# Analyticity in the Upper Half-Plane

*Why a causal impulse response automatically extends its Fourier transform to an analytic function in the upper half of the complex-frequency plane — and why that extension decays there.*

## The Fourier integral, restricted to t ≥ 0

Chapter 10 established that a causal LTI system has h(t) = 0 for t < 0.
Its transfer function χ(ω), originally defined for real ω by

```
χ(ω) = ∫_{−∞}^{∞} h(t) e^{iωt} dt
```

therefore collapses to an integral over the positive half-line only:

```
χ(ω) = ∫_{0}^{∞} h(t) e^{iωt} dt
```

This single-sided form is the entire engine of the K–K derivation.
Everything follows from asking: what happens when we allow ω to become
complex?

## Promoting ω to a complex variable

Let z = ω + iη with η = Im z. Substituting,

```
e^{izt} = e^{i(ω + iη)t} = e^{iωt} · e^{−ηt}
```

The first factor is a bounded oscillation on the real axis. The second
factor, `e^{−ηt}`, is the convergence enforcer. Because the integration
range is t ∈ [0, ∞),

- if **η > 0** (upper half-plane), `e^{−ηt}` decays exponentially as
  t → ∞, and the integral converges absolutely whenever h(t) is of no
  worse than polynomial growth;
- if **η < 0** (lower half-plane), `e^{−ηt}` grows exponentially, the
  integral diverges for any non-trivial h, and χ(z) cannot be defined
  this way.

So the causal cut-off to t ≥ 0 produces an asymmetry: the Fourier
integral defines χ(z) **naturally** in the upper half-plane and **not at
all** in the lower half-plane. That asymmetry is the geometric form of
causality.

## Analyticity

Inside the strip η > 0 the integral not only converges but converges
uniformly on compact subsets (differentiate under the integral sign is
justified by the same exponential decay). Standard Morera / Fubini
arguments then give:

```
χ(z) = ∫_{0}^{∞} h(t) e^{izt} dt    is holomorphic for Im z > 0.
```

No singularities in the open UHP. In the closed UHP (Im z ≥ 0) one may
still have poles or branch points *on* the real axis — these correspond
to undamped resonances or marginal stability — but they are the only
singularities allowed. Any dissipation pushes the poles strictly into
the lower half-plane.

This is the (3) ⇒ (1) arrow of Titchmarsh's theorem, made concrete: a
time-domain equality (h(t) = 0 for t < 0) is promoted to a
complex-analytic property (χ analytic in Im z > 0) by inspection of a
single exponential factor.

## Decay at infinity in the UHP

Analyticity alone is not enough for the K–K contour argument in Chapter
12. We also need χ(z) → 0 as |z| → ∞ inside the UHP, fast enough that
the large semicircular arc of the closing contour contributes nothing.
This is the **boundedness** half of the Hardy-space H² condition, and it
is what licenses Jordan's lemma (Chapter 9) on the closing arc.

Fortunately the Fourier integral makes this decay automatic for physical
response functions. Two complementary mechanisms contribute:

1. **Riemann–Lebesgue.** If h ∈ L¹(0, ∞), then on the real axis
   χ(ω) → 0 as |ω| → ∞. Essentially, infinitely fast oscillation
   averages any integrable h to zero.
2. **Exponential suppression off-axis.** Along any ray into the UHP
   at angle 0 < θ < π, we have Im z = |z| sin θ > 0, and the factor
   `e^{−(Im z) t}` forces χ(z) to decay **exponentially** in |z| as
   soon as we move off the real axis. The integral is dominated by
   a small neighbourhood of t = 0, and `|χ(z)| ≤ |h(0⁺)| / |z|` for
   large |z| in any such ray (assuming h is continuous at 0⁺).

Polynomial decay on the real axis plus exponential decay off it is far
more than Jordan's lemma needs (`|χ(z)| → 0` along the arc is enough);
the K–K contour closure will go through cleanly.

## What can still go wrong at infinity

For some response functions h(t) has a delta-function piece at t = 0
(an instantaneous response), and χ(ω) does **not** vanish as |ω| → ∞ —
it tends to a non-zero constant. Classic examples:

- Refractive index: `n(ω) → 1` as ω → ∞.
- Dielectric function: `ε(ω) → 1`.
- Magnetic permeability: `μ(ω) → 1`.

These are still causal, still analytic in the UHP, but the large-arc
contribution in the K–K contour no longer vanishes. The fix is either
to work with the **deviation** `n(ω) − 1`, `ε(ω) − 1`, `μ(ω) − 1` —
which does decay — or to use a **subtracted dispersion relation**
(Chapter 15). Both approaches restore convergence without giving up any
physics.

## The engineering convention flips the sign

In the engineering / signal-processing convention the Fourier kernel is
`e^{+jωt}` rather than `e^{−iωt}`, and the role of the Laplace variable
is played by `s = jω`. The same causality argument now says: the Laplace
transform H(s) = ∫₀^∞ h(t) e^{−st} dt is analytic for **Re s > 0**,
i.e. the right half of the s-plane. Translated back to ω = s/j, "right
half-plane in s" becomes "**lower** half-plane in ω". Engineers therefore
state K–K with χ(ω) analytic for Im ω < 0, and every ±i in Chapter 12's
derivation flips sign.

Mathematically the two conventions are related by ω → −ω (or complex
conjugation of the kernel), and no physics is lost. Knowing where the
flip lives prevents sign-hunting bugs when porting a formula between a
physics paper and a control-theory textbook.

## What we have so far

At this point we have earned:

- χ(ω) extends to an analytic function χ(z) on Im z > 0.
- χ(z) → 0 as |z| → ∞ anywhere on or above the real axis (with the
  decay-at-infinity caveats noted).
- Any singularities on the real axis are simple poles (resonances), and
  will be treated by principal-value prescriptions.

Chapter 12 now takes these three facts, applies Cauchy's theorem on a
closed UHP contour, and extracts the K–K relations in both the two-sided
and one-sided forms.

## See also

- [09_jordan_lemma.md](09_jordan_lemma.md)
- [10_causality_LTI.md](10_causality_LTI.md)
- [12_derivation_KK.md](12_derivation_KK.md)