The Cauchy-Riemann Equations

Suppose a function of a complex variable looks, near a point, indistinguishable from multiplication by a fixed complex number. When does such a function actually exist? How can we detect it using only ordinary real calculus? This question sits at the heart of complex analysis. Complex differentiability is a dramatically stronger condition than its real counterpart. Yet it can be tested by two unassuming partial differential equations. These equations relate the real and imaginary parts of a function. The Cauchy-Riemann equations translate a geometric demand into a concrete algebraic constraint. The geometric demand is that complex differentiability requires a function's derivative be the same regardless of the direction of approach. The constraint applies to partial derivatives. This translation is what makes complex analysis computationally tractable. Instead of checking a limit along every possible direction in the plane, we check two real equations. The equations also expose the deep rigidity of complex-differentiable functions. This foreshadows why such functions are automatically smooth. It also explains why they are conformal away from critical points. Finally, it shows why they are determined everywhere by their values on arbitrarily small open sets. The big picture is this: a complex function f=u+ivf = u + iv can be viewed as a map R2→R2\mathbb{R}^2 \to \mathbb{R}^2. It is real-differentiable in the ordinary multivariable sense whenever uu and vv have continuous partial derivatives. Complex differentiability asks for something stronger. This real derivative, a 2×22\times 2 real matrix, must actually represent multiplication by a complex number. The Cauchy-Riemann equations are precisely the condition on that matrix's entries. They are what makes this true. This single idea unifies the algebra of complex numbers with the geometry of the plane. This article gives the precise definition of complex differentiability. It states and proves the theorem characterizing it via the Cauchy-Riemann equations. Both the necessary and the full sufficient direction are covered, with the correct extra hypothesis. The article works through two detailed non-symmetric examples. It exhibits a classical counterexample showing why continuity of the partials cannot be dropped. It closes by placing the equations in their historical and modern context. They serve as the doorway to the theory of Holomorphic Functions.

Key takeaways
  • Complex differentiability at a point means a single complex limit exists; this is far stronger than real differentiability of u,vu,v as a pair of two-variable functions.
  • Writing f=u+ivf = u+iv with z=x+iyz=x+iy, the Cauchy-Riemann equations are ux=vyu_x = v_y and uy=−vxu_y = -v_x.
  • The equations are necessary for complex differentiability at a point, always, with no extra hypothesis.
  • They are sufficient only together with real-differentiability of u,vu,v (guaranteed e.g. by continuous partials) — dropping this hypothesis breaks the theorem.
  • Geometrically, the equations say the real Jacobian matrix of ff is a scaled rotation matrix, i.e. multiplication by a complex number.
  • Key idea: complex differentiability equals real differentiability plus the extra algebraic constraint that the derivative matrix commutes with multiplication by ii.

Where this lives today

The Cauchy–Riemann equations are the gateway to the entire theory of Holomorphic Functions: once verified on an open set, they unlock power series representations, the Cauchy integral formula, and the fact that a complex-differentiable function is automatically infinitely differentiable — a phenomenon with no counterpart in real analysis. In physics, the equations are the mathematical signature of two-dimensional irrotational, incompressible fluid flow and of electrostatic potential problems: the real and imaginary parts of a holomorphic function are harmonic conjugates, giving a direct bridge between complex analysis and Laplace's equation, which underlies heat flow, elasticity, and potential theory. In engineering, conformal maps built from holomorphic functions — precisely the maps satisfying these equations — are used to solve two-dimensional problems in aerodynamics (airfoil design via the Joukowski transform) and electromagnetics by transplanting complicated domains onto simple ones like disks or half-planes. In pure mathematics, the equations reappear in higher dimensions as the model for elliptic partial differential equations, and their failure or generalization (the ∂ˉ\bar\partial-operator) is central to several complex variables and to modern index theory. They also underlie numerical conformal mapping algorithms used in computer graphics and mesh generation, and appear in signal processing through the Hilbert transform, which encodes the same real/imaginary compatibility condition for analytic signals. Understanding these equations is inseparable from understanding Complex Functions and Limits, on which the entire definition of complex differentiability rests.

