Cauchy-Riemann Equations

⚡ Key Takeaways

One-line statement: A complex function $f(z) = u(x,y) + iv(x,y)$ is complex-differentiable at a point only if its real and imaginary parts satisfy $u_x = v_y$ and $u_y = -v_x$ there. Key result: These two partial differential equations, together with continuity of the partials, are equivalent to holomorphy. Where it applies: Complex analysis, conformal mapping, potential theory, fluid dynamics, electrostatics. Who/when: d'Alembert (1752), Euler (1777), Cauchy (1814), Riemann (1851).

💡 The Core Insight

For a real-linear map $\mathbb{R}^2 \to \mathbb{R}^2$ to act as multiplication by a single complex number, it must scale all directions equally and rotate them by the same angle. The Cauchy–Riemann equations are exactly the algebraic condition that the derivative — the local linear approximation — has this special form. Complex differentiability $=$ real differentiability $+$ the constraint that the limit of the difference quotient is independent of the direction of approach.

DefinitionComplex Differentiability

Let $\Omega \subseteq \mathbb{C}$ be open and $f : \Omega \to \mathbb{C}$ . We say $f$ is complex-differentiable at $z_0 \in \Omega$ if the limit

1

\quad f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}

exists, where $h \in \mathbb{C}$ approaches $0$ from any direction in the plane. If $f$ is complex-differentiable at every point of $\Omega$ , it is called holomorphic (or analytic) on $\Omega$ . Derivation hint: This is the literal transcription of the real derivative definition, but $h$ now ranges over $\mathbb{C}$ , so the same limiting value must emerge no matter how $h \to 0$ .

DefinitionThe Cauchy–Riemann Equations

Write $f(z) = u(x,y) + i\,v(x,y)$ where $z = x + iy$ , and $u, v : \Omega \to \mathbb{R}$ are the real and imaginary parts. The Cauchy–Riemann equations are

2

\quad \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.

Derivation hint: Obtained by computing $f'(z_0)$ in [1] twice — once with $h$ real, once with $h$ purely imaginary — and equating the real and imaginary parts of the two results. Here $u_x \equiv \partial u/\partial x$ , etc. Every symbol: $u$ = real part, $v$ = imaginary part, $x = \operatorname{Re} z$ , $y = \operatorname{Im} z$ , and the subscripts denote partial derivatives.

TheoremCauchy–Riemann Criterion for Holomorphy

Let $f = u + iv$ be defined on an open set $\Omega$ .

(Necessity) If $f$ is complex-differentiable at $z_0$ , then the partial derivatives $u_x, u_y, v_x, v_y$ exist at $z_0$ and satisfy [2] there. Moreover

3

\quad f'(z_0) = u_x + i\,v_x = v_y - i\,u_y.

Derivation hint: Substitute the real and imaginary limit evaluations of $f'(z_0)$ into the definition; the first equality uses $h$ real, the second uses $h = ik$ imaginary.

(Sufficiency) If $u$ and $v$ have continuous first partial derivatives in a neighbourhood of $z_0$ and satisfy [2] at $z_0$ , then $f$ is complex-differentiable at $z_0$ .

TheoremPolar and Wirtinger Forms

In polar coordinates $z = re^{i\theta}$ , with $f = u(r,\theta) + iv(r,\theta)$ , the equations become

4

\quad \frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}, \qquad \frac{1}{r}\frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial r}.

Derivation hint: Apply the chain rule with $x = r\cos\theta$ , $y = r\sin\theta$ to convert the Cartesian forms in [2]. Using the Wirtinger derivatives $\partial_z = \tfrac12(\partial_x - i\partial_y)$ and $\partial_{\bar z} = \tfrac12(\partial_x + i\partial_y)$ , all of [2] collapses to the single elegant equation

5

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

Derivation hint: Expand $\partial_{\bar z} f = \tfrac12[(u_x - v_y) + i(v_x + u_y)]$ ; setting both bracketed real expressions to zero reproduces [2].

ProofNecessity of the Equations

Assume $f'(z_0)$ exists. We exploit the directional independence of the limit in [1].

