Analytic Function — Definition, Formula & Examples

An analytic function is a function that can be expressed as a convergent power series in a neighborhood of every point in its domain. In complex analysis, a function of a complex variable is analytic if and only if it is holomorphic (complex-differentiable).

A function $f: U \to \mathbb{C}$ , where $U \subseteq \mathbb{C}$ is open, is analytic at a point $z_0 \in U$ if there exists $r > 0$ such that $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ converges for all $z$ with $|z - z_0| < r$ . The function is analytic on $U$ if it is analytic at every point of $U$ .

Key Formula

f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n, \quad a_n = \frac{f^{(n)}(z_0)}{n!}

Where:

$z_0$ = The center point of the power series expansion
$a_n$ = The nth coefficient of the series
$f^{(n)}(z_0)$ = The nth derivative of f evaluated at z_0

How It Works

To determine whether a complex function is analytic, you can check whether it satisfies the Cauchy–Riemann equations. Write

f(z) = u(x,y) + iv(x,y)

where

z = x + iy

, and verify that

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

and

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

, with

u

and

v

having continuous partial derivatives. If these conditions hold throughout an open set, the function is analytic there. A landmark result of complex analysis is that analyticity, holomorphicity, and satisfying the Cauchy–Riemann equations are all equivalent for functions on open subsets of

\mathbb{C}

Worked Example

Problem: Show that

f(z) = z^2

is analytic on all of

\mathbb{C}

and find its power series centered at

z_0 = 1

Step 1: Write

z = (z - 1) + 1

and expand

z^2

in powers of

(z - 1)

z^2 = [(z-1) + 1]^2 = (z-1)^2 + 2(z-1) + 1

Step 2: Identify the coefficients using the formula

a_n = f^{(n)}(1)/n!

. We have

f(1) = 1

f'(1) = 2

, and

f''(1) = 2

, so

a_0 = 1

a_1 = 2

a_2 = 1

, and

a_n = 0

for

n \geq 3

f(z) = 1 + 2(z-1) + 1 \cdot (z-1)^2

Step 3: This finite series converges for all

z \in \mathbb{C}

, confirming

f(z) = z^2

is entire (analytic everywhere).

Answer:

f(z) = z^2

is analytic on

\mathbb{C}

with power series

1 + 2(z-1) + (z-1)^2

about

z_0 = 1

Why It Matters

Analytic functions are central to complex analysis, which underpins much of electrical engineering (AC circuit analysis, signal processing via Laplace and Fourier transforms) and theoretical physics (quantum field theory, fluid dynamics). Understanding analyticity is also essential for evaluating contour integrals and applying the residue theorem.

Common Mistakes

Mistake: Assuming that a real function that is infinitely differentiable must be analytic.

Correction: In real analysis,

C^\infty

does not imply analytic. The classic counterexample is

f(x) = e^{-1/x^2}

(with

f(0)=0

), whose Taylor series at

x=0

converges to

0

, not to

f(x)

. In complex analysis, however, being once complex-differentiable on an open set guarantees analyticity.