Ask the tutor: Why does complex differentiability at just one point tell us so little, while complex differentiability on an entire open set tells us so much?

DefinitionComplex differentiability and the Cauchy–Riemann equations

Let Ω⊆C\Omega \subseteq \mathbb{C} be open, let f:Ω→Cf : \Omega \to \mathbb{C}, and let z0∈Ωz_0 \in \Omega. We say ff is complex differentiable at z0z_0 if the limit

f′(z0):=lim⁡h→0f(z0+h)−f(z0)h,h∈C∖{0},f'(z_0) := \lim_{h \to 0} \frac{f(z_0+h) - f(z_0)}{h}, \qquad h \in \mathbb{C}\setminus\{0\},

exists (the limit being taken as h→0h \to 0 through complex values, i.e. from every direction in the plane simultaneously). Write z=x+iyz = x+iy and identify ff with a pair of real-valued functions of two real variables,

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

The Cauchy–Riemann equations at a point (x0,y0)(x_0,y_0) are the pair of partial differential equations

∂u∂x(x0,y0)=∂v∂y(x0,y0),∂u∂y(x0,y0)=−∂v∂x(x0,y0).\frac{\partial u}{\partial x}(x_0,y_0) = \frac{\partial v}{\partial y}(x_0,y_0), \qquad \frac{\partial u}{\partial y}(x_0,y_0) = -\frac{\partial v}{\partial x}(x_0,y_0).

Intuition

Think of ff as a transformation of the plane: it takes the point (x,y)(x,y) to (u(x,y),v(x,y))(u(x,y), v(x,y)). Ordinary (real) differentiability at a point means this transformation is well-approximated, near that point, by a linear map — its Jacobian matrix

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

A generic 2×22\times 2 real matrix represents a linear map that can stretch the plane unevenly in different directions and can also reflect it. Complex differentiability demands something much more special: the linear approximation must be multiplication by a single complex number a+bia+bi, which as a real-linear map is represented by the matrix

(a−bba).\begin{pmatrix} a & -b \\ b & a \end{pmatrix}.

Matrices of this form are exactly the matrices representing rotation-and-scaling — they preserve angles and orientation. Comparing this template matrix entry-by-entry with the Jacobian JJ forces ux=vyu_x = v_y (the two diagonal entries equal) and uy=−vxu_y = -v_x (the off-diagonal entries negatives of each other). These are exactly the Cauchy-Riemann equations. This is why complex differentiable maps are conformal wherever the derivative is nonzero: their derivative, viewed as a real linear map, is never a generic stretch — it is always a spinning, uniform dilation. Another way to see the same content: differentiating "in the direction of hh" should not depend on whether hh is real or purely imaginary. Approaching along the real axis (hh real) gives derivative ux+ivxu_x + i v_x. Approaching along the imaginary axis (h=ikh = ik, kk real) gives derivative 1i(uy+ivy)=vy−iuy\frac{1}{i}(u_y + iv_y) = v_y - i u_y. Setting these equal — the real parts and imaginary parts separately — again produces exactly ux=vyu_x = v_y and uy=−vxu_y = -v_x.

Figure
The Jacobian of a complex-differentiable map is a rotation-scaling matrix
General real-linear map (no C-R): square → parallelogram J = [[2, 1],[0.3, 1]] (u_x≠v_y, angles not preserved) Cauchy–Riemann map (rotation × scale): square → rotated square J = [[a, -b],[b, a]], a=1.4·cos30°, b=1.4·sin30° Left: u_x=2, v_y=1, u_y=1, v_x=0.3 — C-R equations fail. Right: u_x=v_y≈1.21, u_y=-v_x≈-0.70 — C-R equations hold exactly.
TheoremCauchy–Riemann characterization of complex differentiability

Let Ω⊆C\Omega \subseteq \mathbb{C} be open, f=u+iv:Ω→Cf = u+iv : \Omega \to \mathbb{C}, and z0=x0+iy0∈Ωz_0 = x_0+iy_0 \in \Omega.

