# Subtracted Dispersion Relations

*When χ(ω) does not decay fast enough at infinity to close the Kramers–Kronig contour, subtract one (or more) known reference values to manufacture the needed decay — the price is one external input per subtraction.*

## The convergence problem

The derivation in Chapter 12 depended on the large semicircular arc Γ_R
contributing zero in the limit R → ∞. That step required χ(z) → 0 as
|z| → ∞ in the UHP. Many physical response functions fail this test:

| Quantity | Large-ω limit | Decays? |
|---|---|---|
| Refractive index n(ω) | 1 | No |
| Dielectric function ε(ω) | 1 (dielectric) or ε_∞ | No |
| Magnetic permeability μ(ω) | 1 | No |
| Forward scattering amplitude f(ω) | O(ω) or constant | No |
| Acoustic compliance | often O(1) | No |

In each case the plain K–K integral is divergent, not because the
physics is non-causal but because a non-zero instantaneous response —
a delta at t = 0 in h(t) — adds a constant to χ that K–K cannot
reconstruct from any amount of finite-frequency data. The DC-kill
property of the Hilbert transform (H[constant] = 0, Chapter 13) is
exactly this obstruction.

The fix is **subtraction**: replace χ(ω) by a modified function that
does decay, apply the standard contour argument to it, then translate
back.

## Once-subtracted form

Pick a convenient **anchor frequency** ω_a on the real axis (where
either χ(ω_a) is known experimentally, or the physics is simple — often
ω_a = 0 or ω_a → ∞ is taken). Define the **subtracted** response

```
χ_s(ω) = [χ(ω) − χ(ω_a)] / (ω − ω_a).
```

Three facts about χ_s:

1. χ_s is **still analytic in the UHP**. The numerator is analytic
   there and vanishes at ω = ω_a, killing the apparent pole from the
   denominator — so χ_s is analytic on the real axis at ω = ω_a as
   well (removable singularity).
2. χ_s **decays one order faster** than χ at infinity. If χ(z) tends
   to a non-zero constant χ(∞), then χ_s(z) ~ [χ(∞) − χ(ω_a)] / z,
   which is O(1/|z|). Jordan's lemma now closes the contour.
3. Real-axis continuity and principal-value bookkeeping carry over
   unchanged.

Applying the full Chapter 12 derivation to χ_s(ω), the resulting K–K
relation is

```
χ_s(ω) = (1/iπ) P ∫_{−∞}^{∞} χ_s(ω′)/(ω′ − ω) dω′.
```

Multiplying back through by (ω − ω_a) and adding χ(ω_a),

```
χ(ω) = χ(ω_a) + (ω − ω_a)/(iπ) · P ∫_{−∞}^{∞} [χ(ω′) − χ(ω_a)] / [(ω′−ω)(ω′−ω_a)] dω′.
```

Splitting into real/imaginary parts gives the **once-subtracted K–K
relations**:

```
χ′(ω) − χ′(ω_a) = ((ω − ω_a)/π) · P ∫_{−∞}^{∞}  χ″(ω′)/[(ω′ − ω)(ω′ − ω_a)] dω′
χ″(ω) − χ″(ω_a) = −((ω − ω_a)/π) · P ∫_{−∞}^{∞}  χ′(ω′)/[(ω′ − ω)(ω′ − ω_a)] dω′
```

The subtracted kernel `1/[(ω′−ω)(ω′−ω_a)]` falls off as 1/ω′² at large
|ω′|, so the integral converges whenever χ is merely bounded at
infinity.

### Anchor at ω_a = 0

The most common choice, for optical response functions with a real
χ(0), is ω_a = 0:

```
χ′(ω) − χ′(0) = (ω/π) P ∫_{−∞}^{∞}  χ″(ω′) / [ω′(ω′ − ω)] dω′
```

For a real h(t) this folds onto the positive axis as

```
χ′(ω) − χ′(0) = (2ω²/π) P ∫₀^∞  χ″(ω′) / [ω′(ω′² − ω²)] dω′
```

(with the usual crossing-symmetry manipulation). For optical n(ω) one
applies this to `n(ω) − 1` — the subtracted anchor happens to coincide
with the natural asymptote — giving the familiar "refractive-index
dispersion relation" used in optical-constants tabulations.

