In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and particularly in the field of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the Weierstrass factorization theorem asserts that every
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
can be represented as a (possibly infinite) product involving its
zeroes
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
. The theorem may be viewed as an extension of 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 polynomial ...
, which asserts that every polynomial may be factored into linear factors, one for each root.
The theorem, which is named for
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.
A generalization of the theorem extends it to
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
s and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's
zeros and poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if i ...
, and an associated non-zero
holomorphic function
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 derivativ ...
.
Motivation
The consequences of 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 polynomial ...
are twofold.
[.]
Firstly, any finite sequence
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
has an associated
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 exa ...
that has
zeroes
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
precisely at the points of that
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 ...
,
Secondly, any polynomial function
in the complex plane has a
factorization
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
where is a non-zero constant and are the zeroes of .
The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of additional terms in the product is demonstrated when one considers
where the sequence
is not
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
. It can never define an entire function, because the
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
does not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra.
A necessary condition for convergence of the infinite product in question is that for each z, the factors
must approach 1 as
. So it stands to reason that one should seek a function that could be 0 at a prescribed point, yet remain near 1 when not at that point and furthermore introduce no more zeroes than those prescribed.
Weierstrass' ''elementary factors'' have these properties and serve the same purpose as the factors
above.
The elementary factors
Consider the functions of the form
for
. At
, they evaluate to
and have a flat slope at order up to
. Right after
, they sharply fall to some small positive value. In contrast, consider the function
which has no flat slope but, at
, evaluates to exactly zero. Also note that for ,
:
![First_5_Weierstrass_factors_on_the_unit_interval](https://upload.wikimedia.org/wikipedia/commons/1/15/First_5_Weierstrass_factors_on_the_unit_interval.svg)
The ''elementary factors'',
also referred to as ''primary factors'',
[, chapter 2.]
are functions that combine the properties of zero slope and zero value (see graphic):
:
For and
, one may express it as
and one can read off how those properties are enforced.
The utility of the elementary factors lies in the following lemma:
Lemma (15.8, Rudin) for ,
:
The two forms of the theorem
Existence of entire function with specified zeroes
Let
be a sequence of non-zero complex numbers such that
.
If
is any sequence of nonnegative integers such that for all
,
:
then the function
:
is entire with zeros only at points
. If a number
occurs in the sequence
exactly times, then function has a zero at
of multiplicity .
* The sequence
in the statement of the theorem always exists. For example, we could always take
and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence , will not break the convergence.
* The theorem generalizes to the following:
sequences
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 called t ...
in
open subsets
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
(and hence
regions
In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
) of the
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 pl ...
have associated functions that are
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 derivativ ...
in those subsets and have zeroes at the points of the sequence.
* Also the case given by the fundamental theorem of algebra is incorporated here. If the sequence
is finite then we can take
and obtain:
.
The Weierstrass factorization theorem
Let be an entire function, and let
be the non-zero zeros of repeated according to multiplicity; suppose also that has a zero at of order (a zero of order at means ).
Then there exists an entire function and a sequence of integers
such that
:
Examples of factorization
The trigonometric functions
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and
cosine have the factorizations
while the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
has factorization
is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
. The cosine identity can be seen as special case of
for
.
Hadamard factorization theorem
If is an entire function of finite
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
and is the order of the zero of at , then it admits a factorization
:
where is a polynomial of degree , and is the integer part of .
See also
*
Mittag-Leffler's theorem In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factor ...
*
Wallis product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
:\begin
\frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\fr ...
, which can be derived from this theorem applied to the sine function
*
Blaschke product
In complex analysis, the Blaschke product is a bounded analytic function in the open unit disc constructed to have zeros at a (finite or infinite) sequence of prescribed complex numbers
:''a''0, ''a''1, ...
inside the unit disc, with the property ...
Notes
External links
*
* {{webarchive , url=https://web.archive.org/web/20181130113058/https://giphy.com/gifs/math-visualization-algorithm-xThuW9Pyh8jXvfbrUc , date=30 November 2018 , title=Visualization of the Weierstrass factorization of the sine function due to Euler
Theorems in complex analysis