(Necessity, no extra hypothesis.) If ff is complex differentiable at z0z_0, then the partial derivatives ux,uy,vx,vyu_x,u_y,v_x,v_y exist at (x0,y0)(x_0,y_0), satisfy the Cauchy–Riemann equations there, and

f′(z0)=ux(x0,y0)+i vx(x0,y0).f'(z_0) = u_x(x_0,y_0) + i\,v_x(x_0,y_0).

(Sufficiency, with the differentiability hypothesis.) Conversely, suppose uu and vv are real-differentiable at (x0,y0)(x_0,y_0) (i.e. each admits a first-order Taylor approximation with error o(∣h∣)o(|h|) there) and satisfy the Cauchy–Riemann equations at (x0,y0)(x_0,y_0). Then ff is complex differentiable at z0z_0, with derivative f′(z0)=ux(x0,y0)+ivx(x0,y0)f'(z_0) = u_x(x_0,y_0)+iv_x(x_0,y_0). In particular, if u,vu,v have partial derivatives that are continuous on an open set Ω\Omega and satisfy the Cauchy–Riemann equations throughout Ω\Omega, then ff is complex differentiable at every point of Ω\Omega.

Proof

We use the direct-limit technique for necessity, and the linear-approximation (differential) technique for sufficiency.

Necessity. Assume f′(z0)f'(z_0) exists as a limit over complex h→0h \to 0. Because the limit exists along every path, it exists in particular along the two coordinate directions, and both must agree with f′(z0)f'(z_0). Along the real axis. Let h=t∈Rh = t \in \mathbb{R}, t→0t \to 0. Then

f(z0+t)−f(z0)t=u(x0+t,y0)−u(x0,y0)t+i v(x0+t,y0)−v(x0,y0)t.\frac{f(z_0+t)-f(z_0)}{t} = \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}.

As t→0t \to 0 this converges to ux(x0,y0)+ivx(x0,y0)u_x(x_0,y_0) + i v_x(x_0,y_0) — and the very existence of this limit forces the two real one-variable difference quotients to converge separately (a complex sequence converges iff its real and imaginary parts converge). Hence ux(x0,y0)u_x(x_0,y_0) and vx(x0,y0)v_x(x_0,y_0) exist, and

f′(z0)=ux(x0,y0)+i vx(x0,y0).(1)f'(z_0) = u_x(x_0,y_0) + i\,v_x(x_0,y_0). \tag{1}

Along the imaginary axis. Let h=ith = it, t∈R∖{0}t \in \mathbb{R}\setminus\{0\}, t→0t\to 0. Then

f(z0+it)−f(z0)it=1i[u(x0,y0+t)−u(x0,y0)t+i v(x0,y0+t)−v(x0,y0)t].\frac{f(z_0+it)-f(z_0)}{it} = \frac{1}{i}\left[\frac{u(x_0,y_0+t)-u(x_0,y_0)}{t} + i\,\frac{v(x_0,y_0+t)-v(x_0,y_0)}{t}\right].

Again existence of the overall limit (equal to f′(z0)f'(z_0), since the full complex limit exists and every sub-path must agree with it) forces the two real difference quotients inside the bracket to converge, so uy(x0,y0)u_y(x_0,y_0) and vy(x0,y0)v_y(x_0,y_0) exist. Using 1/i=−i1/i = -i,

f′(z0)=−i(uy(x0,y0)+i vy(x0,y0))=vy(x0,y0)−i uy(x0,y0).(2)f'(z_0) = -i\big(u_y(x_0,y_0) + i\,v_y(x_0,y_0)\big) = v_y(x_0,y_0) - i\,u_y(x_0,y_0). \tag{2}

Comparing the real and imaginary parts of (1) and (2):

ux(x0,y0)=vy(x0,y0),vx(x0,y0)=−uy(x0,y0),u_x(x_0,y_0) = v_y(x_0,y_0), \qquad v_x(x_0,y_0) = -u_y(x_0,y_0),

which are exactly the Cauchy–Riemann equations. This proves necessity, with no hypothesis beyond complex differentiability itself.

