# Complex Functions and Analyticity

*What it means for a function ℂ → ℂ to be differentiable, why that is a dramatically stronger condition than real differentiability, and why analyticity is the property that encodes causality.*

## Complex-valued functions of a complex variable

A complex function is a map `f: D → ℂ` with `D ⊆ ℂ`. Writing
`z = x + iy` and `f(z) = u(x, y) + i v(x, y)` splits it into two real
fields. But viewing f as a function of the single complex variable z
imposes far more than viewing (u, v) as a pair of real functions of
(x, y).

## Complex differentiability

The derivative of f at z₀ is defined by exactly the same-looking limit as
in real calculus,

```
f′(z₀) = lim_{h → 0}  [f(z₀ + h) − f(z₀)] / h
```

but h is now a **complex** increment. The limit must exist and give the
same value regardless of the direction from which h → 0 in the complex
plane. This directional-independence requirement is what makes complex
differentiability so restrictive: a real-differentiable map of two
variables has two independent partial derivatives, but a
complex-differentiable function has only one — every directional
derivative is forced to agree.

Writing `h = h_x + i h_y` and taking h along the real axis and the
imaginary axis in turn converts this directional equality into the
**Cauchy–Riemann equations** (Chapter 4):

```
∂u/∂x =  ∂v/∂y
∂u/∂y = −∂v/∂x
```

These two equations, plus continuity of the partial derivatives, are
equivalent to complex differentiability at an interior point.

## Holomorphy

f is **holomorphic on an open set D** if it is complex-differentiable at
every point of D. Holomorphy is the central notion of complex analysis.
Any function holomorphic on all of ℂ is called **entire** — examples:
polynomials, `exp z`, `sin z`, `cos z`.

## Analyticity = holomorphy: the miracle

In real calculus, "differentiable" is much weaker than "infinitely
differentiable" which is weaker still than "equal to its Taylor series".
The standard counterexamples (`|x|`, the smooth bump functions that are
C^∞ but not analytic) show these are genuinely distinct.

**In complex calculus, these distinctions collapse.** If f is
holomorphic on D (i.e., complex-differentiable once in a neighbourhood
of every point), then:

1. f has derivatives of all orders on D — every holomorphic function is
   automatically C^∞.
2. For every point z₀ ∈ D, f has a convergent power-series representation

```
f(z) = Σ_{n = 0}^{∞}  a_n (z − z₀)^n,   a_n = f^{(n)}(z₀) / n!
```

   valid in the largest open disk centred at z₀ and contained in D.

A function representable locally by a convergent power series is called
**analytic**. The theorem above — holomorphic ⇔ analytic — is a theorem
unique to complex analysis, proved via the Cauchy integral formula
(Chapter 6). Because of it, the two words are used interchangeably in
practice. The present series follows that convention.

## Morera's theorem — the converse

Cauchy's theorem says: f holomorphic on a simply connected domain
⇒ ∮_γ f dz = 0 for every closed contour γ in the domain. **Morera's
theorem** is the converse:

> If f is continuous on an open set D, and ∮_γ f dz = 0 for every
> triangular (equivalently, every closed, piecewise-smooth) contour γ in
> D, then f is holomorphic on D.

Morera's theorem is the practical tool for **proving** analyticity
without having to check differentiability directly — in particular for
showing that limits of sequences of analytic functions, and integrals of
analytic kernels with respect to a parameter, remain analytic. It will
be used in Chapter 11 to show that the one-sided Fourier transform of a
causal, suitably bounded time-domain response is analytic in the upper
half-plane.

## Zeros, isolation, and rigidity

Analytic functions are **rigid**. If f is analytic on a connected open
set D and the set of points where f(z) = 0 has a limit point in D, then
f ≡ 0 on all of D. Equivalently, the zeros of a non-zero analytic
function are **isolated**. Contrast with smooth functions on ℝ, where
zeros can accumulate arbitrarily.

A related consequence is the **identity theorem**: if two analytic
functions agree on any set with a limit point in their common domain,
they agree everywhere on that connected domain. This is why "analytic
continuation" is unique — a function defined by a formula on a tiny
patch of ℂ has at most one analytic extension to any larger connected
domain.

## Meromorphic functions

A function f is **meromorphic on D** if it is holomorphic on D except at
an isolated set of points where it has **poles** — singularities at
which 1/f is holomorphic and vanishes (Chapter 7). Rational functions
p(z)/q(z) are meromorphic on ℂ, with poles at the zeros of q. The
Lorentz susceptibility

```
χ(ω) = ω_p² / (ω₀² − ω² − iγω)
```

is meromorphic in ω, with two simple poles at the roots of
`ω² + iγω − ω₀² = 0`. Both roots have negative imaginary part for γ > 0
— i.e., they lie in the lower half-plane — which is exactly the
condition that makes χ analytic in the **upper** half-plane. Causality,
in the Lorentz model, is the statement that the damping term `−iγω`
pushes every pole into the lower half-plane.

## Why analyticity encodes causality

A linear, time-invariant system has impulse response h(t) and transfer
function

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

If h(t) = 0 for t < 0 (causality), then for **complex** ω = ω_R + iω_I
the integrand carries a factor `e^{−ω_I t}` which decays for t > 0
whenever ω_I > 0. The integral therefore converges absolutely for every
ω in the upper half-plane, and term-wise differentiation under the
integral shows χ(ω) is analytic there. Chapter 11 makes this precise
and states the converse (Titchmarsh's theorem).

The entire K–K derivation rests on this bridge: **causality of h(t) ⇔
analyticity of χ(ω) in the upper half-plane**. Once that bridge is in
place, the Cauchy integral formula (Chapter 6) applied on a
large semicircular contour in the UHP, with an indentation around the
pole at ω′ = ω on the real axis, yields the two K–K integrals
mechanically.

## See also

- [04_cauchy_riemann.md](04_cauchy_riemann.md)
- [06_cauchy_theorems.md](06_cauchy_theorems.md)
- [07_residues.md](07_residues.md)