Step 1 — Approach along the real axis. Let $h = t$ with $t \in \mathbb{R}$ , $t \to 0$ . Then

6

\quad f'(z_0) = \lim_{t\to 0}\frac{u(x_0+t,y_0)-u(x_0,y_0)}{t} + i\,\frac{v(x_0+t,y_0)-v(x_0,y_0)}{t} = u_x + i\,v_x.

Derivation hint: With $h$ real, the difference quotient splits into real-axis partial derivatives of $u$ and $v$ by definition.

Step 2 — Approach along the imaginary axis. Let $h = is$ with $s \in \mathbb{R}$ , $s\to 0$ . Since $1/(is) = -i/s$ ,

7

\quad f'(z_0) = \lim_{s\to 0}\frac{-i}{s}\big[u(x_0,y_0+s)-u(x_0,y_0) + i(v(x_0,y_0+s)-v(x_0,y_0))\big] = v_y - i\,u_y.

Derivation hint: Factor $1/(is) = -i/s$ and distribute over the $u$ and $v$ increments; the $-i \cdot i = 1$ converts the $v$ term.

Step 3 — Equate. The two computed values of $f'(z_0)$ must agree:

8

\quad u_x + i\,v_x = v_y - i\,u_y.

Derivation hint: Both expressions equal the same complex number $f'(z_0)$ , so set them equal.

Step 4 — Match real and imaginary parts. Real parts give $u_x = v_y$ ; imaginary parts give $v_x = -u_y$ , i.e. $u_y = -v_x$ . These are precisely [2]. $\blacksquare$

∎

ProofSufficiency (with continuous partials)

Assume $u,v$ have continuous partials near $z_0$ and satisfy [2] at $z_0$ . Let $h = \Delta x + i\Delta y$ .

Step 1 — Real differentiability. Continuity of the partials implies $u$ and $v$ are (totally) differentiable as functions $\mathbb{R}^2 \to \mathbb{R}$ :

9

\quad \Delta u = u_x \Delta x + u_y \Delta y + o(|h|), \qquad \Delta v = v_x \Delta x + v_y \Delta y + o(|h|).

Derivation hint: Continuous first partials is the standard sufficient condition for total differentiability in multivariable calculus.

Step 2 — Assemble the increment. Then $\Delta f = \Delta u + i\Delta v$ . Substitute the CR relations $v_y = u_x$ and $v_x = -u_y$ :

10

\quad \Delta f = (u_x + i v_x)\Delta x + (u_y + i v_y)\Delta y + o(|h|) = (u_x + i v_x)(\Delta x + i\Delta y) + o(|h|).

Derivation hint: Verify $(u_y + i v_y) = i(u_x + i v_x)$ using $u_y = -v_x$ and $v_y = u_x$ ; this lets you factor out $h = \Delta x + i\Delta y$ .

Step 3 — Form the quotient. Dividing by $h$ ,

11

\quad \frac{\Delta f}{h} = (u_x + i v_x) + \frac{o(|h|)}{h} \xrightarrow{h\to 0} u_x + i v_x.

Derivation hint: The error term $o(|h|)/h \to 0$ since $|o(|h|)/h| = o(|h|)/|h| \to 0$ . The limit is direction-independent. Hence $f'(z_0)$ exists and equals $u_x + iv_x$ . $\blacksquare$

∎

⚠️ Continuity of partials is essential for sufficiency

The mere existence of the partial derivatives satisfying [2] at a single point does not guarantee complex differentiability there. Without continuity (or total real-differentiability), the increment may fail to be linearizable, and the directional limit can depend on the path. See the counterexample below.

mathematician · Bernhard Riemann | 1826–1866

Though the equations bear two names, it was Riemann who, in his revolutionary 1851 Göttingen dissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Größe, elevated them from a computational curiosity to the defining principle of function theory. Riemann viewed a holomorphic function geometrically — as a conformal, structure-preserving map — and used the equations to connect complex analysis with potential theory and harmonic functions. Augustin-Louis Cauchy had earlier (1814, 1825) used the equations in the context of contour integration. Riemann's geometric vision, including Riemann surfaces, transformed the field, despite his tragically short life ended by tuberculosis at 39.

