Cauchy–Riemann Equations

Equivalent forms

f(z) = u(x,y) + i\,v(x,y)

\frac{\partial f}{\partial \bar z} = 0

f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}

Two innocent partial-derivative equations are equivalent to the entire edifice of complex differentiability — and they force u and v to each be harmonic (∇²u = ∇²v = 0).

Symbol	Name	SI unit	Dimension	Range
$u_x$	∂u/∂xoutput Partial of the real part in x	1	1	-5 – 5
$v_y$	∂v/∂y Partial of the imaginary part in y	1	1	-5 – 5
$x$	x Real coordinate of z = x + iy	1	1	-3 – 3
$y$	y Imaginary coordinate of z = x + iy	1	1	-3 – 3

Unit systems

Symbol	SI	CGS	Imperial
$x$	1	1	1
$y$	1	1	1
$u_x$	1	1	1
$v_y$	1	1	1

Where it holds

Functions

f = u + iv

with u, v differentiable; analyticity needs the partials continuous as well.

Dimensional analysis

\partial u/\partial x \to [u]/[x]

\partial v/\partial y \to [v]/[y]

u, v, x, y are all real numbers of the same kind, so every term is dimensionless and the equalities are consistent.

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Discovery

Augustin-Louis Cauchy / Bernhard Riemann · 1851

D'Alembert and Euler met these relations in 18th-century fluid flow. Cauchy built complex integration on them in the 1820s, and Riemann's 1851 dissertation made them the foundation of geometric function theory — analytic functions as angle-preserving (conformal) maps.

Try this

What does it take for a function of a complex number to have a derivative? Two tiny equations that secretly force the function to preserve every angle it touches.

Feed a grid through an analytic map and watch the squares stay square — proof that the Cauchy–Riemann conditions hold.

Research status: stable

Real-world applications

Conformal mapping for 2D potential flow and aerofoil design
Electrostatics and heat flow in two dimensions
Image and mesh warping that preserves angles
String theory and 2D conformal field theory

Common misconceptions

Differentiable real-and-imaginary parts imply analytic — you also need the CR equations linking them.
Analytic just means smooth — it is far stronger: analytic functions are infinitely differentiable and locally power series.
Conjugation z̄ is analytic — it satisfies none of the CR equations.

Experimental verification

Realized physically in 2D ideal fluid flow and electrostatics, where the complex potential's real and imaginary parts give the potential and stream function; flow-net experiments confirm the orthogonal harmonic structure.

Derivation

Require

f'(z) = lim \Delta f/\Delta z

to be the same whether

\Delta z

is real

(\Delta x)

or imaginary

(i\Delta y)

.

Along

\Delta x

:

f' = u_x + i v_x

.

Along

i\Delta y

:

f' = (u_y + i v_y)/i = v_y - i u_y

.

Equating real and imaginary parts gives

u_x = v_y

and

u_y = -v_x

.

Limiting cases

CR equations hold + continuous partials⟶ f is analyticf has a complex derivative independent of direction.

f = z

̄ (conjugate)⟶ CR fails everywhere

u_x = 1

but

v_y = -1

— conjugation is nowhere complex-differentiable.

CR holds⟶

\nabla ^{2}u = \nabla ^{2}v = 0

Both real and imaginary parts are harmonic conjugates.

What if…

What if only u is given?

If u is harmonic, the CR equations let you reconstruct its harmonic conjugate v up to a constant — and hence the analytic f.

What if I use polar form?

CR becomes $u_r = (1/r)v_\theta and v_r = -(1/r)u_\theta$ — handy for power and log functions.

1

Check f(z) = z²

Given ·

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

Find · Do the CR equations hold?

Steps

$u = x^{2}-y^{2}$ , $v = 2xy$
$u_x = 2x$ , $v_y = 2x$ → equal $\checkmark$
$u_y = -2y$ , $v_x = 2y \to u_y = -v_x \checkmark$

Result ·

u_x = 2x = v_y

and

u_y = -2y = -v_x \checkmark

— analytic everywhere

2

Check f(z) = z̄

Given ·

f:: z̄ $= x - iy$

Find · Is it analytic?

Steps

$u = x$ , $v = -y$
$u_x = 1$ , $v_y = -1 \to not$ equal
CR violated $\to no$ complex derivative

Result ·

u_x = 1

but

v_y = -1 \to CR

fails; not analytic anywhere

Analytic (Holomorphic) Functions→Euler's Formula→Cauchy's Integral Formula→Laplacian→