u \in [t_{0}, T]$ , $r \geq 2 a$ we write $x : = r n$ , $t : = u + r / c$ , $R : = ∣ x - y ∣ = ((r - n \cdot y)^{2} + ∣ y ∣^{2} - (n \cdot y)^{2})^{1 / 2}$ , $N : = (x - y) / R$ , $σ : = n \cdot y$ , $v_{y} : = u + σ / c$ and $δ_{y} : = (r - σ - R) / c$ , so that $t - R / c = v_{y} + δ_{y}$ ; $Γ_{ρ}, Γ_{ρ t}, Γ_{ρ t t}, Γ_{J}, Γ_{J t}, Γ_{J t t}$ denote the suprema of $∣ ρ ∣, ∣ ρ_{t} ∣, ∣ ρ_{t t} ∣, ∣ J ∣, ∣ J_{t} ∣, ∣ J_{t t} ∣$ on ${\overline{B}}_{a} \times [t_{0}, T]$ , and $λ (B_{a})$ the Lebesgue measure of the ball.

[F1]

A measurable $h$ is integrable exactly when $\int ∣ h ∣ < + \infty$ , and then the integral is computed from the positive and negative parts or, for complex $h = u + i v$ , as $\int u + i \int v$ ; $L^{1}$ denotes the integrable class and $\int_{E} f = \int f χ_{E}$ is the integral over a measurable set. (Integrable real and complex functions, and their integrals, The class $L^{1} (μ)$ of integrable functions, Integral over a measurable subset)

[F2]

The integral is linear, $∣ \int f ∣ \leq \int ∣ f ∣$ , $0 \leq f \leq g$ measurable implies $\int f \leq \int g$ , $\int c f = c \int f$ for $c \geq 0$ , and integrable functions that agree almost everywhere have the same integral. (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree)

[F3]

Under ${A C}_{ω}$ the ball $B_{a}$ is Lebesgue measurable with $0 < λ (B_{a}) < + \infty$ ; every continuous map $R^{n} \to R^{m}$ is Borel measurable; and sums, scalar multiples and pointwise products of measurable functions are measurable. (Euclidean balls have positive finite Lebesgue measure, Continuous functions on Euclidean spaces are Borel measurable, Arithmetic and lattice operations preserve measurability whenever they are defined)

[F4]

Mean value theorem: if $ϕ$ is continuous on $[p, q]$ and differentiable on $(p, q)$ then $ϕ (q) - ϕ (p) = ϕ^{'} (ξ) (q - p)$ for some $ξ \in (p, q)$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ )

[F5]

Parameter integrals over a fixed compact measurable set of finite measure: if $K \subseteq R^{n}$ is compact, measurable and of finite measure, if $a < b$ , and if the real-valued $f$ and $\partial_{t} f$ are continuous on a neighbourhood of $K \times [a, b]$ , then $t \mapsto \int_{K} f (x, t) d x$ is differentiable at every $t \in (a, b)$ with derivative $\int_{K} \partial_{t} f (x, t) d x$ and has the corresponding one-sided relative derivatives at the endpoints of $[a, b]$ ; the lemma is stated pointwise in its one real parameter and explicitly asserts nothing for moving parameter sets or for singularities crossing $K$ . Applying it once per parameter direction with its parameter renamed to $x_{1}, x_{2}, x_{3}$ or $t$ (all other variables held fixed), and iterating on the resulting integrals, yields the corresponding partial derivatives of a parameter integral whose integrand is smooth in all variables on a neighbourhood of the fixed $K$ . (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface)

[F6]

Chain rule and algebra of derivatives for the composite maps appearing below, such as $y \mapsto (y, u + n \cdot y / c)$ and $(x, t) \mapsto (y, t - ∣ x - y ∣ / c)$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ) The spatial gradient, divergence and curl are the coordinate operators specified in the Statement (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Divergence and curl of a $C^{1}$ vector field).