Sufficiency. Assume u,vu,v are real-differentiable at (x0,y0)(x_0,y_0): writing h=s+ith = s+it with s,t∈Rs,t \in \mathbb{R} small, real-differentiability of uu means

u(x0+s,y0+t)−u(x0,y0)=uxs+uyt+ε1(s,t),u(x_0+s,y_0+t) - u(x_0,y_0) = u_x s + u_y t + \varepsilon_1(s,t),

where ux,uyu_x,u_y are evaluated at (x0,y0)(x_0,y_0) and ε1(s,t)=o(s2+t2)\varepsilon_1(s,t) = o(\sqrt{s^2+t^2}) as (s,t)→(0,0)(s,t)\to(0,0); similarly

v(x0+s,y0+t)−v(x0,y0)=vxs+vyt+ε2(s,t),ε2=o(s2+t2).v(x_0+s,y_0+t) - v(x_0,y_0) = v_x s + v_y t + \varepsilon_2(s,t), \qquad \varepsilon_2 = o(\sqrt{s^2+t^2}).

Add the second equation times ii to the first:

f(z0+h)−f(z0)=(ux+ivx)s+(uy+ivy)t+ε(s,t),ε=ε1+iε2=o(∣h∣),f(z_0+h) - f(z_0) = (u_x + iv_x)s + (u_y+iv_y)t + \varepsilon(s,t), \qquad \varepsilon = \varepsilon_1 + i\varepsilon_2 = o(|h|),

since ∣h∣=s2+t2|h| = \sqrt{s^2+t^2}. Now use the Cauchy–Riemann equations uy=−vxu_y = -v_x, vy=uxv_y = u_x to rewrite the second coefficient:

uy+ivy=−vx+iux=i(ux+ivx).u_y + iv_y = -v_x + i u_x = i(u_x + iv_x).

Substituting,

f(z0+h)−f(z0)=(ux+ivx)s+i(ux+ivx)t+ε(s,t)=(ux+ivx)(s+it)+ε(s,t)=(ux+ivx) h+ε(h).f(z_0+h)-f(z_0) = (u_x+iv_x)s + i(u_x+iv_x)t + \varepsilon(s,t) = (u_x+iv_x)(s+it) + \varepsilon(s,t) = (u_x+iv_x)\,h + \varepsilon(h).

Therefore

f(z0+h)−f(z0)h−(ux+ivx)=ε(h)h,\frac{f(z_0+h)-f(z_0)}{h} - (u_x+iv_x) = \frac{\varepsilon(h)}{h},

and since ∣ε(h)/h∣=∣ε(h)∣/∣h∣→0|\varepsilon(h)/h| = |\varepsilon(h)|/|h| \to 0 as h→0h \to 0 (this is exactly what ε(h)=o(∣h∣)\varepsilon(h)=o(|h|) means, and it holds regardless of the direction from which h→0h\to 0, because the estimate ε(s,t)=o(s2+t2)\varepsilon(s,t) = o(\sqrt{s^2+t^2}) was direction-independent), we conclude

lim⁡h→0f(z0+h)−f(z0)h=ux(x0,y0)+ivx(x0,y0).\lim_{h\to 0} \frac{f(z_0+h)-f(z_0)}{h} = u_x(x_0,y_0) + i v_x(x_0,y_0).

This limit exists and is finite, so ff is complex differentiable at z0z_0 with the stated derivative.

Final clause. If u,vu,v have continuous partial derivatives on all of Ω\Omega, then by a standard fact from multivariable calculus (the mean value theorem applied to each variable separately shows a function with continuous partials is real-differentiable at every point — this is a prerequisite fact from real analysis, not proved here but not circular, since it makes no reference to complex differentiability), u,vu,v are real-differentiable at every point of Ω\Omega. Combined with the Cauchy–Riemann equations holding throughout Ω\Omega, the sufficiency argument just given applies at every point, so ff is complex differentiable throughout Ω\Omega. ■\blacksquare

∎

Worked examples

Example 1: f(z)=z2+zˉf(z) = z^2 + \bar z, tested at a generic non-symmetric point. Write z=x+iyz = x+iy. Then

f(z)=(x+iy)2+(x−iy)=(x2−y2+x)+i(2xy−y).f(z) = (x+iy)^2 + (x-iy) = (x^2-y^2+x) + i(2xy - y).

