Analytic Functions

Discover complex differentiability and the Cauchy-Riemann equations

Analytic Functions

Being differentiable in the complex sense is a strict test, and most functions fail it. The ones that pass, the analytic functions, get a lot of structure for free.

1. What a Tiny Square Becomes

Drag the small square and shrink it. Under an analytic function its image becomes a rotated, scaled square; under z̄ or |z|² it flips, shears, or flattens however far you zoom.

z-plane: drag the square

Its image under f(z) = z², magnified ×3.2 about f(z₀)

90.0°

angle at the white corner

1.89 / 1.89

stretch of bottom / left edge

0.000, 0.000

uₓ − v_y, u_y + vₓ (both 0 ⇔ C–R)

1.89 / 0.00

|∂f/∂z| / |∂f/∂z̄|

f(z)

Square size (zoom in by shrinking it)0.316

Cauchy–Riemann holds: the square only rotates by 32.0° and scales by 1.89.

Colored edges: the exact image of the square, magnified so it stays the same size as you shrink it. Dashed: the Jacobian's prediction. As the square shrinks the two agree, so what you see is the derivative itself. Any real-linear map can be written dz ↦ a·dz + b·dz̄; it is a rotation and scaling exactly when b = ∂f/∂z̄ = 0, which is the Cauchy–Riemann equations in one line. Non-analytic functions are shown in rose.

Go deeper:Cauchy-Riemann Equations

In complex analysis, a function is analytic (or holomorphic) if it is complex differentiable in a neighborhood of every point in its domain. This seemingly simple requirement has profound consequences!

The key to understanding analyticity is the Cauchy-Riemann equations - a pair of partial differential equations that relate the real and imaginary parts of a complex function.

For a function f(z) = u(x,y) + iv(x,y) to be analytic, it must satisfy the Cauchy-Riemann equations:

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

Zoomed in far enough, any smooth map of the plane looks like its Jacobian matrix [[uₓ, u_y], [vₓ, v_y]]. Complex differentiability asks for more: the Jacobian must be multiplication by a single complex number f′(z₀), which rotates by arg f′(z₀) and scales by |f′(z₀)|. A matrix [[p, −q], [q, p]] does exactly that, and matching it entry by entry gives the two equations above.

Another way to say it: every real-linear map can be written dz ↦ a·dz + b·dz̄, with a = ∂f/∂z and b = ∂f/∂z̄. The dz̄ part is what flips and shears the square in the demo, and the Cauchy–Riemann equations are the single condition ∂f/∂z̄ = 0.

2. Harmonic Functions Visualizer

Choose an analytic function and compare its real part u with its imaginary part v. Both are harmonic, and their level curves cross at right angles.

Choose an analytic function:

Square: f(z) = z²

Real part: u(x,y) = x² - y² (hyperbolic paraboloid)

Imaginary part: v(x,y) = 2xy (hyperbolic paraboloid)

Real Part: u(x,y)

Blue = low values, Red = high values

Imaginary Part: v(x,y)

Blue = low values, Red = high values

Harmonic Property Test at (1, 0.5):

∇²u = 0.0000

✓ u is harmonic (∇²u ≈ 0)

∇²v = 0.0000

✓ v is harmonic (∇²v ≈ 0)

🎨 Reading the Heatmaps:

Color represents the value of the function: blue (negative/low) → red (positive/high)
Contour lines (if visible) show where the function has constant values
For analytic functions, u and v level curves are orthogonal (perpendicular)

⚗️ Harmonic Functions:

A function h is harmonic if ∇²h = ∂²h/∂x² + ∂²h/∂y² = 0
For analytic f = u + iv, both u and v are harmonic
Harmonic functions have no local maxima or minima (only on boundaries)
Real-world applications: steady-state heat distribution, electric potential, gravitational potential

Go deeper:Harmonic Functions

The real and imaginary parts of an analytic function are harmonic functions - they satisfy Laplace's equation:

∇²u = ∂²u/∂x² + ∂²u/∂y² = 0

Harmonic functions have many beautiful properties: they have no local maxima or minima, the level curves of u and its harmonic conjugate v are orthogonal, and they model steady-state heat distribution.

3. Analyticity Tester

Test each function and see which pass. Start with z and eᶻ, then try the conjugate z̄ and |z|².

Select a function to test:

Square

f(z) = z²

✓

Status: Analytic

Where: Entire complex plane (entire function)

Reason: Polynomial functions are always analytic

Cauchy-Riemann Analysis:

Real part:

u(x,y) = x² - y²

Imaginary part:

v(x,y) = 2xy

∂u/∂x

∂u/∂y

-2y

∂v/∂x

∂v/∂y

Equation 1: ∂u/∂x = ∂v/∂y2x = 2x ✓

Equation 2: ∂u/∂y = -∂v/∂x-2y = -2y ✓

Domain Coloring Visualization:

Analytic functions produce smooth, continuous color patterns (except at poles)

📚 Key Insights:

Polynomials (z, z², z³, etc.) are analytic everywhere (entire functions)
Rational functions (1/z, (z²+1)/(z-1)) are analytic except at poles
Transcendental functions (e^z, sin(z), cos(z)) are typically entire
Conjugation and real/imaginary projection destroy analyticity
|z|, |z|², Re(z), Im(z) are NOT analytic (except at isolated points)

Go deeper:Which Functions Are Analytic?

Not all complex functions are analytic! Test various functions to see which satisfy the Cauchy-Riemann equations. Some surprising examples:

f(z) = z is analytic everywhere ✓
f(z) = z̄ (conjugate) is NOT analytic ✗
f(z) = |z|² is complex differentiable only at z = 0, so it is analytic nowhere ✗
f(z) = eᶻ is analytic everywhere ✓

Analyticity is Rare: Most functions from ℝ² → ℝ² are not analytic. Only those that satisfy C-R equations have this special property.

Powerful Consequences: Analytic functions have infinite derivatives, can be represented as power series, and satisfy the Cauchy Integral Theorem - topics we'll explore next!

Key Takeaways

Analytic means complex differentiable near every point
The Cauchy-Riemann equations are the test for complex differentiability
The real and imaginary parts of an analytic function are harmonic (∇²u = 0, ∇²v = 0)
Most smooth maps of the plane, like z̄, are not analytic

Spot a mistake or have an idea?