# Derivation of the Kramers–Kronig Relations

*The contour-integral derivation: one application of Cauchy's theorem, one indented semicircle, one Sokhotski–Plemelj move, and the two integral relations drop out.*

## Setup

From Chapter 11 we have a response function χ(z) satisfying:

- **Analytic** in the open upper half-plane Im z > 0.
- **Continuous** onto the real axis (modulo isolated real poles, handled
  below by a small indentation).
- **Decay at infinity**: χ(z) → 0 as |z| → ∞ with Im z ≥ 0, fast enough
  for Jordan's lemma to kill any large-arc contribution.

Fix a real point ω ∈ ℝ at which we wish to relate χ′(ω) and χ″(ω).
Construct the following closed contour C in the complex ω′-plane
(traversed counter-clockwise):

1. The real axis from −R to ω − ε.
2. A small semicircle γ_ε of radius ε, centred at ω, indented into the
   **upper** half-plane (so that ω is excluded from the interior).
3. The real axis from ω + ε to +R.
4. A large semicircle Γ_R of radius R in the upper half-plane closing
   back to −R.

Inside C the function ω′ ↦ χ(ω′) / (ω′ − ω) is analytic — the only
potential singularity, the simple pole at ω′ = ω, has been indented out.
Cauchy's theorem therefore gives

```
∮_C  χ(ω′) / (ω′ − ω)  dω′  =  0.
```

The rest of the derivation is bookkeeping: take each of the four pieces
in turn and collect limits as ε → 0, R → ∞.

## Piece 1: the large arc Γ_R

On Γ_R we write ω′ = R e^{iθ} with θ ∈ (0, π). The factor 1/(ω′ − ω)
is O(1/R), and χ(ω′) → 0 as R → ∞ by assumption. Jordan's lemma
(Chapter 9), or a direct estimate `|∫_{Γ_R} | ≤ π R · max|χ|/R = π·max|χ|`,
then gives

```
lim_{R→∞} ∫_{Γ_R}  χ(ω′) / (ω′ − ω)  dω′  =  0.
```

The large arc contributes nothing. (If χ does not decay — e.g. n(ω)
with n(∞) = 1 — this step fails and we must instead use a **subtracted**
relation; see Chapter 15.)

## Piece 2: the small semicircle γ_ε — Sokhotski–Plemelj

Near ω, parametrise γ_ε as ω′ = ω + ε e^{iφ} with φ running from π down
to 0 (the semicircle is traversed **clockwise** when the outer contour
is CCW and the indentation is into the UHP). Then dω′ = iε e^{iφ} dφ,
ω′ − ω = ε e^{iφ}, and

```
∫_{γ_ε}  χ(ω′) / (ω′ − ω)  dω′
    = ∫_π^0  [χ(ω + ε e^{iφ}) / (ε e^{iφ})] · (iε e^{iφ}) dφ
    = i ∫_π^0 χ(ω + ε e^{iφ}) dφ.
```

As ε → 0, χ(ω + ε e^{iφ}) → χ(ω) uniformly in φ (continuity), so

```
lim_{ε→0}  ∫_{γ_ε}  χ(ω′) / (ω′ − ω) dω′  =  i χ(ω) · (0 − π)  =  −i π χ(ω).
```

This is the **half-residue rule** — a pole on the contour contributes
`±iπ` times the residue, with sign set by which side of the real axis
we indent into. This is Sokhotski–Plemelj (Chapter 8) in disguise, and
it is the geometric heart of why K–K has the coupling constant 1/π.