So u(x,y)=x2−y2+xu(x,y) = x^2-y^2+x and v(x,y)=2xy−yv(x,y) = 2xy-y. Compute all four partials:

ux=2x+1,uy=−2y,vx=2y,vy=2x−1.u_x = 2x+1, \quad u_y = -2y, \quad v_x = 2y, \quad v_y = 2x-1.

The Cauchy–Riemann equations demand ux=vyu_x = v_y and uy=−vxu_y = -v_x:

2x+1=2x−1(never true — a contradiction independent of x,y),2x+1 = 2x-1 \quad\text{(never true — a contradiction independent of }x,y\text{)},
−2y=−2y(always true).-2y = -2y \quad\text{(always true)}.

The first equation fails for every (x,y)(x,y), since 2x+1=2x−12x+1=2x-1 would force 1=−11=-1. Hence f(z)=z2+zˉf(z) = z^2+\bar z satisfies the Cauchy–Riemann equations nowhere in C\mathbb{C}, and by the necessity direction of the theorem, ff is complex differentiable at no point. This matches the intuition that zˉ\bar z (complex conjugation) is orientation-reversing (a reflection), while any complex-differentiable map's derivative must be a rotation-scaling — reflections are never of that form unless the scale is zero, and adding an analytic piece z2z^2 cannot cancel that structural obstruction, as the computation confirms directly.

Example 2: f(z)=ex(cos⁡y+isin⁡y)f(z) = e^{x}(\cos y + i\sin y) (i.e. f(z)=ezf(z)=e^z), checked carefully at the asymmetric point z0=1+2iz_0 = 1 + 2i. Here u(x,y)=excos⁡yu(x,y) = e^x\cos y, v(x,y)=exsin⁡yv(x,y) = e^x \sin y, both built from elementary functions and hence smooth (all partials exist and are continuous everywhere). Compute:

ux=excos⁡y,uy=−exsin⁡y,vx=exsin⁡y,vy=excos⁡y.u_x = e^x\cos y, \qquad u_y = -e^x\sin y, \qquad v_x = e^x\sin y, \qquad v_y = e^x\cos y.

Check Cauchy–Riemann at the point (x0,y0)=(1,2)(x_0,y_0) = (1,2) (so z0=1+2iz_0 = 1+2i, chosen non-symmetric — neither on an axis nor at a "nice" angle):

ux(1,2)=ecos⁡2,vy(1,2)=ecos⁡2⇒ux=vy ✓,u_x(1,2) = e\cos 2, \qquad v_y(1,2) = e\cos 2 \quad \Rightarrow \quad u_x = v_y \ \checkmark,
uy(1,2)=−esin⁡2,−vx(1,2)=−esin⁡2⇒uy=−vx ✓.u_y(1,2) = -e\sin 2, \qquad -v_x(1,2) = -e\sin 2 \quad \Rightarrow \quad u_y = -v_x \ \checkmark.

Both equations hold, and since u,vu,v have continuous partial derivatives everywhere (their formulas involve only ex,cos⁡y,sin⁡ye^x,\cos y,\sin y, all smooth), the sufficiency clause applies at every point of C\mathbb{C}, not just z0z_0. Hence f(z)=ezf(z)=e^z is complex differentiable throughout C\mathbb{C}, with derivative

f′(1+2i)=ux(1,2)+ivx(1,2)=ecos⁡2+i esin⁡2=e1+2i=ez0,f'(1+2i) = u_x(1,2) + iv_x(1,2) = e\cos 2 + i\,e\sin 2 = e^{1+2i} = e^{z_0},

confirming the familiar formula ddzez=ez\frac{d}{dz}e^z = e^z at this specific, deliberately asymmetric point. Numerically, e≈2.71828e \approx 2.71828, cos⁡2≈−0.41615\cos 2 \approx -0.41615, sin⁡2≈0.90930\sin 2 \approx 0.90930, so f′(1+2i)≈−1.13120+2.47173if'(1+2i) \approx -1.13120 + 2.47173i.

