In
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
, an
entire function, also called an
integral function, is a complex-valued
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
that is
holomorphicIn 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 planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis...
. Typical examples of entire functions are the
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
s and the
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change ...
, and any sums, products and compositions of these, including the
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations...
and the
trigonometric functionIn 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
sineMaurice Sinet, known as Siné is a French cartoonist.As a young man he studied drawing and graphic arts, while earning a living as a cabaret singer. After his military service he started publishing his drawings and also worked as a photo-retoucher for porn magazines. His first published drawing...
and cosine and their
hyperbolic counterpartsIn mathematics, the 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", etc., in analogy to the derived...
the hyperbolic sine and hyperbolic cosine functions.
A
transcendental entire function is an entire function that is not a polynomial (see
transcendental functionA 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...
).
Every entire function can be represented as a
power seriesIn mathematics, a power series is an infinite series of the formwhere a
n represents the coefficient of the nth term, c is a constant, and x varies around c...
which
converges compactlyIn mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. It is associated with the compact-open topology.-Definition:...
.
In
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
, an
entire function, also called an
integral function, is a complex-valued
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
that is
holomorphicIn 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 planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis...
. Typical examples of entire functions are the
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
s and the
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change ...
, and any sums, products and compositions of these, including the
error functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations...
and the
trigonometric functionIn 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
sineMaurice Sinet, known as Siné is a French cartoonist.As a young man he studied drawing and graphic arts, while earning a living as a cabaret singer. After his military service he started publishing his drawings and also worked as a photo-retoucher for porn magazines. His first published drawing...
and cosine and their
hyperbolic counterpartsIn mathematics, the 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", etc., in analogy to the derived...
the hyperbolic sine and hyperbolic cosine functions.
A
transcendental entire function is an entire function that is not a polynomial (see
transcendental functionA 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 seriesIn mathematics, a power series is an infinite series of the formwhere a
n represents the coefficient of the nth term, c is a constant, and x varies around c...
which
converges compactlyIn 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 theoremIn 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
zeroesIn 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 aswhere g is a holomorphic function g such that g is not zero.Generally, the...
. Neither the
natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718
281828...
nor the
square rootIn mathematics, a square root of a number x is a number r such that r
2 = x, or, in other words, a number r whose square is x....
functions can be
continued analyticallyIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function...
to an entire function.
The entire functions on the complex plane form a
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
(in fact a
Prüfer domainIn 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...
).
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
degreeWhen a polynomial is expressed as a sum or difference of terms , the exponent of the term with the highest exponent is the degree of the polynomial. The degree of a term is the sum of the powers of each variable in the term. The words degree and order are used interchangeably.For example, the...
at most
n.
The special case
n = 0 is called
Liouville's theoremIn 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...
: any
boundedIn 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 < ∞ such thatfor all x in X....
entire function must be constant. Liouville's theorem may be used to elegantly prove the
fundamental theorem of algebraIn mathematics, 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 which is entire on the whole
Riemann sphereIn mathematics, the Riemann sphere is a way of extending the plane of complex numbers with one additional point at infinity, in a way that makes expressions such as...
(complex plane
and the point at infinity) is constant. Thus any non-constant entire function must have a
singularityIn 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 infinityThe point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line,...
, either a
poleIn the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at...
for a polynomial or an
essential singularityIn complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.Basically, the category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into...
for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function
f and any complex
w there is a
sequenceIn 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...
with and .
Picard's little theoremIn 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, except possibly one. The latter exception is illustrated by the
exponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change ...
, which never takes on the value 0.
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 of an entire function is defined using the limit superior as:
where is the distance from and is the maximum
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
of when 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 choosen strictly increasing sequence of positive integers . Any such sequence defines an entire series g(z); and if it is conveniently choosen, the inequality also holds, for all real .
Other examples
J. E. Littlewood chose the
Weierstrass 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 integral250px|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 functionIn mathematics, the reciprocal Gamma function is the functionwhere denotes the Gamma function. Since the Gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function...
. The exponential function and the error function are special cases of the
Mittag-Leffler functionIn 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 formula
In complex analysis, a discipline within mathematics, 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 theorem
In 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 theorem
In 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,...