🧠 Intuition

Picture the derivative as a local microscope. Zooming in near $z_0$ , any differentiable map $\mathbb{R}^2 \to \mathbb{R}^2$ looks linear — described by the Jacobian

J = \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}.

Multiplication by a complex number $a + bi$ corresponds to the special matrix $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ , which is a uniform scaling by $\sqrt{a^2+b^2}$ combined with a rotation by $\arg(a+bi)$ . Forcing $J$ into this shape demands $u_x = v_y$ ( $= a$ ) and $v_x = -u_y$ ( $= b$ ) — exactly the Cauchy–Riemann equations. Geometrically: holomorphic maps preserve angles (they are conformal) wherever $f'(z) \neq 0$ . They cannot stretch one direction more than another; the local picture is a rotation and a dilation, never a shear.

Figure

The Jacobian of a holomorphic map: rotation-scaling, never shear

ExampleVerifying holomorphy of

f(z) = z^2

Given: $f(z) = z^2$ . Find $u, v$ and verify the Cauchy–Riemann equations; compute $f'(z)$ .

Step 1. Expand: $z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy$ . So

u(x,y) = x^2 - y^2, \qquad v(x,y) = 2xy.

Step 2. Compute the four partials:

u_x = 2x, \quad u_y = -2y, \quad v_x = 2y, \quad v_y = 2x.

Step 3. Check [2]:

u_x = 2x = v_y \ \checkmark \qquad u_y = -2y = -v_x \ \checkmark

The equations hold for all $(x,y)$ , and the partials are continuous everywhere, so $f$ is entire (holomorphic on all of $\mathbb{C}$ ).

Step 4. By [3], $f'(z) = u_x + iv_x = 2x + i\,2y = 2(x+iy) = 2z$ .

\boxed{f'(z) = 2z \quad \text{(dimensionless; holomorphic on all of } \mathbb{C})}

ExampleWhere is

f(z) = |z|^2

differentiable?

Given: $f(z) = |z|^2 = x^2 + y^2$ , so $u = x^2 + y^2$ , $v = 0$ .

Step 1. Partials: $u_x = 2x$ , $u_y = 2y$ , $v_x = 0$ , $v_y = 0$ .

Step 2. Apply [2]:

u_x = v_y \implies 2x = 0 \implies x = 0,

u_y = -v_x \implies 2y = 0 \implies y = 0.

Step 3. Both hold only at the single point $z = 0$ . The partials are continuous, so $f$ is complex-differentiable only at the origin, with

f'(0) = u_x + iv_x \big|_{(0,0)} = 0.

Step 4. Because differentiability holds at an isolated point (not on any open set), $f$ is nowhere holomorphic.

\boxed{f \text{ is differentiable only at } z=0, \ f'(0)=0; \text{ holomorphic nowhere.}}

⚠️ Satisfying CR ≠ holomorphic

A frequent error is to conclude that any function satisfying [2] at a point is differentiable there. The correct chain is: complex differentiability $\Rightarrow$ CR equations (always), but CR equations $\Rightarrow$ complex differentiability only when the partials are continuous (or $u,v$ are totally real-differentiable). Pointwise satisfaction of CR without regularity is insufficient.

CounterexampleCR satisfied at a point, yet not differentiable

Consider

12

\quad f(z) = \begin{cases} \dfrac{\bar z^2}{z}, & z \neq 0,\\[4pt] 0, & z = 0.\end{cases}

Derivation hint: Write $z = re^{i\theta}$ so $\bar z^2 / z = r e^{-3i\theta}$ ; examine the difference quotient $f(h)/h = \bar h^2/h^2 = e^{-4i\theta}$ along rays $\theta = \text{const}$ . At $z=0$ , one verifies $u_x = u_y = v_x = v_y = 0$ along the axes, so the Cauchy–Riemann equations are satisfied at the origin. Yet the difference quotient $f(h)/h = \overline{h}^2/h^2 = e^{-4i\theta}$ depends on the angle $\theta$ of approach — taking value $1$ along the real axis but $-1$ along the line $\theta = \pi/4$ . The limit does not exist, so $f$ is not complex-differentiable at $0$ . The failure: $u,v$ are not totally differentiable at the origin.

