Analytic Functions

Analytic functions, also called holomorphic functions, are the central objects of study in complex analysis. These functions are complex-differentiable in a neighborhood of every point in their domain. The remarkable properties of analytic functions, including their representation as power series and the constraints imposed by the Cauchy-Riemann equations, make them exceptionally well-behaved.

Definition

Complex Differentiability

A function ƒ from the complex numbers to the complex numbers is complex differentiable at a point (z_0) if the following limit exists:

ƒ′*((z_0))=(lim_h→0)((ƒ*((z_0)+h)−ƒ((z_0)))/h)

where h is a complex number that can approach 0 from any direction in the complex plane. This is far more restrictive than real differentiability.

Analytic (Holomorphic) Functions

A function ƒ is analytic (or holomorphic) on an open set U if f is complex differentiable at every point in U. A function is entire if it is analytic on the entire complex plane.

The Cauchy-Riemann Equations

Write z=x+i*y and ƒ(z)=u(x,y)+i*v(x,y), where u and v are real-valued functions. Complex differentiability imposes strong constraints on u and v.

Theorem

If ƒ=u+i*v is complex differentiable at a point, then the partial derivatives satisfy the Cauchy-Riemann equations:

∂(u)/∂(x)=∂(v)/∂(y) and ∂(u)/∂(y)=−∂(v)/∂(x)

Conversely, if u and v have continuous first partial derivatives and satisfy these equations, then ƒ is analytic.

Fundamental Properties

Power Series Representation

A function is analytic at a point (z_0) if and only if it can be represented as a convergent power series in a neighborhood of (z_0):

ƒ(z)=(∑_n=0^∞)((a_n))*(z−(z_0))n

This is a deep result: complex differentiability implies analyticity (representation by power series), which is far stronger than what happens in real analysis.

Infinite Differentiability

If ƒ is analytic on an open set U, then f has derivatives of all orders on U, and each derivative is itself analytic. The coefficients in the power series are given by:

(a_n)=(ƒ((z_0))n)/n!

Harmonic Components

If ƒ=u+i*v is analytic, then both u and v are harmonic functions, meaning they satisfy Laplace's equation:

∇2*u=∂2(u)/∂(x2)+∂2(u)/∂(y2)=0

The functions u and v are called harmonic conjugates of each other.

Examples

Polynomials

Every polynomial p(z)=(a_0)+(a_1)*z+⋯+(a_n)*zn is entire (analytic on all of C).

Exponential Function

The complex exponential is entire:

ez=(∑_n=0^∞)((zn)/(n!))

With z=x+i*y, we have the important formula:

ez=ex*(cos(y)+i*sin(y))

Trigonometric Functions

The complex sine and cosine are entire functions defined by:

sin(z)=(e(i*z)−e(−i*z))/(2*i),cos(z)=(e(i*z)+e(−i*z))/2

Rational Functions

A rational function p(z)/q(z) is analytic everywhere except at the zeros of q(z). These points are called poles.

Logarithm

The complex logarithm is multi-valued, but each branch is analytic. The principal branch is:

(log_)(z)=ln(z)+i*arg(z)

where the argument is typically restricted to a range like (-π,π]. The logarithm is analytic on the cut plane (C minus a ray from 0 to infinity).

Key Theorems

Cauchy Integral Theorem

If ƒ is analytic in a simply connected domain D, and C is any closed contour in D, then:

(∮_C)(ƒ(z)*d(z))=0

This remarkable result has no analog in real analysis.

Cauchy Integral Formula

If ƒ is analytic inside and on a simple closed contour C, and (z_0) is inside C, then:

ƒ((z_0))=1/(2*π*i)*(∮_C)(ƒ(z)/(z−(z_0))*d(z))

The value of an analytic function at any interior point is determined by its values on the boundary. This is a form of rigidity unique to analytic functions.

Liouville Theorem

A bounded entire function must be constant. More precisely, if ƒ is entire and there exists M such that |ƒ(z)|≤M for all z, then ƒ is constant.

Identity Theorem

If two analytic functions agree on a set with an accumulation point in their common domain, they must be identical throughout any connected component containing that point.

This means that an analytic function is completely determined by its values on an arbitrarily small region.

Maximum Modulus Principle

If ƒ is analytic and non-constant on a connected open set U, then |ƒ| has no maximum in U. Any maximum must occur on the boundary.

This principle is extremely useful for establishing bounds on analytic functions.

Zeros of Analytic Functions

The zeros of an analytic function have special structure.

Isolated Zeros

If ƒ is analytic and not identically zero on a connected open set, then its zeros are isolated. Around each zero (z_0), we can write:

ƒ(z)=(z−(z_0))m*g(z)

where m is a positive integer (the order of the zero) and g is analytic with g((z_0)) not equal to zero.

Analytic Continuation

Analytic continuation is the process of extending an analytic function beyond its original domain of definition while preserving analyticity.

The key fact is that such an extension, if it exists, is unique. This allows us to speak of THE analytic continuation of a function.

For example, the Riemann zeta function is defined initially for Re(s)>1, but can be analytically continued to all of C except s=1.

Real vs Complex Analysis

The contrast between real and complex differentiability is striking:

In real analysis: A function can be once differentiable but not twice differentiable. There exist infinitely differentiable functions that are not analytic (e.g., exp(−1/x2) extended to be 0 at x=0).

In complex analysis: One complex derivative implies infinitely many. Complex differentiability implies analyticity (power series representation).

Applications

Fluid Dynamics

In 2D incompressible, irrotational fluid flow, the velocity potential and stream function form the real and imaginary parts of an analytic function. The Cauchy-Riemann equations correspond to the incompressibility and irrotationality conditions.

Electrostatics

In 2D electrostatics, the electric potential satisfies Laplace's equation and can be treated as the real part of an analytic function. The imaginary part gives the electric field lines.

Conformal Mapping

An analytic function with non-zero derivative is conformal (angle-preserving). Conformal maps are used to transform complicated domains into simpler ones where problems can be solved more easily.

Number Theory

Analytic functions like the Riemann zeta function are central to number theory. The distribution of prime numbers is intimately connected to the zeros of zeta(s).

Evaluation of Integrals

Many difficult real integrals can be computed using complex contour integration and the residue theorem. Analytic function theory provides powerful tools for evaluating integrals that are intractable by real methods.

Summary

An analytic (holomorphic) function is one that is complex differentiable in a neighborhood of each point in its domain. This condition is equivalent to: satisfying the Cauchy-Riemann equations; being representable as a convergent power series; having derivatives of all orders. Key theorems include Cauchy's integral theorem (closed contour integrals are zero), Cauchy's integral formula (interior values determined by boundary), Liouville's theorem (bounded entire functions are constant), and the maximum modulus principle (maxima occur on boundaries). The rigidity of analytic functions makes them powerful tools in physics, engineering, and number theory.