Dispersion relations for a finitely subtracted analytic response

For $n\ge 1$, by [F1] applied in a disc contained in the neighbourhood of ${\omega }_0$ on which $\chi$ is holomorphic, $\chi (z)=P(z)+(z-{\omega }_0{)}^n\tilde{g}(z)$ for a holomorphic $\tilde{g}$ there; hence $f$ has the holomorphic removable continuation $\tilde{g}$ at ${\omega }_0$, and elsewhere $f=(\chi -P)\mathrm{/}(z-{\omega }_0{)}^n$ is holomorphic on $H$ because the denominator does not vanish; the two definitions agree on the punctured intersection, so $f$ is holomorphic on $H$ and near ${\omega }_0$, continuous on the closure of $H$ (the removable continuation at ${\omega }_0$ together with the continuity of $\chi$ and polynomiality of $P$), of class $C^1$ on $\mathbb{R}$ by hypothesis, and $\mathrm{\mid }f(z)\mathrm{\mid }\le C^{\mathrm{\prime }}(1+\mathrm{\mid }z{\mathrm{\mid }}^{n-1})\mathrm{/}\mathrm{\mid }z-{\omega }_0{\mathrm{\mid }}^n=O(1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$ uniformly on $\mathrm{Im}z\ge 0$ outside a compact set.

Fix real $\omega$, $R>\mathrm{\mid }\omega \mathrm{\mid }+2$ and $0<\eta <1$, and form the closed indented contour $\mathrm{\Gamma }$ consisting of the real segments from $\omega -R$ to $\omega -\eta$ and from $\omega +\eta$ to $\omega +R$, the upper semicircle of radius $\eta$ about $\omega$ traversed from $\omega -\eta$ to $\omega +\eta$, and the large upper semicircle of radius $R$ about $\omega$ traversed from $\omega +R$ to $\omega -R$. For $\delta >0$ the domain $D_{\delta }=\{z:\mathrm{Im}z>-\delta \mathrm{/}2\}\setminus \{\omega -iv:v\ge 0\}$ is open and star-shaped about $\omega +iA$ for every $A>0$, the contour $\mathrm{\Gamma }$ lies in $D_{\delta }$, and $z\mapsto f(z+i\delta )\mathrm{/}(z-\omega )$ is holomorphic on $D_{\delta }$ because $\mathrm{Im}(z+i\delta )>\delta \mathrm{/}2>0$ and the denominator avoids its slit zero; hence [F2] gives ${\int }_{\mathrm{\Gamma }}f(z+i\delta )\mathrm{/}(z-\omega )dz=0$.

Constitutive branch: for histories $E_1,E_2$ with the stated bounded continuous derivatives through order $m$ that agree on $(-\mathrm{\infty },t]$, locality [F4] gives equality of all derivatives through order $m$ at every $s<t$, because a sufficiently small neighbourhood of $s$ lies in $(-\mathrm{\infty },t)$. Their continuity then gives $E_1^{(j)}(t)=E_2^{(j)}(t)$ by taking $s\uparrow t$, and the convolutions agree, $(K\ast E_1)(t)=(K\ast E_2)(t)={\int }_0^{\mathrm{\infty }}K(s)E_i(t-s)ds$, because $t-s\le t$ for $s\ge 0$; hence the response ${\sum }_{j=0}^m{A}_j{E}^{(j)}+K\ast E$ evaluated at $t$ is the same for $E_1$ and $E_2$: the response is causal.