## Piece 3: the two real-axis segments — principal value

The two real segments combine, in the joint limit ε → 0, R → ∞, into the
Cauchy principal value:

```
lim_{ε→0, R→∞} [∫_{−R}^{ω−ε} + ∫_{ω+ε}^{R}]  χ(ω′)/(ω′−ω) dω′
    =  P ∫_{−∞}^{∞}  χ(ω′) / (ω′ − ω) dω′.
```

## Assemble

Summing the four contributions and setting the total to zero,

```
P ∫_{−∞}^{∞}  χ(ω′)/(ω′−ω) dω′   −   iπ χ(ω)   +   0   =   0
```

which rearranges to

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

Using `1/i = −i`,

```
χ(ω)  =  (−i / π)  P ∫_{−∞}^{∞}  χ(ω′) / (ω′ − ω) dω′.
```

This is the complex K–K relation. It says χ(ω) on the real axis is
reconstructed from itself by a Hilbert transform — every real point
knows its neighbours via the principal-value integral. At this level
real and imaginary parts are still tangled together.

## Split into real and imaginary parts

Write χ = χ′ + iχ″, both real on the real axis. Then the integrand
splits likewise, and the prefactor `−i/π` rotates real ↔ imaginary:

```
χ′(ω) + iχ″(ω)
  = (−i/π) P ∫  [χ′(ω′) + iχ″(ω′)] / (ω′ − ω) dω′
  = (−i/π) P ∫ χ′(ω′)/(ω′−ω) dω′  +  (1/π) P ∫ χ″(ω′)/(ω′−ω) dω′.
```

Matching real and imaginary parts on both sides gives the **two-sided
Kramers–Kronig relations**:

```
χ′(ω)  =  (1/π)  P ∫_{−∞}^{∞}  χ″(ω′) / (ω′ − ω) dω′
χ″(ω) = −(1/π)  P ∫_{−∞}^{∞}  χ′(ω′) / (ω′ − ω) dω′
```

χ′ and χ″ are Hilbert transforms of each other.

## Folding to positive frequencies

If h(t) is real, χ has crossing symmetry (Chapter 10):

```
χ′(−ω) =  χ′(ω)         χ″(−ω) = −χ″(ω).
```

Fold the integration range (−∞, 0) onto (0, ∞) by substituting
ω′ → −ω′ in the negative-ω′ half and using the parity rules. For the
first relation, combining the two halves,

```
χ′(ω) = (1/π) P ∫₀^∞ χ″(ω′) [1/(ω′−ω) − 1/(−ω′−ω)] dω′
      = (1/π) P ∫₀^∞ χ″(ω′) [ (ω′+ω) + (ω′−ω) ] / (ω′² − ω²) dω′
      = (2/π) P ∫₀^∞ ω′ χ″(ω′) / (ω′² − ω²) dω′.
```

The same manipulation on the second relation, using χ′ even and χ″ odd,
gives

```
χ″(ω) = −(1/π) P ∫₀^∞ χ′(ω′) [1/(ω′−ω) + 1/(ω′+ω)] dω′   (… after care with signs)
      = −(2ω/π) P ∫₀^∞ χ′(ω′) / (ω′² − ω²) dω′.
```

So the **one-sided Kramers–Kronig relations** are

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

These are the forms quoted in every optics and dielectric-spectroscopy
textbook. Note the mechanical origin of each factor of two (the fold),
the factor of ω in the second integrand (comes from χ′ being even), and
the principal-value denominator `ω′² − ω²` (pole at ω′ = +ω only, after
the fold uses up the ω′ = −ω partner).

## Step audit

One-line audit of each ingredient used:

- **Analyticity in UHP** (Chapter 11): justified `∮ = 0` via Cauchy.
- **Decay at infinity** (Chapter 11): killed Γ_R via Jordan.
- **Sokhotski–Plemelj** (Chapter 8): gave the −iπχ(ω) from γ_ε.
- **Principal value** (Chapter 8): assembled the real-axis segments.
- **Crossing symmetry** (Chapter 10): folded two-sided → one-sided.

Remove any one of the five and the derivation collapses. Nothing else
was needed.

## See also

- [08_principal_value.md](08_principal_value.md)
- [11_analyticity_upper_half_plane.md](11_analyticity_upper_half_plane.md)
- [13_hilbert_transform.md](13_hilbert_transform.md)
- [15_subtracted_dispersion.md](15_subtracted_dispersion.md)