52  Complex functions

Differentiability in the complex plane and the Cauchy-Riemann equations

A function f of a complex variable maps complex numbers to complex numbers. Writing z = x + iy and separating f(z) = u(x,y) + iv(x,y), the function is a pair of real-valued functions of two real variables. The condition on u and v that makes f genuinely complex-differentiable — the Cauchy-Riemann equations — is what the rest of the chapter develops.

52.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • f(z) = u + iv: f of z equals u plus i v

Definitions to keep handy

These are the words we keep coming back to. If one feels slippery, come back here and steady it before you push on.

  • complex function: A rule that takes a complex number in and gives a complex number out.

  • complex differentiable: The derivative exists no matter which direction you approach from in the complex plane.

  • Cauchy-Riemann equations: The conditions on u(x,y) and v(x,y) that make f(z)=u+iv complex differentiable.

  • analytic: Complex differentiable throughout a region; this unlocks the big theorems.

  • harmonic: A function whose Laplacian is zero; it describes steady-state potentials.

Here is the main move this chapter is making, in plain terms. You do not need to be fast. You just need to keep the thread.

  • Coming in: Complex numbers extend the real line to a plane. Functions of a complex variable map that plane to itself. The condition for a complex function to be differentiable — the Cauchy-Riemann equations — is far more restrictive than real differentiability, and functions that satisfy it have extraordinary properties: infinitely differentiable, representable by power series, and connected to the solutions of Laplace’s equation.

  • Leaving with: A complex function f(z) = u(x,y) + iv(x,y) is differentiable at z₀ if the limit of [f(z)−f(z₀)]/(z−z₀) exists regardless of how z approaches z₀. This requires u_x = v_y and u_y = −v_x (the Cauchy-Riemann equations). Functions that are differentiable throughout an open domain are analytic. The real and imaginary parts of any analytic function are harmonic — they satisfy Laplace’s equation — which connects complex analysis to potential theory, heat conduction, and fluid flow.

52.2 Complex numbers: brief review

Every complex number z = x + iy is a point in the Argand plane, with real part x on the horizontal axis and imaginary part y on the vertical.

Polar form. z = r e^{i\theta}, where r = |z| = \sqrt{x^2+y^2} is the modulus and \theta = \arg z is the argument — the angle from the positive real axis, measured counterclockwise. Euler’s formula e^{i\theta} = \cos\theta + i\sin\theta connects the polar and rectangular forms.

De Moivre’s theorem. (re^{i\theta})^n = r^n e^{in\theta}. Integer powers of complex numbers are most efficiently computed in polar form.

Conjugate. \bar{z} = x - iy. Identities: z\bar{z} = |z|^2; \text{Re}(z) = (z+\bar{z})/2; \text{Im}(z) = (z-\bar{z})/(2i).

52.3 Complex functions

A complex function f: \mathbb{C} \to \mathbb{C} maps each z = x+iy to a complex value f(z) = u(x,y) + iv(x,y).

Example. f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + 2ixy, so u = x^2-y^2, v = 2xy.

Example. f(z) = \bar{z} = x - iy, so u = x, v = -y. This function is nowhere differentiable in the complex sense — a consequence of the Cauchy-Riemann equations derived in the next section.

Limits and continuity work exactly as in real analysis, with one additional clause: f is continuous at z_0 if \lim_{z\to z_0}f(z) = f(z_0), where the limit must be the same from every direction of approach.

52.4 Complex differentiability

The derivative of f at z_0 is:

f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}

The limit must exist and agree no matter how z approaches z_0 — a far stronger requirement than real differentiability.

52.4.1 Deriving the Cauchy-Riemann equations

The key idea is that the limit [f(z)-f(z_0)]/(z-z_0) must give the same answer regardless of the direction of approach. The two simplest directions — horizontal and vertical — already force a constraint on u and v.

Approach z_0 = x_0+iy_0 along two different directions and demand the limit is the same.

Horizontal approach (z = x+iy_0, z-z_0 = \Delta x):

Write f(z) - f(z_0) = [u(x_0+\Delta x, y_0) - u(x_0, y_0)] + i[v(x_0+\Delta x, y_0) - v(x_0, y_0)]. Dividing by the real number \Delta x keeps real and imaginary parts separate: \frac{f(z)-f(z_0)}{\Delta x} = \frac{[u(x_0+\Delta x,\,y_0)-u(x_0,y_0)]}{\Delta x} + i\frac{[v(x_0+\Delta x,\,y_0)-v(x_0,y_0)]}{\Delta x}

Taking \Delta x \to 0, each fraction is just a partial derivative: f'(z_0) = u_x + iv_x

Vertical approach (z = x_0+iy, z-z_0 = i\Delta y):

Now the increment is purely imaginary, so we divide by i\Delta y. Dividing by i is the same as multiplying by -i, which swaps real and imaginary parts. Carrying through: \frac{f(z)-f(z_0)}{i\Delta y} = \frac{1}{i}\cdot\frac{[u(x_0,y_0+\Delta y)-u(x_0,y_0)]+i[v(x_0,y_0+\Delta y)-v(x_0,y_0)]}{\Delta y}

Using 1/i = -i and taking \Delta y \to 0: f'(z_0) = v_y - iu_y

Equating real and imaginary parts of u_x + iv_x = v_y - iu_y:

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

These are the Cauchy-Riemann (C-R) equations. They are necessary for complex differentiability. If the partial derivatives are continuous in a neighbourhood of z_0, they are also sufficient.