[F7]

A $C^{k}$ map has continuous word partial derivatives through order $k$ , and $C^{\infty}$ means this for every $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Finite sums, products and composites preserve these classes ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition). On $(0, \infty)$ , $(q^{α})^{'} = α q^{α - 1}$ and real powers agree with rational powers (Continuity and derivatives of positive-base real powers, The exponential definition of real powers agrees with the existing rational powers); iterating this rule makes $\sqrt{q} = q^{1 / 2}$ and $1 / q$ smooth there. Closed bounded Euclidean sets are compact, continuous functions on compact sets are uniformly continuous and attain finite extrema (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F8]

Zero-data wave Cauchy problem of speed $c$ from time $t_{0}$ with source $h \in C^{\infty} (R^{3} \times [t_{0}, T])$ in the extension sense: its solution is $u (x, t) = c^{2} \int_{t_{0}}^{t} (t - v) M_{c (t - v)} h (\cdot, v) (x) d v$ on $R^{3} \times [t_{0}, T]$ , it is $C^{\infty}$ there, and it satisfies $c^{- 2} \partial_{t}^{2} u - Δ u = h$ ; here $M_{r} h (x, v) = \frac{1}{4 π} \int_{S^{2}} h (x + r n, v) d Ω (n)$ . (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support)

[F9]

Polar-coordinates formula: for every nonnegative Borel $f : R^{3} \to [0, \infty]$ , $\int_{R^{3}} f (y) d y = \int_{0}^{\infty} \int_{S^{2}} f (r ω) r^{2} d σ (ω) d r$ ; here $σ$ is the polar surface measure of the cited formula, which equals the chart surface measure $d Ω$ of [F13] (Agreement with the existing polar sphere measure), so the two symbols are used interchangeably; a signed or complex $f$ is handled by applying the formula to positive and negative parts and subtracting. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Agreement with the existing polar sphere measure)

[F10]

One-dimensional $L^{1}$ change of variables: for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R$ and $f \in L^{1} (V)$ , $\int_{V} f = \int_{U} f \circ T ∣ T^{'} ∣$ . (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions)

[F11]

Divergence theorem for a bounded $C^{1}$ Euclidean domain $Ω$ and $F \in C^{1} (\overline{Ω}; R^{n})$ : $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ , with the outward normal. (Divergence on a bounded C1 Euclidean domain)

[F12]

The Euclidean norm satisfies the triangle and Cauchy-Schwarz inequalities (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation). Cross product and inner product on $R^{3}$ : $u \times v$ is bilinear and antisymmetric, $n \cdot (u \times v) = \det (n, u, v)$ , and for $∣ n ∣ = 1$ the identities $n \cdot (n \times w) = 0$ and $n \times (n \times w) = n (n \cdot w) - w$ hold, so $∣ n \times (n \times w) ∣ = ∣ n \times w ∣$ . (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ )

[F13]

Surface integral on the compact embedded $C^{1}$ hypersurface $S^{2}$ : for a nonnegative Borel $f$ the integral $\int_{S^{2}} f d Ω$ is defined by finitely many regular parametrizations and an ambient partition of unity, the empty surface integrates to $0$ , the definition is independent of the chosen presentation, the total surface measure $\int_{S^{2}} 1 d Ω$ is finite, and the integral is monotone in $f$ ; the definition carries the ${A C}_{ω}$ convention already declared. (Surface integration on compact C1 hypersurfaces)

[F14]

$C^{1}$ change of variables for nonnegative Borel functions: under ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R^{m}$ and every nonnegative Borel $h : V \to [0, \infty]$ , $\int_{V} h (y) d y = \int_{U} h (T (z)) ∣ \det D T (z) ∣ d z$ with equality in $[0, \infty]$ ; in particular the translation $T (z) = x + z$ of $R^{3}$ is a $C^{1}$ diffeomorphism with $\det D T = 1$ , and a signed integrand is handled through its positive and negative parts by linearity. (Borel change of variables from the compact-support formula and Radon uniqueness, The Lebesgue integral is linear on $L^{1} (μ)$ )

Proof

technique · direct; part (a) differentiates the retarded integrals and identifies them with the wave Cauchy solution, part (b) expands the field integrals uniformly in the far zone and derives the continuity identity, part (c) passes to the surface integral

1.1F5F6F7F3

Smoothness of the potentials: for $x \in U$ and $y \in {\overline{B}}_{a}$ one has $R = ∣ x - y ∣ \geq ∣ x ∣ - a > 0$ , so the maps $(x, t, y) \mapsto ρ (y, t - R / c) / R$ and $(x, t, y) \mapsto J (y, t - R / c) / R$ are $C^{\infty}$ on a neighbourhood of $U \times R \times {\overline{B}}_{a}$ by [F6] and [F7]. Apply [F5] to the fixed compact set $K = {\overline{B}}_{a}$ once for each parameter direction with its one real parameter renamed to $x_{1}$ , $x_{2}$ , $x_{3}$ and $t$ (the remaining variables held fixed) and iterate on the resulting integrals: every differentiated integrand is again continuous on a neighbourhood of $K$ , so each application is covered by the lemma and $φ$ and each component of $A$ have continuous partial derivatives of every order on $U \times R$ , these differentiated parameter integrals are jointly continuous: on each compact parameter neighbourhood the continuous differentiated integrands are uniformly continuous on its product with $K$ , so the integral of their difference is bounded by $λ (K)$ times their uniformly vanishing difference [F2]. Hence the potentials are $C^{\infty}$ on $U \times R$ by [F7], so the derivatives defining the fields are ordinary derivatives even at $∣ x ∣ = 2 a$ .

