Analytic Holomorphic Functions: Explained with Examples

A function f is analytic (holomorphic) at z_0 if it is complex-differentiable at every point of some disc around z_0. The complex derivative f′(z_0) = lim_{h→0} [f(z_0+h) − f(z_0)]/h must exist with h approaching through all complex directions, a far stronger demand than real differentiability.

The reward is the miracle of complex analysis: once differentiable in a region, f is automatically infinitely differentiable and equals its Taylor series there. Analyticity unlocks Cauchy's theorem, power series, and residues, turning hard real integrals into short complex computations.

Key Concepts

• Complex derivative: the difference-quotient limit with h complex.
• Analytic at a point: differentiable throughout a neighborhood; always a neighborhood property.
• Entire: analytic on all of C, e.g. polynomials, e^z, sin z.
• Singularity: a point where analyticity fails, e.g. z = 0 for 1/z.
• Power series link: analytic in a disc exactly when equal to a convergent power series there.

How It Works

1. Test the quotient along h = t (real) and h = it (imaginary); both limits must agree.
2. f(z) = z²: [(z_0+h)² − z_0²]/h = 2z_0 + h → 2z_0 from every direction; entire, with f′ = 2z.
3. f(z) = z̄: real direction gives 1, imaginary gives −1; differentiable nowhere.
4. f(z) = |z|²: differentiable only at 0, hence analytic nowhere.

Worked Example

Verify f(z) = z² has derivative 2 + 2i at z_0 = 1 + i. The quotient is [(1+i+h)² − (1+i)²]/h = 2(1+i) + h → 2 + 2i from any direction. Check h = 0.01: (1.01+i)² = 0.0201 + 2.02i, minus (1+i)² = 2i gives 0.0201 + 0.02i; over 0.01 that is 2.01 + 2i, within 0.01 of 2 + 2i. Check h = 0.01i: (1+1.01i)² = −0.0201 + 2.02i, minus 2i gives −0.0201 + 0.02i; over 0.01i that is 2 + 2.01i, again within 0.01 of 2 + 2i. Both directions agree, as analyticity requires.

Common Misconceptions

• "Differentiable at a point ⇒ analytic there." Analyticity needs a neighborhood; |z|² is differentiable at 0 yet analytic nowhere.
• "Complex differentiable ≈ real differentiable." Real once-differentiable functions can be rough; complex once-differentiable implies infinitely differentiable.
• "Cauchy–Riemann at a point suffices." Necessity yes; sufficiency needs the equations plus continuous partials in a neighborhood.

Real-World Applications

• The Joukowsky transform maps cylinder airflow to airfoil airflow in aerodynamics.
• Analytic signals define instantaneous frequency in signal processing.
• 2D Laplace problems for heat and ideal fluids are solved via analytic functions.

Analyticity, differentiability throughout a neighborhood, forces infinite differentiability and power-series expansions. The failures of z̄ and |z|² show how special analytic functions are.