CounterexampleWhy real-differentiability of u,vu,v cannot be dropped from the sufficiency direction

The sufficiency direction requires u,vu,v to be real-differentiable at the point (equivalently, to admit a genuine linear approximation with o(∣h∣)o(|h|) error) — merely having partial derivatives that satisfy the Cauchy–Riemann equations is not enough. Here is a function where the Cauchy–Riemann equations hold at the origin, both partials exist there, yet ff fails to be complex differentiable at the origin. Define

f(z)={zˉ2zz≠00z=0.f(z) = \begin{cases} \dfrac{\bar z^2}{z} & z \ne 0 \\ 0 & z = 0. \end{cases}

Write z=x+iy=reiθz = x+iy = re^{i\theta}. Then zˉ2/z=z‾⋅z‾/z=z‾⋅z/z‾\bar z^2/z = \overline{z}\cdot \overline{z}/z = \overline{z} \cdot \overline{z/z}... more directly: zˉ2/z=r2e−2iθ⋅1reiθ⋅r\bar z^2 / z = r^2 e^{-2i\theta}\cdot \frac{1}{r e^{i\theta}}\cdot r. Let us just compute cleanly: for z≠0z\ne 0,

zˉ2z=zˉ2z⋅zˉzˉ=zˉ3∣z∣2.\frac{\bar z^2}{z} = \frac{\bar z^2}{z}\cdot\frac{\bar z}{\bar z}= \frac{\bar z^3}{|z|^2}.

Alternatively and more simply, using polar form z=reiθz = re^{i\theta}: zˉ=re−iθ\bar z = re^{-i\theta}, so

zˉ2z=r2e−2iθreiθ=r e−3iθ.\frac{\bar z^2}{z} = \frac{r^2 e^{-2i\theta}}{re^{i\theta}} = r\,e^{-3i\theta}.

So f(reiθ)=re−3iθ=rcos⁡3θ−i rsin⁡3θf(re^{i\theta}) = r e^{-3i\theta} = r\cos 3\theta - i\, r\sin 3\theta for z≠0z\ne 0. Check partial derivatives exist at the origin and satisfy Cauchy–Riemann there. Along the real axis (y=0y=0, so θ=0\theta = 0 for x>0x>0): f(x)=xf(x) = x for x>0x>0, giving u(x,0)=x,v(x,0)=0u(x,0)=x, v(x,0)=0; along θ=π\theta=\pi for x<0x<0: r=−xr=-x, f=−x⋅e−3iπ=−x⋅(−1)=xf = -x\cdot e^{-3i\pi} = -x\cdot(-1) = x real, consistent with u(x,0)=xu(x,0)=x for all real x≠0x\ne0, and u(0,0)=0u(0,0)=0, so

ux(0,0)=lim⁡x→0u(x,0)−0x=lim⁡x→0xx=1,vx(0,0)=lim⁡x→0v(x,0)−0x=0.u_x(0,0) = \lim_{x\to 0}\frac{u(x,0)-0}{x} = \lim_{x\to0}\frac{x}{x} = 1, \qquad v_x(0,0) = \lim_{x\to0}\frac{v(x,0)-0}{x}=0.

Along the imaginary axis (x=0x=0, θ=π/2\theta = \pi/2 for y>0y>0, r=yr=y): f(iy)=y e−3iπ/2=y⋅i=iyf(iy) = y\, e^{-3i\pi/2} = y\cdot i = iy (since e−3iπ/2=cos⁡(−3π/2)+isin⁡(−3π/2)=0+ie^{-3i\pi/2}=\cos(-3\pi/2)+i\sin(-3\pi/2) = 0 + i), so u(0,y)=0u(0,y)=0, v(0,y)=yv(0,y)=y, giving

uy(0,0)=0,vy(0,0)=1.u_y(0,0) = 0, \qquad v_y(0,0) = 1.

