Analytic Functions

Discover complex differentiability and the Cauchy-Riemann equations

Analytic Functions

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.

1. Cauchy-Riemann Equations Checker

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

These equations ensure that the function is differentiable in the complex sense, not just as a mapping from ℝ² → ℝ².

Square: f(z) = z²
u = x² - y², v = 2xy. C-R equations satisfied everywhere ✓

Test Point: z = 1.00 + 1.00i

Partial Derivatives at z = 1.00 + 1.00i:

∂u/∂x
2.000
∂u/∂y
-2.000
∂v/∂x
2.000
∂v/∂y
2.000

Cauchy-Riemann Equations:

∂u/∂x = ∂v/∂y   →   2.000 ≟ 2.000
✓ Pass
∂u/∂y = -∂v/∂x   →   -2.000 ≟ -2.000
✓ Pass
✓ ANALYTIC
Square satisfies the Cauchy-Riemann equations at this point.

💡 Understanding the Check:

  • We compute partial derivatives numerically using finite differences (h = 0.0001)
  • Both C-R equations must be satisfied for the function to be analytic at this point
  • Tolerance of 0.01 accounts for numerical error in derivative computation
  • A function is analytic in a region if C-R holds at every point in that region

2. Harmonic Functions Visualizer

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, their level curves are orthogonal, and they model steady-state heat distribution.

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

3. Analyticity Tester

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 NOT analytic (except at z=0) ✗
  • f(z) = eᶻ is analytic everywhere ✓

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
2x
∂u/∂y
-2y
∂v/∂x
2y
∂v/∂y
2x
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)

Key Takeaways

1. Cauchy-Riemann Equations: The fundamental test for complex differentiability. They connect the partial derivatives of u and v in a specific way.

2. Harmonic Functions: Real and imaginary parts of analytic functions are harmonic (∇²u = 0, ∇²v = 0). This connection is unique to complex analysis!

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

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