1.2F12F1

Elementary estimates in the far zone: for $y \in B_{a}$ , $r \geq 2 a$ and $n \in S^{2}$ , expanding $R^{2} = (r - σ)^{2} + ∣ y ∣^{2} - σ^{2}$ with $σ = n \cdot y$ gives $∣ R - (r - σ) ∣ = ∣ ∣ y ∣^{2} - σ^{2} ∣ / (R + r - σ) \leq a^{2} / r$ because $R + r - σ \geq r - σ + ∣ r - σ ∣ \geq r$ ; consequently $∣ r - R ∣ \leq a + a^{2} / r$ , $∣ N - n ∣ = ∣ (r - R) n - y ∣ / R \leq (a + a^{2} / r + a) / (r - a) \leq 5 a / r$ , $∣ 1 / R - 1 / r ∣ \leq (a + a^{2} / r) / (r (r - a)) \leq 3 a / r^{2}$ , and $∣ δ_{y} ∣ \leq a^{2} / (r c)$ , all uniformly in $y, n$ .

2.1step 1.1F5F6F12

Field formulas: by [F6], $\nabla_{x} R = N$ , $\nabla_{x} (t - R / c) = - N / c$ and $\nabla_{x} \times [J (y, t - R / c)] = - N \times J_{t} (y, t - R / c) / c$ , while $\nabla_{x} (1 / R) = - N / R^{2}$ ; hence $\nabla_{x} [ρ (y, t - R / c) / R] = - N [ρ / R^{2} + ρ_{t} / (c R)]$ and $\nabla_{x} \times [J (y, t - R / c) / R] = - N \times [J / R^{2} + J_{t} / (c R)]$ . Each of these first derivatives of the integrands is continuous on a neighbourhood of ${\overline{B}}_{a}$ , so by [F5] the derivatives may be taken under the integral: $- \nabla φ - \partial_{t} A$ and $\nabla \times A$ are given on $U$ , in particular for $∣ x ∣ \geq 2 a$ , by the two displayed zero-extended integrals of the statement, since $E = - \nabla φ - \partial_{t} A$ and $B = \nabla \times A$ .

2.2step 1.1F9F14F1F3

Polar-coordinates identity: fix $x \in U$ and $t \in R$ , and let $h$ be either $ρ$ or one component $J_{k}$ ; then $y \mapsto h (y, t - ∣ x - y ∣ / c) / ∣ x - y ∣$ vanishes off ${\overline{B}}_{a}$ and $∣ x - y ∣ \geq ∣ x ∣ - a > 0$ there, so the quotient on ${\overline{B}}_{a}$ , extended by zero elsewhere, defines a bounded compactly supported Borel function $f$ on all of $R^{3}$ . The translation $T (z) : = x + z$ is a $C^{1}$ diffeomorphism of $R^{3}$ with $\det D T = 1$ , so [F14] applied to the positive and negative parts of $f$ gives $\int_{R^{3}} f (y) d y = \int_{R^{3}} f (x + z) d z$ ; applying [F9] to the positive and negative parts of $z \mapsto f (x + z)$ , which at $z = ρ ω$ equals $h (x + ρ ω, t - ρ / c) / ρ$ and therefore has inner angular integral $\frac{1}{ρ} \int_{S^{2}} h (x + ρ n, t - ρ / c) d Ω (n) = \frac{4 π}{ρ} M_{ρ} h (x, t - ρ / c)$ , gives $\frac{1}{4 π} \int_{R^{3}} \frac{h (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y = \int_{0}^{\infty} r M_{r} h (x, t - r / c) d r$ with $M_{r} h (x, v) = \frac{1}{4 π} \int_{S^{2}} h (x + r n, v) d Ω (n)$ , because the inner angular integral is $4 π M_{r} h$ and the radial factor is $r^{2} / r = r$ .