On the fixed contour $\mathrm{\Gamma }$ the functions $z\mapsto f(z+i\delta )$ converge uniformly to $z\mapsto f(z)$ as $\delta \downarrow 0$ by the uniform continuity [F2] of the continuous $f$ on a compact neighbourhood of $\mathrm{\Gamma }$ in the closed half-plane, so [F2] transfers the identity of step 1.2 to ${\int }_{\mathrm{\Gamma }}f(z)\mathrm{/}(z-\omega )dz=0$; the large arc contributes at most $\pi R\cdot C\mathrm{/}(R(R-\mathrm{\mid }\omega \mathrm{\mid }))=O(1\mathrm{/}R)\to 0$ as $R\to \mathrm{\infty }$ because $\mathrm{\mid }f(z)\mathrm{\mid }\le C\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ holds for $\mathrm{\mid }z\mathrm{\mid }\ge 1$ in the closed half-plane, while the small arc, parametrized by $z=\omega +\eta e^{i\theta }$ with $\theta$ decreasing from $\pi$ to $0$, contributes $-i\pi f(\omega )+o(1)$ as $\eta \downarrow 0$ by continuity of $f$ at $\omega$; the two real segments, whose tails are absolutely integrable since $f(\xi )\mathrm{/}(\xi -\omega )=O(\mathrm{\mid }\xi {\mathrm{\mid }}^{-2})$, therefore satisfy $\mathrm{PV}{\int }_{\mathbb{R}}f(\xi )\mathrm{/}(\xi -\omega )d\xi =i\pi f(\omega )$, the principal value existing because the constant term $f(\omega )\mathrm{/}(\xi -\omega )$ cancels under the symmetric truncation and the $C^1$ hypothesis bounds the divided difference.

Transfer function: for the monochromatic history $E(t)=E_0{e}^{-i\omega t}$ with constant $E_0$, one has $E^{(j)}(t)=(-i\omega {)}^j{E}_0{e}^{-i\omega t}$ and $(K\ast E)(t)={\chi }_K(\omega )E_0{e}^{-i\omega t}$ by the defining formula of [F3] with $z=\omega$; hence the response is $(Q(\omega )+{\chi }_K(\omega ))E_0{e}^{-i\omega t}$ with $Q(z)={\sum }_{j=0}^m{A}_j(-iz{)}^j$, the analytic transfer function is $Q(z)+{\chi }_K(z)$ on $\mathrm{Im}z>0$, with its real boundary values just computed. The formal evaluation on $e^{-izt}$ gives the same expression because $e^{izs}$ decays on the causal integral, but this complex-frequency history is not claimed to belong to the globally bounded history class.

Taking real and imaginary parts of the identity $i\pi f(\omega )=\mathrm{PV}{\int }_{\mathbb{R}}f(\xi )\mathrm{/}(\xi -\omega )d\xi$ gives $\mathrm{Re}f(\omega )=\frac{1}{\pi }\mathrm{PV}{\int }_{\mathbb{R}}\frac{\mathrm{Im}f(\xi )}{\xi -\omega }d\xi$ and $\mathrm{Im}f(\omega )=-\frac{1}{\pi }\mathrm{PV}{\int }_{\mathbb{R}}\frac{\mathrm{Re}f(\xi )}{\xi -\omega }d\xi$; multiplying by the real number $(\omega -{\omega }_0{)}^n$ and adding the complex polynomial $P$ yields the two displayed formulas for $\mathrm{Re}\chi$ and $\mathrm{Im}\chi$, since $\chi =P+(\cdot -{\omega }_0{)}^{n}f$ on the closed half-plane.

For $n\ge 1$ and $\omega ={\omega }_0$ the factor $(\omega -{\omega }_0{)}^n$ vanishes and $P({\omega }_0)=\chi ({\omega }_0)$ by Taylor's formula at ${\omega }_0$, so the displayed formulas are read as that Taylor value; the imaginary numerator subtracts the full complex polynomial $P$, which therefore cannot be omitted. For $n=0$ one has $P=0$, $f=\chi$, factor $1$, and the growth hypothesis is exactly $O(1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$, so the same contour argument of steps 1.2, 2.1 and 3.1 applies with the principal-value convention at ${\omega }_0$ as well.

Subtracting the declared polynomial $Q$ from $Q+{\chi }_K$ leaves ${\chi }_K$, whose entrywise principal-value dispersion formulas are exactly those proved in [F3]; this branch therefore requires no holomorphic continuation through a real subtraction point and no growth hypothesis beyond those of [F3]; general passivity and storage remain separate model conditions, and the declared polynomial $Q$ of this branch need not coincide with the Taylor polynomial $P$ of the first branch. This proves both branches of the Statement, the first with no choice beyond the cited complex-analysis suppliers and the second with the declared Countable Choice and Dependent Choice inherited from [F3].