The Cauchy-Riemann Equations

Part 9, Chapter 9: Complex Analysis, Holomorphic Functions

Learning objectives

State and derive the Cauchy-Riemann equations from the direction-independent limit
Use the equations as an algebraic test for holomorphicity
Reconstruct the harmonic conjugate $v$ from a harmonic $u$ on a simply connected domain
Recognize that real and imaginary parts of a holomorphic function are harmonic

The Cauchy-Riemann equations are the algebraic content of holomorphicity. They convert the direction-independent limit definition into two simple equations involving partial derivatives of the real and imaginary parts. With them, testing whether $f = u + iv$ is holomorphic becomes a high-school exercise in computing four partial derivatives. They also reveal the deepest single fact of complex analysis: real and imaginary parts of holomorphic functions are harmonic, satisfying Laplace's equation. That observation unifies complex analysis with potential theory, electrostatics, steady-state heat flow, and 2D fluid mechanics.

The equations

Let $f(z) = u(x, y) + i v(x, y)$ where $z = x + iy$ and $u, v$ are real-valued. The Cauchy-Riemann equations are:

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

Derivation in one paragraph

If $f$ is holomorphic at $z_0$ , the limit defining $f'(z_0)$ must give the same number along every direction. Take $h = t \in \mathbb{R}$ (horizontal): $f'(z_0) = u_x(z_0) + i v_x(z_0)$ ivx(z0). Take $h = it$ (vertical): $f'(z_0) = \frac{1}{i}(u_y(z_0) + i v_y(z_0)) = v_y(z_0) - i u_y(z_0)$ ivy(z0))=vy(z0)−iuy(z0). Equating real and imaginary parts yields $u_x = v_y$ and $v_x = -u_y$ , which are the Cauchy-Riemann equations. Conversely, if $u, v$ have continuous partials and satisfy CR on an open set, then $f = u + iv$ is holomorphic there.

The widget lets you compare an input grid in $z$ to its image under $f(z)$ . Try $f(z) = z^2$ : the image grid lines remain perpendicular wherever $f'(z) \neq 0$ , a visible consequence of CR.

Harmonic functions and harmonic conjugates

If $f = u + iv$ is holomorphic, then differentiating CR (and using equality of mixed partials) gives $u_{xx} + u_{yy} = (v_y)_x + (-v_x)_y = v_{yx} - v_{xy} = 0$ (vy)x+(−vx)y=vyx−vxy=0. So $u$ satisfies Laplace's equation: $\nabla^2 u = 0$ . Similarly $\nabla^2 v = 0$ . Functions with zero Laplacian are called harmonic.

Going the other way is the more useful direction. Given a harmonic $u$ on a simply connected domain, the Cauchy-Riemann equations can be integrated to find $v$ (the harmonic conjugate), unique up to a constant. Concretely: solve $v_y = u_x$ for $v$ by integrating with respect to $y$ , then use $v_x = -u_y$ to pin down the constant of integration. The construction always succeeds on simply connected domains; on multiply connected ones a global conjugate may not exist (e.g. $u = \ln|z|$ on $\mathbb{C} \setminus \{0\}$ has no single-valued conjugate).

Where this shows up

Electrostatics in 2D: The electric potential $\varphi$ in a charge-free region satisfies Laplace's equation. Its conjugate $\psi$ encodes the stream function, and $f = \varphi + i\psi$ is holomorphic. Conformal mapping techniques then turn complicated boundary problems into simple ones (rectangle → disk → half-plane).

Steady-state heat conduction: Equilibrium temperature distributions in 2D are harmonic. Engineering analyses of heat sinks, microchip cooling layouts, and printed-circuit thermal modelling use harmonic conjugates and conformal mapping to reduce the geometry to manageable canonical domains.

Aerodynamics: The Joukowski transformation $w = z + 1/z$ maps a circle to an airfoil; the holomorphic flow potential around the circle pulls back to the flow around the airfoil. The Cauchy-Riemann equations guarantee the velocity field is divergence-free and curl-free.

Image processing: Quasiconformal maps (controlled deviation from CR) underlie modern morphing and face-warping algorithms. The CR equations measure exactly how far a deformation is from being angle-preserving.

Pause and think: If $u(x, y) = x^2 - y^2$ is to be the real part of a holomorphic $f$ , what must $v$ satisfy? Compute $u_x = 2x$ x=2x; then $v_y = 2x$ y=2x, so $v = 2xy + g(x)$ . Now use $v_x = -u_y = 2y$ 2y to force $g'(x) = 0$ . Conclusion: $v = 2xy + C$ , and $f(z) = z^2 + iC$ .

Try it

Verify CR for $f(z) = e^z$ using $u = e^x \cos y$ and $v = e^x \sin y$ . Compute all four partials and check both equations.

Predict first: is $f(z) = |z|^2$ holomorphic anywhere? Apply CR with $u = x^2 + y^2$ , $v = 0$ . Find the set of points where both CR equations hold.

Given the harmonic $u(x, y) = e^x \cos y$ , find a harmonic conjugate $v$ . (Hint: the answer should give you $f(z) = e^z$ up to a constant.)

True or false: every smooth real-valued function of two variables is the real part of some holomorphic function. (Hint: the function must first satisfy Laplace's equation. Most smooth functions do not.)

A trap to watch for

Cauchy-Riemann alone is not quite sufficient for holomorphicity. There exist pathological functions where CR holds at a single point but $f$ fails to be holomorphic there (the partial derivatives exist but are discontinuous). The precise sufficient condition is: $u, v$ have continuous first partials AND satisfy CR on an open set. In practice this never bites, the functions you write down with elementary operations have continuous partials wherever they are defined, but the textbook hypothesis is "continuous partials + CR," not "CR alone." Beginners sometimes drop the continuity assumption and get burned by counterexamples like $f(z) = e^{-1/z^4}$ (with $f(0) = 0$ ), which satisfies CR at $0$ but is not holomorphic at $0$ .

What you now know

You can test holomorphicity by computing four partial derivatives and checking two algebraic equations, recover a holomorphic function from its real or imaginary part, and recognize that complex analysis is the algebraic side of two-dimensional potential theory. The next section turns to integration: Cauchy's theorem and integral formula, which extract the global rigidity hidden in the local CR equations.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, §9.3.

Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill, §2.1.

Stein, E. M., Shakarchi, R. (2003). Complex Analysis. Princeton University Press, ch. 1.

Conway, J. B. (1978). Functions of One Complex Variable I (2nd ed.). Springer, §3.2.

Brown, J. W., Churchill, R. V. (2014). Complex Variables and Applications (9th ed.). McGraw-Hill, §21-26.

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