Analytic (Holomorphic) Functions

Equivalent forms

f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n

Complex-differentiable once ⇒ smooth, power-series exact, and harmonic — the rigidity that makes complex analysis so powerful for physics.

Symbol	Name	SI unit	Dimension	Range
$u$	Real partoutput Real part of f = u + iv	1	1	—
$v$	Imaginary part Imaginary part of f = u + iv	1	1	—
$x$	Real coordinate Real part of z	1	1	-3 – 3
$y$	Imaginary coordinate Imaginary part of z	1	1	-3 – 3

Where it holds

Defined on open subsets of ℂ. A function analytic on a domain is determined everywhere by its values on any small disk (identity theorem) — enabling analytic continuation.

Dimensional analysis

u_x \to (\partial of u)/(\partial x)

v_y \to (\partial of v)/(\partial y)

u and v share f's dimension and x,y share z's; both sides scale as f/z.

\checkmark

Discovery

Cauchy, Riemann, Weierstrass · 1851

Cauchy laid the integral foundations; Riemann (1851 thesis) gave the geometric, conformal-mapping viewpoint via the Cauchy–Riemann equations; Weierstrass built the theory from power series. Their three approaches — integrals, geometry, and series — turned out to describe exactly the same class of functions.

Research status: stable

Real-world applications

Conformal mapping to solve 2D electrostatics and aerodynamics
Analytic continuation (e.g. the Riemann zeta function, dispersion relations)
Fluid flow and heat-conduction potentials

Common misconceptions

Complex-differentiable is far stronger than real-differentiable in x and y separately
| $z|^{2} and$ conjugate z̄ are NOT analytic (they fail Cauchy–Riemann)
'Analytic' and 'holomorphic' are synonyms for functions on ℂ

Experimental verification

A mathematical concept; physically, 2D electrostatic potentials and ideal fluid flows are real/imaginary parts of analytic functions, and conformal maps solve real boundary-value problems.

Derivation

Require the limit

f'(z)=lim (f(z+h)-f(z))/h

to be the same for

h\to 0

along any direction.

Taking h real vs h imaginary and equating gives

\partial f/\partial x = (1/i)\partial f/\partial y

.

Writing

f=u+iv

and separating real/imaginary parts yields the Cauchy–Riemann equations

u_x=v_y

,

u_y=-v_x

.

Cross-differentiating shows

u_xx+u_yy=0

and likewise for v — both are harmonic.

Cauchy's integral formula then proves f is infinitely differentiable and analytic (equals its Taylor series).

What if…

What if we

try f(z)=z

̄ (complex conjugate)?

$u=x$ , $v=-y$ gives $u_x=1$ but $v_y=-1$ — Cauchy–Riemann fails, so z̄ is nowhere analytic despite being smooth in x,y.

What if you know f on a tiny disk?

The identity theorem fixes f on the whole connected domain — this rigidity is what analytic continuation exploits.

1

Test analyticity of f(z)=z²

Given ·

z^{2}=(x^{2}-y^{2})+i(2xy)

Find · does it satisfy Cauchy–Riemann?

Steps

$u=x^{2}-y^{2}$ , $v=2xy$
$u_x=2x$ , $v_y=2x \checkmark$ ; $u_y=-2y$ , $-v_x=-2y \checkmark$
CR satisfied $\to z^{2} is$ entire

Result · Yes:

u_x=2x=v_y

and

u_y=-2y=-v_x

— analytic everywhere

Cauchy–Riemann Equations→Cauchy's Integral Formula→laplace equation