Cauchy–Riemann Equations

The Cauchy–Riemann equations are a pair of first-order partial differential equations central to complex analysis. For a function $f(z)=u(x,y)+iv(x,y)$ , where $z=x+iy$ , they state that

\boxed{u_x=v_y,\qquad u_y=-v_x.}

They are necessary for complex differentiability and, when the real and imaginary parts are differentiable as functions of two real variables, are also sufficient. They express the compatibility required for a real two-dimensional differential to act as multiplication by a single complex number. (people.math.sc.edu)

Definition and derivation

Let $f$ be defined on an open set $D\subseteq\mathbb C$ , and write

f(x+iy)=u(x,y)+iv(x,y),

with $u$ and $v$ real-valued. Subscripts denote partial derivatives, such as $u_x=\partial u/\partial x$ . The complex derivative at $z_0$ is

f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}.

This limit must have the same value for every way in which the complex increment $h$ approaches zero. (ocw.mit.edu)

Taking real increments $h=t$ gives

f'(z_0)=u_x+iv_x.

Taking imaginary increments $h=it$ gives

f'(z_0)=\frac{u_y+iv_y}{i}=v_y-iu_y.

All partial derivatives here are evaluated at $(x_0,y_0)$ . Equating real and imaginary parts yields the Cauchy–Riemann equations. When the derivative exists, it can therefore be computed as

\boxed{f'(z_0)=u_x+iv_x=u_x-iu_y.}

Comparing these two directions proves necessity; by itself, it does not establish convergence along every possible approach. (ocw.mit.edu)

Necessity, sufficiency, and holomorphicity

The precise pointwise criterion is:

A function $f=u+iv$ is complex differentiable at $z_0$ if and only if $u$ and $v$ are differentiable in the real, multivariable sense at $(x_0,y_0)$ and satisfy the Cauchy–Riemann equations there.

Real differentiability means that the function admits a linear approximation with an error $o(|h|)$ , not merely that its coordinate partial derivatives exist. Under the equations, this approximation becomes

f(z_0+h)=f(z_0)+(u_x+iv_x)h+o(|h|),

which proves complex differentiability. (people.math.sc.edu)

A frequently used sufficient condition is that the four first partial derivatives exist in a neighborhood of $(x_0,y_0)$ , are continuous at that point, and satisfy the equations there. Continuous partial derivatives ensure the required real differentiability. (complexanalysis.org)

A function is holomorphic on $D$ when it is complex differentiable at every point of $D$ . Thus, for $u,v\in C^1(D)$ , holomorphicity is equivalent to satisfaction of the equations throughout $D$ . In one complex variable, holomorphic functions are also analytic: locally they possess convergent power series representations. Differentiability at a single point is not equivalent to holomorphicity on a neighborhood. (users.math.msu.edu)

Examples and limitations

A polynomial

For the polynomial $f(z)=z^2$ ,

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

Consequently,

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

The partial derivatives are continuous everywhere, so $f$ is holomorphic throughout the plane, with $f'(z)=2z$ . (users.math.msu.edu)

Complex conjugation

For $f(z)=\overline z=x-iy$ ,

u_x=1,\qquad v_y=-1.

The first equation fails everywhere. Complex conjugation is therefore nowhere complex differentiable, although it is a smooth real mapping of the plane. (ocw.mit.edu)

Equations satisfied without differentiability

Consider

f(z)= \begin{cases} \dfrac{\overline z^{\,2}}{z},&z\ne0,\\[4pt] 0,&z=0. \end{cases}

This is a continuous function at zero because $|f(z)|=|z|$ . At zero its partial derivatives satisfy

u_x=v_y=1,\qquad u_y=v_x=0.

Nevertheless,

\frac{f(z)}{z}=\frac{\overline z^{\,2}}{z^2}

equals $1$ along the real axis and $-1$ along the line $z=t(1+i)$ . Hence $f'(0)$ does not exist. Even continuity, existence of all four partial derivatives, and satisfaction of the equations at one point do not suffice without an additional differentiability hypothesis. (complexanalysis.org)

Geometric interpretation

Viewing $f$ as the real mapping $(x,y)\mapsto(u,v)$ , its Jacobian matrix has the form

Df= \begin{pmatrix} u_x&u_y\\ v_x&v_y \end{pmatrix} = \begin{pmatrix} a&-b\\ b&a \end{pmatrix}, \qquad a=u_x,\quad b=v_x.