## Twice-subtracted form

If χ(ω) is *even* faster-growing at infinity — say it tends to a finite
constant but differs from that constant by a term growing like ω —
then one subtraction is not enough. Subtract twice: pick two anchors
ω_a and ω_b, and define

```
χ_{ss}(ω) = [χ(ω) − χ(ω_a) − (ω − ω_a) χ′_ω(ω_a)]  /  (ω − ω_a)².
```

(Here χ′_ω means d χ/d ω, not the real part.) This has a double zero at
ω_a, kills the divergent pole, and decays as 1/ω² faster than χ — so if
χ(z) ~ z at infinity, χ_{ss} ~ 1/z, and the contour closes. The same
Cauchy machinery produces

```
χ(ω) − χ(ω_a) − (ω − ω_a) χ′_ω(ω_a)
  =  ((ω − ω_a)² / iπ) · P ∫ [χ(ω′) − χ(ω_a) − (ω′ − ω_a) χ′_ω(ω_a)] / [(ω′ − ω)(ω′ − ω_a)²] dω′.
```

Splitting real/imaginary parts gives a **twice-subtracted** dispersion
relation. It requires **two** pieces of external input (χ(ω_a) and its
derivative, or two independent anchor values χ(ω_a), χ(ω_b)), but it
converges for any χ growing at most linearly at infinity.

Twice-subtracted relations are the workhorse of **forward scattering
amplitudes** in high-energy physics, where the amplitude f(ω) can grow
like ω (Pomeron exchange), making the unsubtracted and once-subtracted
relations formally divergent.

## General pattern: n subtractions

Every subtraction:

- lowers the large-|ω| tail by one power of ω;
- introduces one extra polynomial-in-ω multiplier out front (ω − ω_a,
  (ω − ω_a)², …);
- requires one extra external input (χ(ω_a), χ′(ω_a), …);
- leaves the physical content unchanged — every subtracted form is
  equivalent to the original whenever the original converges.

The number of subtractions needed is fixed by the leading high-frequency
behaviour of χ: if χ(ω) ~ c ω^k as |ω| → ∞ with k ≥ 0, then k + 1
subtractions are needed to drag the kernel below 1/ω.

## Trade-off

Subtractions are not free. Each one **replaces** genuine spectral
information by an external anchor value. The unsubtracted K–K relation
reconstructs χ on the real axis from χ″ alone; the once-subtracted one
requires χ″ **and** χ(ω_a); the twice-subtracted one requires χ″ and
two anchor inputs. The more subtractions, the more one relies on
off-spectrum physics (limits, derivatives at anchor points) rather than
on the measured spectrum itself.

In practice you pick the **minimum** number of subtractions that
controls the tail — usually one for optics and dielectric response,
two for hadronic forward amplitudes — and choose the anchor where the
external information is most trustworthy (ω_a = 0 if static data is
clean, ω_a = ∞ if a model asymptote is clean).

## Connection to sum rules

Subtracted dispersion relations and sum rules are two sides of the same
coin. A sum rule is the statement that a given asymptotic coefficient
of χ′(ω) equals a moment of χ″ (Chapter 14); a subtraction is the
statement that if that asymptotic coefficient is non-zero you must
anchor it externally. Whenever a physical argument makes an asymptotic
coefficient vanish (an f-sum saturation, a superconvergence relation),
a previously-needed subtraction becomes unnecessary — and conversely, a
subtraction you are forced to introduce indicates the sum rule at that
order is violated (or, more usefully, anchored rather than predicted).

## See also

- [12_derivation_KK.md](12_derivation_KK.md)
- [14_sum_rules.md](14_sum_rules.md)
- [19_minimum_phase_DSP.md](19_minimum_phase_DSP.md)