2.3step 1.2F11F6F2

Continuity identity: fix $n, u$ and let $ρ_{0} > a$ ; the map $F (y) : = J (y, u + n \cdot y / c)$ is $C^{1}$ on ${\overline{B}}_{ρ_{0}}$ , and [F6] gives $div F (y) = {div}_{y} J (y, v_{y}) + \frac{n}{c} \cdot J_{t} (y, v_{y})$ with $v_{y} = u + n \cdot y / c$ . By [F11] on $B_{ρ_{0}}$ , whose boundary flux vanishes because $J = 0$ for $∣ y ∣ > a$ , $\int_{B_{ρ_{0}}} div F d y = 0$ ; substituting the chain-rule expression and using (H2), $0 = \int_{B_{ρ_{0}}} [- ρ_{t} (y, v_{y}) + \frac{1}{c} n \cdot J_{t} (y, v_{y})] d y$ . The integrands vanish for $∣ y ∣ > a$ by (H1), so [F2] lets the integrals be taken over $B_{a}$ ; the second is $\frac{1}{c} n \cdot V (n, u)$ by the definition of $V$ , whence $\int_{B_{a}} ρ_{t} (y, v_{y}) d y = \frac{1}{c} n \cdot V (n, u)$ .

3.1step 2.2F10F8F1

Identification with the Cauchy solution: fix $x \in U$ . For $t > t_{0}$ the radial integrand of step 2.2 vanishes when $r > c (t - t_{0})$ , because then its source time is less than $t_{0}$ . Apply [F10] to the affine diffeomorphism $v \mapsto c (t - v)$ from $(t_{0}, t)$ onto $(0, c (t - t_{0}))$ , with absolute derivative $c$ ; the integrand is bounded on this finite interval, hence integrable. Endpoint values do not affect the Lebesgue integrals, and this gives $\int_{0}^{\infty} r M_{r} h (x, t - r / c) d r = c^{2} \int_{t_{0}}^{t} (t - v) M_{c (t - v)} h (x, v) d v$ . At $t = t_{0}$ both sides are zero by (H1), so the identity holds throughout $[t_{0}, T]$ . Multiplying it by $μ_{0} c^{2}$ for $h = ρ$ , and by $μ_{0}$ for $h = J_{k}$ , identifies $φ$ and $A_{k}$ respectively with the zero-data Cauchy solutions of [F8] with sources $μ_{0} c^{2} ρ$ and $μ_{0} J_{k}$ . This holds on $U \times [t_{0}, T]$ and in particular at $∣ x ∣ = 2 a$ .

3.2step 1.2step 2.3F2F3F4F12

Uniform error estimates: fix $n, u, r$ , let $r e t$ denote the argument $v_{y} + δ_{y}$ and $0$ the argument $v_{y}$ , and write $F_{t o t} : = c^{2} N ρ_{r e t} / R^{2} + c N ρ_{t, r e t} / R - J_{t, r e t} / R$ and $F_{l e a d} : = \frac{1}{r} (c n ρ_{t, 0} - J_{t, 0})$ , so that the field formula of step 2.1 and the definition of $V$ give $E (r n, u + r / c) - E_{l e a d} (n, u, r) = \frac{μ_{0}}{4 π} \int_{B_{a}} (F_{t o t} - F_{l e a d}) d y$ by the linearity of the integral. Every source factor is bounded by its supremum on ${\overline{B}}_{a} \times [t_{0}, T]$ , and (H1) makes $ρ$ and $J$ vanish on the open set $R^{3} \times ((- \infty, t_{0}) \cup (T, \infty))$ , hence makes all their partial derivatives vanish there too, so these suprema also dominate the retarded arguments $v_{y} + δ_{y}$ ; the mean value theorem [F4] gives $∣ ρ_{t, r e t} - ρ_{t, 0} ∣ \leq Γ_{ρ t t} ∣ δ_{y} ∣$ and $∣ J_{t, r e t} - J_{t, 0} ∣ \leq Γ_{J t t} ∣ δ_{y} ∣$ : when the difference is nonzero, apply [F4] to its scalar projection onto the unit direction of that difference, and use Cauchy-Schwarz from [F12], and using $R^{- 1} \leq 2 / r$ , $R^{- 2} \leq 4 / r^{2}$ and $∣ N / R - n / r ∣ \leq ∣ N - n ∣ / R + ∣ 1 / R - 1 / r ∣ \leq 13 a / r^{2}$ yields $∣ F_{t o t} - F_{l e a d} ∣ \leq Γ_{E}^{'} / r^{2}$ , with $Γ_{E}^{'} : = 4 c^{2} Γ_{ρ} + 13 c a Γ_{ρ t} + a^{2} Γ_{ρ t t} + 3 a Γ_{J t} + a^{2} Γ_{J t t} / c$ for all $r \geq 2 a$ ; hence, by the triangle inequality for the integral and the finiteness of $λ (B_{a})$ , $∣ E (r n, u + r / c) - E_{l e a d} (n, u, r) ∣ \leq Γ_{E} / r^{2}$ with $Γ_{E} : = \frac{μ_{0}}{4 π} Γ_{E}^{'} λ (B_{a})$ . The same computation applied to $B (r n, u + r / c) = - \frac{μ_{0}}{4 π} \int_{B_{a}} N \times [J_{r e t} / R^{2} + J_{t, r e t} / (c R)] d y$ and to $n \times E_{l e a d} / c = - \frac{μ_{0}}{4 π} \int_{B_{a}} \frac{1}{c r} n \times J_{t, 0} d y$ gives $∣ B (r n, u + r / c) - n \times E_{l e a d} (n, u, r) / c ∣ \leq Γ_{B} / r^{2}$ , for $Γ_{B} : = \frac{μ_{0} λ (B_{a})}{4 π} (4 Γ_{J} + 13 a Γ_{J t} / c + a^{2} Γ_{J t t} / c^{2})$ .