This linear map is multiplication by $a+ib=f'(z)$ . If $f'(z)\ne0$ , it combines a rotation through $\arg f'(z)$ with uniform scaling by $|f'(z)|$ . Thus a holomorphic function is conformal wherever its derivative is nonzero: it preserves oriented angles between intersecting smooth curves. (ocw.mit.edu)

The determinant is

\det Df=a^2+b^2=|f'(z)|^2.

It is positive at such points, so the local mapping preserves orientation. At a zero of $f'$ , the differential degenerates, and the usual local conformality conclusion does not apply. (ocw.mit.edu)

Harmonic functions and harmonic conjugates

Differentiating the equations and equating mixed partial derivatives gives

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

Thus the real and imaginary parts of a holomorphic function are harmonic functions: they satisfy Laplace’s equation. Holomorphic functions have the smoothness needed for this calculation. (ocw.mit.edu)

A function $v$ for which $u+iv$ is holomorphic is called a harmonic conjugate of $u$ . The equations prescribe

v_x=-u_y,\qquad v_y=u_x.

Locally, every harmonic function has such a conjugate. A global conjugate exists on a simply connected plane domain and is unique up to an additive real constant. Without simple connectivity, global existence can fail: $u(z)=\log|z|$ on $\mathbb C\setminus\{0\}$ has local conjugates given by branches of the argument, but no single-valued conjugate on the whole punctured plane. (kleinex.mit.edu)

The equations also imply

\nabla u\cdot\nabla v=0,\qquad |\nabla u|=|\nabla v|.

Accordingly, regular level curves of conjugate harmonic functions intersect orthogonally. (ocw.mit.edu)

Alternative formulations

In polar coordinates $z=re^{i\theta}$ , with $r>0$ , the chain rule transforms the equations into

\boxed{u_r=\frac1r v_\theta,\qquad v_r=-\frac1r u_\theta.}

This formulation is useful on sectors and annuli; it cannot be applied directly at the origin, where polar coordinates are singular. (complexanalysis.org)

The Wirtinger differential operators are

\frac{\partial}{\partial z} =\frac12\left(\frac{\partial}{\partial x} -i\frac{\partial}{\partial y}\right), \qquad \frac{\partial}{\partial\overline z} =\frac12\left(\frac{\partial}{\partial x} +i\frac{\partial}{\partial y}\right).

Direct substitution shows that

\frac{\partial f}{\partial\overline z} =\frac12\bigl[(u_x-v_y)+i(v_x+u_y)\bigr].

The Cauchy–Riemann equations are therefore equivalent to $\partial f/\partial\overline z=0$ . With real differentiability, this says that the first-order differential contains a $dz$ term but no $d\overline z$ term. (people.math.sc.edu)

Applications

In two-dimensional incompressible, irrotational fluid flow, a velocity potential $\phi$ and stream function $\psi$ can be combined into a holomorphic complex potential

W(z)=\phi(x,y)+i\psi(x,y).

With velocity components $V_x=\phi_x$ and $V_y=\phi_y$ , the equations give

\psi_y=V_x,\qquad \psi_x=-V_y, \qquad W'(z)=V_x-iV_y.

This representation connects streamline geometry, potential theory, and conformal transformations. It applies under these flow assumptions, not to arbitrary fluid motion. (ocw.mit.edu)

Historical development

The equations are named after Augustin-Louis Cauchy and Bernhard Riemann, but their origins precede both mathematicians. Related relations appeared in Jean le Rond d’Alembert’s 1752 work on fluid resistance and in Leonhard Euler’s fluid studies of 1757. Euler subsequently treated them in the context of complex differentiation. Cauchy connected them with complex integration and path independence; Riemann incorporated them into his development of complex function theory. Their history therefore spans both fluid mechanics and the foundations of complex analysis. (arxiv.org)

References

Complex Variablespeople.math.sc.edu
The Cauchy-Riemann Equationscomplexanalysis.org
Analytic and Harmonic Functions; the Cauchy-Riemann Equationsusers.math.msu.edu
04 S18 Topic 2: Analytic functionsocw.mit.edu
04 S18 Topic 5: Introduction to harmonic functionsocw.mit.edu
04 S18 Topic 6: Two dimensional hydrodynamics and complex potentialsocw.mit.edu
04 S18 Topic 10: Conformal transformationsocw.mit.edu
A historical review of the Cauchy-Riemann equations and the Cauchy Theoremarxiv.org