Entire Functions

Entire functions, also known as integral functions, represent a fundamental class of complex functions that are holomorphic everywhere in the complex plane. These functions possess remarkable properties that make them central to complex analysis and its applications.

Definition and Basic Properties

An entire function f(z) is defined as a function that:

• Is complex differentiable at every point in the complex plane ℂ
• Has a convergent Taylor series representation around any point
• Has no singularities in the finite complex plane

Notable examples include:

• Exponential function e^z
• polynomial functions
• Trigonometric functions sin(z) and cos(z)
• hyperbolic functions

Characterization Through Series

Every entire function can be expressed as a power series:

f(z) = ∑(n=0 to ∞) aₙzⁿ

This series converges for all complex values of z, with the coefficients aₙ given by:

aₙ = f⁽ⁿ⁾(0)/n!

Important Theorems

Liouville's Theorem

One of the most significant results about entire functions states that any bounded entire function must be constant. This leads to the fundamental theorem of algebra as a corollary.

Picard's Theorems

• Picard's Little Theorem: A non-constant entire function takes every complex value, with at most one exception
• Picard's Great Theorem: Extends this result to functions with essential singularities

Classification and Growth

Entire functions can be classified by their order of growth:

1. Polynomial growth
2. Exponential growth
3. Super-exponential growth

The Weierstrass factorization theorem provides a way to represent entire functions through their zeros.

Applications

Entire functions find applications in:

• complex analysis
• conformal mapping
• analytic number theory
• differential equations
• signal processing

Related Concepts

The study of entire functions connects deeply with:

• meromorphic functions
• analytic continuation
• complex dynamics
• maximum modulus principle

Historical Development

The theory of entire functions was developed significantly by:

• Weierstrass in the 19th century
• Hadamard through his work on factorization
• Nevanlinna through value distribution theory

The systematic study of entire functions has led to profound insights in both pure and applied mathematics, forming a cornerstone of modern complex analysis.