In
complex analysis, an entire function, also called an integral function, is a complex-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on the whole
complex plane. Typical examples of entire functions are
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s and the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, and any finite sums, products and compositions of these, such as the
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s
sine and
cosine and their
hyperbolic counterparts sinh and
cosh, as well as
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s and
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s of entire functions such as the
error function. If an entire function has a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
at , then , taking the limit value at , is an entire function. On the other hand, the
natural logarithm, the
reciprocal function
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
, and the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
are all not entire functions, nor can they be
continued analytically to an entire function.
A
transcendental entire function is an entire function that is not a polynomial.
Properties
Every entire function can be represented as a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
that converges everywhere in the complex plane, hence
uniformly on compact sets. The
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
is infinite, which implies that
or
Any power series satisfying this criterion will represent an entire function.
If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of will be the complex conjugate of the value at . Such functions are sometimes called self-conjugate (the conjugate function,
, being given by
If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane,
up to an imaginary constant. For instance, if the real part is known in a neighborhood of zero, then we can find the coefficients for ''n'' > 0 from the following derivatives with respect to a real variable ''r'':
(Likewise, if the imaginary part is known in a
neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant. (For instance, if the real part is known on part of the unit circle, then it is known on the whole unit circle by
analytic extension
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
, and then the coefficients of the infinite series are determined from the coefficients of the
Fourier series for the real part on the unit circle.) Note however that an entire function is ''not'' determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add ''i'' times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some imaginary number.
The
Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its
zeroes (or "roots").
The entire functions on the complex plane form an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
(in fact a
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 gen ...
). They also form a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
unital associative algebra over the complex numbers.
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
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
.
As a consequence of Liouville's theorem, any function that is entire on the whole
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
(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
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
, either a
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
for a polynomial or an
essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
for a
transcendental entire function. Specifically, by the
Casorati–Weierstrass theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian ...
, for any transcendental entire function and any complex there is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
such that
:
Picard's little theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb ...
is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a
lacunary value In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex number, complex-valued function (mathematics), function defined on a subset of the complex plane is a complex number which is not in the image (mathematics), image ...
of the function. The possibility of a lacunary value is illustrated by the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, which never takes on the value 0. One can take a suitable branch of the logarithm of an entire function that never hits 0, so that this will also be an entire function (according to the
Weierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.
Liouville's theorem is a special case of the following statement:
Growth
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 constant and a strictly increasing sequence of positive integers . Any such sequence defines an entire function , and if the powers are chosen appropriately we may satisfy the inequality for all real . (For instance, it certainly holds if one chooses and, for any integer
one chooses an even exponent
such that
).
Order and type
The order (at infinity) of an entire function
is defined using the
limit superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
as:
where is the disk of radius and
denotes the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when th ...
of
on . The order is a non-negative real number or infinity (except when
for all ). In other words, the order of
is the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of all such that:
The example of
shows that this does not mean ''f''(''z'') = O(exp(, ''z'',
''m'')) if
is of order ''m''.
If
one can also define the ''type'':
If the order is 1 and the type is , the function is said to be "of
exponential type
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function ''e'C'', ''z'', for some real-valued constant ''C'' as , ''z'', → ∞ ...
". If it is of order less than 1 it is said to be of exponential type 0.
If
then the order and type can be found by the formulas
Let
denote the -th derivative of , then we may restate these formulas in terms of the derivatives at any arbitrary point :
The type may be infinite, as in the case of the
reciprocal gamma function
In mathematics, the reciprocal gamma function is the function
:f(z) = \frac,
where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an ...
, or zero (see example below under ).
Examples
Here are some examples of functions of various orders:
Order ''ρ''
For arbitrary positive numbers and one can construct an example of an entire function of order and type using:
Order 0
* Non-zero polynomials
*
Order 1/4
where
Order 1/3
where
Order 1/2
with ''a'' ≠ 0 (for which the type is given by ''σ'' = , ''a'', )
Order 1
* with ()
*
*
*the
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
*the
reciprocal gamma function
In mathematics, the reciprocal gamma function is the function
:f(z) = \frac,
where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an ...
( is infinite)
*
Order 3/2
*
Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
Order 2
* with ()
*The
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathe ...
(σ is infinite).
Order infinity
*
Genus
Entire functions of finite order have
Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
's canonical representation:
where
are those
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of
that are not zero (
),
is the order of the zero of
at
(the case
being taken to mean
),
a polynomial (whose degree we shall call
), and
is the smallest non-negative integer such that the series
converges. The non-negative integer
is called the genus of the entire function
.
If the order ρ is not an integer, then