So at the origin: ux=1=vyu_x=1=v_y and uy=0=−vxu_y=0=-v_x — the Cauchy–Riemann equations hold exactly at the origin. Yet ff is not complex differentiable at 00. Approach the origin along the ray θ\theta fixed, r→0+r\to 0^+:

f(z)−f(0)z−0=re−3iθreiθ=e−4iθ,\frac{f(z)-f(0)}{z-0} = \frac{re^{-3i\theta}}{re^{i\theta}} = e^{-4i\theta},

which is a different complex number for each direction θ\theta (e.g. θ=0\theta=0 gives 11, θ=π/4\theta = \pi/4 gives e−iπ=−1e^{-i\pi}=-1). Since the difference quotient does not converge to a single value as z→0z\to 0 from all directions, f′(0)f'(0) does not exist. The resolution: u,vu,v fail to be real-differentiable at the origin. One can check directly (using the polar formulas u=rcos⁡3θu=r\cos3\theta, v=−rsin⁡3θv=-r\sin3\theta) that no single linear map approximates (u,v)(u,v) near the origin with o(r)o(r) error — the "error" after subtracting the candidate linear part is itself of order rr, not o(r)o(r), because the angular dependence cos⁡3θ,sin⁡3θ\cos3\theta,\sin3\theta does not average away. This is exactly the missing hypothesis: partials existing and satisfying Cauchy–Riemann at an isolated point says nothing about behavior in a full neighborhood, and without the differentiability (or continuity-of-partials) hypothesis, the sufficiency direction of the theorem is false.

Scope, edge cases & common traps

SituationWhat holdsWhat can fail
ff complex differentiable at z0z_0Cauchy–Riemann equations hold at z0z_0 automatically— (necessity is unconditional)
C-R equations hold at z0z_0, partials merely exist (not shown differentiable)Nothing guaranteedff may fail to be complex differentiable at z0z_0 (see counterexample)
C-R equations hold at z0z_0, u,vu,v real-differentiable at z0z_0ff complex differentiable at z0z_0, f′=ux+ivxf'=u_x+iv_xOnly pointwise; says nothing about a neighborhood
C-R equations hold on an open set, partials continuous thereff complex differentiable ("holomorphic") throughout the set—
C-R equations hold only on a lower-dimensional set (e.g. a line)Isolated points/curves of complex differentiability, not holomorphyConfusing "differentiable at a point" with "holomorphic on an open set"

The mathematician behind it

Origins · 1 minAugustin-Louis Cauchy and how this was discovered
AC
Augustin-Louis Cauchy
1789–1857 · Founder of rigorous complex analysis, École Polytechnique and Sorbonne, Paris

Cauchy was one of the most prolific mathematicians in history, producing foundational rigor for calculus (the modern ϵ\epsilon-δ\delta definitions trace to him), the theory of determinants, and above all complex analysis, where he introduced the integral theorem bearing his name and systematically studied complex differentiability using the equations that today carry his name jointly with Riemann, who decades later gave them their modern geometric interpretation in his 1851 dissertation on the foundations of function theory.

History
- 1746 — Jean le Rond d'Alembert uses equations equivalent to the Cauchy–Riemann conditions while studying fluid flow, without a general theory of complex differentiability.
- 1821 — Cauchy publishes his Cours d'Analyse, laying rigorous groundwork for limits and continuity that complex analysis would later depend on.
- 1825 — Cauchy's memoir on integrals between imaginary limits systematically develops the theory of complex-differentiable functions and the equations governing them.
- 1851 — Bernhard Riemann's doctoral dissertation "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse" gives the modern geometric treatment, emphasizing conformality and the equations' role as the defining condition.
- 1900s — the equations become a standard entry point in university courses, cementing "Cauchy–Riemann" as the name for the pair of conditions.
Do the Cauchy–Riemann equations alone prove a function is holomorphic on an open set?

Not by themselves. Holomorphy on an open set Ω\Omega means complex differentiability at every point of Ω\Omega. The theorem's sufficiency direction requires, at each point, not just the two equations but also real-differentiability of u,vu,v there. In practice this extra condition is almost always supplied by continuity of the four partial derivatives throughout Ω\Omega, which is easy to check for elementary functions built from polynomials, exponentials, and trigonometric functions. So the correct statement is: Cauchy–Riemann equations plus continuous partials on Ω\Omega imply holomorphy on Ω\Omega. The equations alone, checked pointwise without any regularity assumption, do not suffice, as the counterexample zˉ2/z\bar z^2/z demonstrates.

Why does complex differentiability force such a strong geometric condition, unlike real differentiability in two variables?

