Entire function
Encyclopedia
In complex analysis
, an entire function, also called an integral function, is a complex-valued function
that is holomorphic
over the whole complex plane
. Typical examples of entire functions are the polynomial
s and the exponential function
, and any sums, products and compositions of these, including the error function
and the trigonometric function
s sine
and cosine and their hyperbolic counterparts
the hyperbolic sine and hyperbolic cosine functions. Neither the natural logarithm
nor the square root
functions can be continued analytically
to an entire function.
A transcendental entire function is an entire function that is not a polynomial (see transcendental function
).
. The Weierstrass factorization theorem
asserts that any entire function can be represented by a product involving its zeroes
.
The entire functions on the complex plane form an integral domain (in fact a Prüfer domain
).
Liouville's theorem
states that any bounded
entire function must be constant. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra
.
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere
(complex plane and the point at infinity) is constant. Thus any non-constant entire function must have a singularity
at the complex point at infinity, either a pole for a polynomial or an essential singularity
for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence
with
and 
.
Picard's little theorem
is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception. The latter exception is illustrated by the exponential function
, which never takes on the value 0.
Liouville's theorem is a special case of the following statement: any entire function f satisfying the inequality
for all z with 
, with n a natural number and M and R positive constants, is necessarily a polynomial, of degree
at most n. Conversely, any entire function f satisfying the inequality
for all z with 
, with n a natural number and M and R positive constants, is necessarily a polynomial, of degree at least n.
is defined using the limit superior as:
where
is the disk of radius 
and 
denotes the supremum norm of 
on 
If 
one can also define the type:
In other words, the order of
is the infimum of all m such that 
as 
. The order need not be finite.
Entire functions may grow as fast as any increasing function: for any increasing function
there exists an entire function 
such that 
for all real 
. Such a function 
may be easily found of the form:
,
for a conveniently chosen strictly increasing sequence of positive integers
. Any such sequence defines an entire series 
; and if it is conveniently chosen, the inequality 
also holds, for all real 
.
as a 'typical' entire function in one of his books. Other examples include the Fresnel integral
s, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and the error function are special cases of the Mittag-Leffler function
.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, an entire function, also called an integral function, is a complex-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
over the whole complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Typical examples of entire functions are the polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s and the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
, and any sums, products and compositions of these, including the error function
Error function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
and the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
and cosine and their hyperbolic counterparts
Hyperbolic function
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...
the hyperbolic sine and hyperbolic cosine functions. Neither the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
nor the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
functions can be continued analytically
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
to an entire function.
A transcendental entire function is an entire function that is not a polynomial (see transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
).
Properties
Every entire function can be represented as a power series that converges uniformly on compact setsCompact convergence
In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...
. The Weierstrass factorization theorem
Weierstrass factorization theorem
In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes...
asserts that any entire function can be represented by a product involving its zeroes
Zero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
.
The entire functions on the complex plane form an integral domain (in fact a Prüfer domain
Prüfer domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules...
).
Liouville's theorem
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
states that any bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
entire function must be constant. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
.
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
(complex plane and the point at infinity) is constant. Thus any non-constant entire function must have a singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
at the complex point at infinity, either a pole for a polynomial or an essential singularity
Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
with
and .Picard's little theorem
Picard theorem
In complex analysis, the term Picard theorem refers to either of two distinct yet related theorems, both of which pertain to the range of an analytic function.-Little Picard:...
is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception. The latter exception is illustrated by the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
, which never takes on the value 0.
Liouville's theorem is a special case of the following statement: any entire function f satisfying the inequality
for all z with , with n a natural number and M and R positive constants, is necessarily a polynomial, of degreeDegree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
at most n. Conversely, any entire function f satisfying the inequality
for all z with , with n a natural number and M and R positive constants, is necessarily a polynomial, of degree at least n.Order and growth
The order (at infinity) of an entire function is defined using the limit superior as:where
is the disk of radius and denotes the supremum norm of on If one can also define the type:In other words, the order of
is the infimum of all m such that as . The order need not be finite.Entire functions may grow as fast as any increasing function: for any increasing function
there exists an entire function such that for all real . Such a function may be easily found of the form:,for a conveniently chosen strictly increasing sequence of positive integers
. Any such sequence defines an entire series ; and if it is conveniently chosen, the inequality also holds, for all real .Other examples
J. E. Littlewood chose the Weierstrass sigma functionWeierstrass sigma function
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.-Weierstrass sigma-function:...
as a 'typical' entire function in one of his books. Other examples include the Fresnel integral
Fresnel integral
250px|thumb|S and C The maximum of C is about 0.977451424. If πt²/2 were used instead of t², then the image would be scaled vertically and horizontally ....
s, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and the error function are special cases of the Mittag-Leffler function
Mittag-Leffler function
In mathematics, the Mittag-Leffler function Eα,β is a special function, a complex function which depends on two complex parameters α and β...
.
See also
- Jensen's formulaJensen's formulaIn the mathematical field known as complex analysis, Jensen's formula, named after Johan Jensen, relates the average magnitude of an analytic function on a circle with the magnitudes of its zeros inside the circle...
- Carlson's theoremCarlson's theoremIn mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem about a summable expansion of an analytic function. It is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for expansions in other bases of...
- Exponential type
- Paley–Wiener theoremPaley–Wiener theoremIn mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley and Norbert Wiener . The original theorems did not use the language of distributions,...