CounterexampleConjugation breaks CR everywhere

The function $f(z) = \bar z = x - iy$ has $u = x$ , $v = -y$ , giving $u_x = 1$ but $v_y = -1$ . Since $u_x \neq v_y$ everywhere, $\bar z$ is nowhere complex-differentiable — despite being perfectly smooth as a map $\mathbb{R}^2 \to \mathbb{R}^2$ . In Wirtinger language, $\partial_{\bar z}\bar z = 1 \neq 0$ , violating [5]. This is the canonical example separating real-smooth from holomorphic.

connection · Harmonic functions and the Laplace equation

If $f = u + iv$ is holomorphic with $C^2$ components, differentiating [2] and using equality of mixed partials yields

13

\quad u_{xx} + u_{yy} = 0, \qquad v_{xx} + v_{yy} = 0.

Derivation hint: Differentiate $u_x = v_y$ w.r.t. $x$ and $u_y = -v_x$ w.r.t. $y$ , then add; mixed partials $v_{yx} = v_{xy}$ cancel. Thus both $u$ and $v$ are harmonic, and $v$ is the harmonic conjugate of $u$ . This is the bridge to Harmonic Functions and Potential Theory, making complex analysis the natural language for 2D electrostatics and ideal fluid flow.

connection · Conformal mappings and Cauchy's theorem

The CR equations underpin the fact that holomorphic functions with $f'\neq 0$ are Conformal Mappings (angle-preserving). They are also the local hypothesis from which Cauchy's Integral Theorem — that closed contour integrals of holomorphic functions vanish — is derived, via Green's theorem applied to $\oint (u+iv)(dx + i\,dy)$ . From there flow Cauchy's Integral Formula, Power Series Representation, and the Residue Theorem.

🔬 Where this appears

⏳ History

- 1752 — Jean le Rond d'Alembert writes down the equations in a treatise on fluid resistance, relating the components of a 2D incompressible, irrotational flow field.

- 1777 — Leonhard Euler uses the equations in connection with definite integrals and the representation of functions of a complex variable.

- 1814 — Augustin-Louis Cauchy employs them in his memoir on definite integrals, foundational to contour integration.

- 1825 — Cauchy's celebrated memoir on integrals between complex limits firmly establishes integration theory built on these relations.

- 1851 — Bernhard Riemann, in his doctoral dissertation, makes the equations the definition of a function of a complex variable, introducing the geometric/conformal viewpoint and Riemann surfaces.

- 1857 — Riemann's work on Abelian functions deepens the connection between holomorphy, harmonic functions, and topology.

- 1900s — Édouard Goursat proves that mere existence of

f'

(without assuming continuity of

f'

) suffices for the integral theorem, refining the regularity requirements.

- 1912–1927 — Wilhelm Wirtinger develops the

\partial_z, \partial_{\bar z}

calculus, giving the equations the compact form

\partial_{\bar z} f = 0

.

- 20th c. — Looman–Menchoff theorem (1923/1936): if

f

is continuous and the partials exist and satisfy CR everywhere on a domain (without assuming continuity of partials), then

f

is holomorphic — a deep sharpening of sufficiency.

open-problem · Regularity, the Looman–Menchoff theorem, and beyond

The precise minimal hypotheses under which the Cauchy–Riemann equations imply holomorphy form a subtle chapter of analysis. The naïve sufficiency theorem requires continuous partials, but the Looman–Menchoff theorem weakens this dramatically: a continuous function whose four partials merely exist and satisfy CR at every point of a domain is automatically holomorphic — yet continuity of $f$ cannot be dropped (the counterexample [12] shows pointwise CR alone is worthless). Pinning down exactly which regularity classes suffice (e.g. Sobolev versions, distributional CR equations as in the $\bar\partial$ -problem) remains a rich area. The inhomogeneous equation $\partial_{\bar z} f = g$ — the $\bar\partial$ (d-bar) problem — generalizes CR to several complex variables, where existence of solutions becomes a deep question intertwined with the geometry of domains (pseudoconvexity, Hörmander's $L^2$ estimates), driving modern multidimensional complex analysis.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also