In real two-variable calculus, differentiability just means a function is well-approximated by some linear map — any of the infinitely many 2×22\times2 matrices is allowed, including stretches, shears, and reflections. Complex differentiability additionally requires the difference quotient to converge to a single number h↦f′(z0)hh\mapsto f'(z_0)h regardless of the direction hh takes to zero. Multiplication by a fixed complex number is a very restrictive kind of real-linear map: it can only rotate and uniformly scale, never shear or stretch unevenly. The Cauchy–Riemann equations are exactly the algebraic fingerprint that a general Jacobian matrix has this restricted rotation-scaling form.

Can the Cauchy–Riemann equations hold along a curve without ff being complex differentiable anywhere near it?

Yes, and this is a genuinely useful cautionary phenomenon. A function like f(z)=∣z∣2=x2+y2f(z)=|z|^2 = x^2+y^2 has u=x2+y2u=x^2+y^2, v=0v=0, giving ux=2xu_x=2x, vy=0v_y=0, so the first equation ux=vyu_x=v_y only holds at x=0x=0; combined with uy=2y=−vx=0u_y=2y=-v_x=0 only at y=0y=0, so the equations hold only at the single point z=0z=0, not along a curve. But other constructions can make the equations hold along a full curve or even a dense set while the function remains complex differentiable only there, never on any open set — so it is never holomorphic anywhere nearby. This shows the equations holding on a "thin" set carries little of the powerful structure that holomorphy on an open set provides.

What is the connection between the Cauchy–Riemann equations and harmonic functions?

If f=u+ivf=u+iv is holomorphic on an open set and u,vu,v have continuous second partial derivatives (which turns out to always be automatic for holomorphic functions, though that fact requires more machinery than the equations alone), differentiating the Cauchy–Riemann equations ux=vyu_x=v_y and uy=−vxu_y=-v_x again and combining gives uxx+uyy=0u_{xx}+u_{yy}=0 and vxx+vyy=0v_{xx}+v_{yy}=0. So both uu and vv satisfy Laplace's equation; they are called harmonic conjugates. This is the bridge connecting complex analysis to potential theory, electrostatics, and steady-state heat flow, all governed by Laplace's equation in two dimensions.

How do the Cauchy–Riemann equations look in polar coordinates?

Writing z=reiθz=re^{i\theta} and f=u+ivf=u+iv as functions of (r,θ)(r,\theta), the equations transform (by the chain rule from x=rcos⁡θ,y=rsin⁡θx=r\cos\theta,y=r\sin\theta) into

ur=1rvθ,vr=−1ruθ(r≠0).u_r = \frac{1}{r}v_\theta, \qquad v_r = -\frac{1}{r}u_\theta \quad (r\ne 0).

These are algebraically equivalent to the Cartesian form wherever r≠0r\ne 0, and are often more convenient for functions naturally expressed in polar form, such as znz^n, log⁡z\log z, or z1/nz^{1/n}, whose real and imaginary parts are cleanest in terms of rr and θ\theta.

Is a function satisfying the Cauchy–Riemann equations automatically continuous?

No — continuity of ff is not automatically implied just from the two partial-derivative equations holding at a point, since those equations only constrain the partials, not the function's behavior along non-axis directions. However, once we also assume u,vu,v are real-differentiable at the point (the correct sufficient hypothesis for the full theorem), continuity of ff at that point follows immediately, because real-differentiability at a point always implies continuity there, by the standard argument that the linear approximation error is o(∣h∣)o(|h|), which forces f(z0+h)→f(z0)f(z_0+h)\to f(z_0) as h→0h\to0.

Why is the counterexample zˉ2/z\bar z^2/z instructive rather than just a technicality?

It shows precisely where the logical gap would be if one tried to weaken the sufficiency theorem's hypothesis from "real-differentiable" to merely "partial derivatives exist." The function's polar form re−3iθre^{-3i\theta} reveals that the difference quotient depends on the approach angle through a factor e−4iθe^{-4i\theta}, a clean, checkable, non-constant function of direction — making the failure completely transparent rather than an abstract pathology. It teaches that checking two numbers (the partials at a point) can never substitute for checking a genuine two-dimensional limit, which is the real content of complex differentiability.

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