Quick check. For f(z) = z^2 we found u = x^2-y^2, v = 2xy earlier. Verify: u_x = 2x = v_y and u_y = -2y = -v_x = -2y. Both equations hold everywhere — consistent with z^2 being entire. Exercise 1 walks through this systematically and uses the result to compute f'(z).

Consequence. For f(z) = \bar{z}: u_x = 1 but v_y = -1. The C-R equations fail everywhere, so \bar{z} is nowhere differentiable — even though its real and imaginary parts are smooth real functions.

When the C-R equations hold, the derivative can be written in any of the equivalent forms: f'(z) = u_x + iv_x = v_y - iu_y

52.5 Analytic functions

f is analytic at z_0 if it is differentiable in some open disk centred at z_0. It is analytic in a domain D if differentiable at every point of D. A function analytic on all of \mathbb{C} is called entire.

Being differentiable at an isolated point does not make a function analytic — analyticity requires differentiability in an open neighbourhood. Exercise 4 shows this with f(z) = |z|^2, which is differentiable only at the origin and analytic nowhere.

52.5.1 Harmonic functions

If f = u + iv is analytic, both u and v satisfy Laplace’s equation: \nabla^2 u = u_{xx} + u_{yy} = 0, \qquad \nabla^2 v = 0

Proof. From the C-R equations: u_{xx} = (u_x)_x = (v_y)_x = v_{yx} and u_{yy} = (u_y)_y = (-v_x)_y = -v_{xy}. Since mixed partials are equal, u_{xx} + u_{yy} = 0.

A function satisfying Laplace’s equation is harmonic. So: the real and imaginary parts of any analytic function are harmonic.

Conversely, given a harmonic u on a simply connected domain — one with no holes, so that any closed loop inside it can be continuously shrunk to a point — there exists a harmonic conjugate v (unique up to a constant) such that f = u + iv is analytic. Finding v from u via the C-R equations is a standard technique.

52.6 Elementary analytic functions

52.6.1 Exponential

e^z = e^x(\cos y + i\sin y), \qquad \frac{d}{dz}e^z = e^z

Entire. Periodic with period 2\pi i: e^{z+2\pi i} = e^z.

52.6.2 Trigonometric

\cos z = \frac{e^{iz}+e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz}-e^{-iz}}{2i}

Both entire. The identities \cos^2 z + \sin^2 z = 1 and all standard trig identities hold. For purely imaginary argument: \cos(iy) = \cosh y, \sin(iy) = i\sinh y — the complex trig and hyperbolic functions are closely related.

52.6.3 Logarithm

\log z = \ln|z| + i\arg z = \ln r + i\theta

Multivalued: \arg z is defined only up to multiples of 2\pi. The principal value \text{Log}\,z = \ln|z| + i\,\text{Arg}\,z uses \text{Arg}\,z \in (-\pi, \pi] and is analytic everywhere except on the branch cut along the negative real axis — the line where \text{Arg}\,z jumps discontinuously by 2\pi as you cross it. Without the cut, \text{Log}\,z cannot be made single-valued and continuous.

52.6.4 Complex powers

z^\alpha = e^{\alpha \log z}

Multivalued in general. Integer powers are single-valued and entire; non-integer powers require a branch cut.

52.7 Conformal mappings: preview

At any point z_0 where f is analytic and f'(z_0) \neq 0, the map f is conformal (Chapter 4): it preserves the angle between any two curves crossing at z_0 (in magnitude and orientation). The scale factor is |f'(z_0)| and the rotation angle is \arg f'(z_0).

Conformal mappings transform solutions of Laplace’s equation from one domain into solutions in another. That is how potential-flow problems, electrostatic field calculations, and heat-conduction problems in irregular geometries are reduced to solvable ones. Chapter 4 develops this fully.

52.8 Exercises

52.8.1 Exercise 1: Cauchy-Riemann equations for f(z) = z^2

Verify that f(z) = z^2 satisfies the Cauchy-Riemann equations everywhere and compute f'(z).

52.8.2 Exercise 2: Finding a harmonic conjugate

Show that u = x^3 - 3xy^2 is harmonic, find its harmonic conjugate v, and identify the analytic function f = u + iv.

52.8.3 Exercise 3: e^z is entire

Verify the Cauchy-Riemann equations for f(z) = e^z and confirm \frac{d}{dz}e^z = e^z.

52.8.4 Exercise 4: Where is f(z) = |z|^2 differentiable?

52.8.5 Exercise 5: Powers via De Moivre

Computing roots and powers in polar form — a technique used throughout this volume, including when locating poles and computing residues.

Compute (1+i)^8 and find all fourth roots of -16.