4.1step 3.2F5F6F7F12

Smoothness of the far-zone moments: the map $(n, u) \mapsto I (n, u) = \int_{B_{a}} J (y, u + n \cdot y / c) d y$ and its partial derivative $\partial_{u} I = \int_{B_{a}} J_{t} (y, u + n \cdot y / c) d y = V (n, u)$ are $C^{\infty}$ in $(n, u) \in R^{3} \times R$ : apply [F5] repeatedly in the directions $n_{1}, n_{2}, n_{3}, u$ with the fixed compact set ${\overline{B}}_{a}$ , each differentiated integrand again being continuous on a neighbourhood of ${\overline{B}}_{a}$ ; the integrals over $B_{a}$ and ${\overline{B}}_{a}$ agree because the smooth sources and all their derivatives vanish on $∣ y ∣ = a$ by continuity from $∣ y ∣ > a$ . In particular $n \mapsto ∣ n \times V (n, u) ∣^{2}$ is continuous for each fixed $u$ .

4.2step 3.2F12F2

Pointwise flux asymptotics: by step 3.2 write $E = E_{l e a d} + ρ_{E}$ and $B = n \times E_{l e a d} / c + ρ_{B}$ at $(r n, u + r / c)$ with $∣ ρ_{E} ∣ \leq Γ_{E} / r^{2}$ , $∣ ρ_{B} ∣ \leq Γ_{B} / r^{2}$ ; then $n \cdot E_{l e a d} = 0$ and $E_{l e a d} \times (n \times E_{l e a d}) = ∣ E_{l e a d} ∣^{2} n$ by [F12], while $∣ E_{l e a d} ∣ = \frac{μ_{0}}{4 π r} ∣ n \times V (n, u) ∣$ because $∣ n \times (n \times V) ∣ = ∣ n \times V ∣$ . Expanding the cross product bilinearly, the error terms are $O (r^{- 3})$ , so $r^{2} n \cdot S (r n, u + r / c) = \frac{μ_{0}}{16 π^{2} c} ∣ n \times V (n, u) ∣^{2} + R (n, u, r)$ with $∣ R (n, u, r) ∣ \leq C / r$ for a constant $C$ depending only on $a, c, μ_{0}$ and the source suprema, uniformly in $n \in S^{2}$ and $u \in [t_{0}, T]$ .

5.1step 4.2step 4.1F13F2∎

Flux limit: by [F13] the surface integral over the compact sphere of a bounded Borel function is finite and monotone, so for every $u \in [t_{0}, T]$ and $r \geq 2 a$ the difference $\int_{S^{2}} r^{2} n \cdot S (r n, u + r / c) d Ω (n) - \frac{μ_{0}}{16 π^{2} c} \int_{S^{2}} ∣ n \times V (n, u) ∣^{2} d Ω (n) = \int_{S^{2}} R (n, u, r) d Ω (n)$ has absolute value at most $\frac{C}{r} \int_{S^{2}} 1 d Ω$ , which tends to $0$ ; the function $n \mapsto ∣ n \times V (n, u) ∣^{2}$ is continuous by step 4.1, so its surface integral is the displayed finite constant. Hence the stated limit at fixed retarded time $u$ follows; no step compares different values of $u$ , so the limit is asserted only at fixed retarded time.

5 · Examples, experiments and counterexamples

None yet.

Em Causal Response And Radiation Prerequisites

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Causal convolution response, transfer function and dispersion relations

Statement

Facts & Assumptions

Proof

Uniform far-zone asymptotics and retarded-time flux limit for compact conserved smooth sources

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources