Entire FunctionEdit

An entire function is a complex-analytic function defined on the whole complex plane. In precise terms, f: C → C is called entire if it is holomorphic at every point of the complex plane, equivalently if around every point z0 it has a convergent power series expansion f(z) = ∑ a_n (z − z0)^n with infinite radius of convergence. This global analyticity makes entire functions the natural objects of study in complex analysis complex analysis and underpins a wide range of results about growth, zeros, and structure.

The class of entire functions includes both polynomials and a rich family of transcendental functions such as the exponential function and trigonometric functions sine function and cosine function. Because they are defined on all of the complex plane, they offer a clean setting for exploring the interaction between local analyticity and global behavior, and they connect to important ideas about how functions grow, how zeros are arranged, and how entire functions can be reconstructed from their zeros.

Definition and basic properties

  • Definition: A function f is entire if it is holomorphic at every point of the complex plane (i.e., on all of C). This implies that about any point z0, f has a power series expansion with infinite radius of convergence. See Taylor series for the general expansion framework.

  • Radius of convergence: For an entire function, the Taylor series about any point has infinite radius of convergence, so the series converges everywhere in the plane.

  • Polynomials vs. transcendental entire functions: Polynomials are the simplest examples of entire functions, and they have a pole at infinity when viewed on the Riemann sphere. By contrast, transcendental entire functions (such as the exponential function or sine function) are not polynomials and exhibit more intricate global behavior.

  • Singularity at infinity: If f is entire, the nature of its singularity at infinity classifies its global growth. A constant function has a removable singularity at infinity, a polynomial has a pole at infinity, and every non-polynomial entire function has an essential singularity at infinity. This perspective is often framed using the compactification of the plane as the Riemann sphere.

  • Zeros: The zeros of an entire function are isolated unless the function is identically zero. This discreteness is a central theme in understanding how zeros constrain the whole function.

  • Liouville-type consequences: Bounded entire functions are constant by Liouville's theorem, which is a foundational result in complex analysis. This sharp constraint on global behavior contrasts with the abundance of entire functions that grow rapidly.

  • Factorization potential: Entire functions are deeply linked to product representations that encode their zeros, a theme developed in the Weierstrass factorization framework. This connects zero locations to the global form of the function and leads to constructive ways to build entire functions with prescribed zeros.

Zeros and factorization

  • Zeros and growth: Because zeros are isolated (unless the function is identically zero), the zero set of an entire function provides a fingerprint of its global behavior. The distribution of zeros interacts with growth and order in ways made precise by various factorization theorems.

  • Weierstrass factorization: A central result is that every entire function can be represented as a product that reflects its zeros, up to an exponential of an entire function. The product is built from canonical factors chosen to ensure convergence. This is known as the Weierstrass factorization theorem and is a powerful bridge between the zero set and the global form of the function. For example, the zeros of the sine function are exactly the multiples of π, and sine can be expressed in a canonical product form reflecting those zeros.

  • Canonical products and primary factors: The idea of canonical products (often involving primary factors) provides a way to regulate convergence when summing or multiplying infinitely many simple factors associated with zeros. See discussions of canonical products in connection with the Weierstrass theory.

  • Examples that illustrate the idea: The function sine function has zeros at {nπ : n ∈ Z}, and up to an exponential factor, its behavior is captured by a product over those zeros. Such product representations reveal how zeros alone can determine a great deal about an entire function’s global structure.

Growth and order

  • Growth measures: To compare how fast entire functions grow, one introduces quantities like M(r) = max{|f(z)| : |z| = r} and the order ρ, defined via the limit superior ρ = limsup_{r→∞} log log M(r) / log r. Functions of finite order have controlled growth, while transcendental entire functions can exhibit very rapid growth.

  • Hadamard factorization: Related to the zero set and growth is the Hadamard factorization, which expresses a broad class of entire functions in a structured product form that mirrors their zeros and a possible exponential factor. This connects growth constraints with a constructive representation.

  • Examples of growth behavior: The exponential function e^z grows rapidly along the positive real axis, while polynomials of degree d grow like a power of z. The interplay of zeros and exponential factors in product representations helps explain how different entire functions manage their growth in all directions of the complex plane.

Examples of important entire functions

  • The exponential function exponential function: Entire and nonzero everywhere; it grows rapidly but has no zeros, making it a fundamental building block in many representations.

  • The sine and cosine: sine function and cosine function are entire, with zeros at regular lattice points (multiples of π for sine, odd multiples of π/2 for cosine). Their product representations reflect these zero patterns and illustrate the factorization ideas.

  • Polynomials: Any polynomial is entire, with a finite set of zeros and a simple kind of global behavior: the associated function on the Riemann sphere has a pole at infinity whose order equals the degree of the polynomial.

  • Other classical examples: Entire transcendental functions such as exponential function and its compositions, as well as various special functions arising in analysis and physics, all live in this broad class.

Singularity at infinity and global behavior

  • Infinity as a lens: The behavior of an entire function as |z| → ∞ is captured by the nature of its singularity at infinity on the Riemann sphere. Constants have removable infinity, polynomials have poles at infinity, and non-polynomial entire functions have essential singularities there, which by the Casorati–Weierstrass theorem implies dense and varied value distribution near infinity.

  • Value distribution: The Casorati–Weierstrass theorem and related results (including Picard-type statements) describe how non-polynomial entire functions can come arbitrarily close to almost any complex value in neighborhoods of the essential singularity at infinity, underscoring the rich global behavior possible in the entire setting.

  • Implications for global theory: The dichotomy between polynomials and transcendental entire functions shapes many central questions in complex analysis, including how zeros, growth, and value distribution fit together in a single global framework on the plane.

See also