Template:Short description In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.
The theorem, which is named for Karl Weierstrass, 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 functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function.Template:Citation needed
MotivationEdit
It is clear that any finite set <math>\{c_n\}</math> of points in the complex plane has an associated polynomial <math display="inline">p(z) = \prod_n (z-c_n)</math> whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function <math>p(z)</math> in the complex plane has a factorization <math display="inline">p(z) = a\prod_n(z-c_n),</math> where Template:Math is a non-zero constant and <math>\{c_n\}</math> is the set of zeroes of <math>p(z)</math>.<ref name="knopp">Template:Citation.</ref>
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 <math display="inline">\prod_n (z-c_n)</math> where the sequence <math>\{c_n\}</math> is not finite. It can never define an entire function, because the infinite product 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. Instead, the theorem replaces these with other factors.
A necessary condition for convergence of the infinite product in question is that for each <math>z</math>, the factors replacing <math> (z-c_n) </math> must approach 1 as <math>n\to\infty</math>. So it stands to reason that one should seek factor functions 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 <math> (z-c_n) </math> above.
Elementary factorsEdit
Consider the functions of the form <math display="inline">\exp\left(-\tfrac{z^{n+1}}{n+1}\right)</math> for <math>n \in \mathbb{N}</math>. At <math>z=0</math>, they evaluate to <math>1</math> and have a flat slope at order up to <math>n</math>. Right after <math>z=1</math>, they sharply fall to some small positive value. In contrast, consider the function <math>1-z</math> which has no flat slope but, at <math>z=1</math>, evaluates to exactly zero. Also note that for Template:Math,
- <math>(1-z) = \exp(\ln(1-z)) = \exp \left( -\tfrac{z^1}{1} - \tfrac{z^2}{2} - \tfrac{z^3}{3} + \cdots \right).</math>
The elementary factors,<ref name="rudin">Template:Citation</ref> also referred to as primary factors,<ref name="boas">Template:Citation, chapter 2.</ref> are functions that combine the properties of zero slope and zero value (see graphic):
- <math>E_n(z) = \begin{cases} (1-z) & \text{if }n=0, \\ (1-z)\exp \left( \frac{z^1}{1}+\frac{z^2}{2}+\cdots+\frac{z^n}{n} \right) & \text{otherwise}. \end{cases} </math>
For Template:Math and <math>n>0</math>, one may express it as <math display="inline">E_n(z)=\exp\left(-\tfrac{z^{n+1}}{n+1}\sum_{k=0}^\infty\tfrac{z^k}{1+k/(n+1)}\right)</math> and one can read off how those properties are enforced.
The utility of the elementary factors <math display="inline">E_n(z)</math> lies in the following lemma:<ref name="rudin"/>
Lemma (15.8, Rudin) for Template:Math, <math>n \in \mathbb{N}</math>
- <math>\vert 1 - E_n(z) \vert \leq \vert z \vert^{n+1}.</math>
Existence of entire function with specified zeroesEdit
Let <math>\{a_n\}</math> be a sequence of non-zero complex numbers such that <math>|a_n|\to\infty</math>. If <math>\{p_n\}</math> is any sequence of nonnegative integers such that for all <math>r>0</math>,
- <math> \sum_{n=1}^\infty \left( r/|a_n|\right)^{1+p_n} < \infty,</math>
then the function
- <math>E(z) = \prod_{n=1}^\infty E_{p_n}(z/a_n)</math>
is entire with zeros only at points <math>a_n</math>.<ref name="rudin"/> If a number <math>z_0</math> occurs in the sequence <math>\{a_n\}</math> exactly Template:Math times, then the function Template:Math has a zero at <math>z=z_0</math> of multiplicity Template:Math.
- The sequence <math>\{p_n\}</math> in the statement of the theorem always exists. For example, we could always take <math>p_n=n</math> and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence Template:Math, will not break the convergence.
- The theorem generalizes to the following: sequences in open subsets (and hence regions) of the Riemann sphere have associated functions that are holomorphic in those subsets and have zeroes at the points of the sequence.<ref name="rudin"/>
Weierstrass factorization theoremEdit
Let Template:Math be an entire function, and let <math>\{a_n\}</math> be the non-zero zeros of Template:Math repeated according to multiplicity; suppose also that Template:Math has a zero at Template:Math of order Template:Math.Template:Efn Then there exists an entire function Template:Math and a sequence of integers <math>\{p_n\}</math> such that
- <math>f(z)=z^m e^{g(z)} \prod_{n=1}^\infty E_{p_n}\!\!\left(\frac{z}{a_n}\right).</math><ref name="conway">Template:Citation</ref>
The case given by the fundamental theorem of algebra is incorporated here. If the sequence <math>\{a_n\}</math> is finite then we can take <math>p_n = 0</math>, <math>m=0</math> and <math>e^{g(z)}=c</math> to obtain <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.
Examples of factorizationEdit
The trigonometric functions sine and cosine have the factorizations <math display=block>\sin \pi z = \pi z \prod_{n\neq 0} \left(1-\frac{z}{n}\right)e^{z/n} = \pi z\prod_{n=1}^\infty \left(1-\left(\frac{z}{n}\right)^2\right)</math> <math display=block>\cos \pi z = \prod_{q \in \mathbb{Z}, \, q \; \text{odd} } \left(1-\frac{2z}{q}\right)e^{2z/q} = \prod_{n=0}^\infty \left( 1 - \left(\frac{z}{n+\tfrac{1}{2}} \right)^2 \right) </math> while the gamma function <math>\Gamma</math> has factorization <math display=block>\frac{1}{\Gamma (z)}=e^{\gamma z}z\prod_{n=1}^{\infty }\left ( 1+\frac{z}{n} \right )e^{-z/n},</math> where <math>\gamma</math> is the Euler–Mascheroni constant.Template:Citation needed The cosine identity can be seen as special case of <math display=block>\frac{1}{\Gamma(s-z)\Gamma(s+z)} = \frac{1}{\Gamma(s)^2}\prod_{n=0}^\infty \left( 1 - \left(\frac{z}{n+s} \right)^2 \right) </math> for <math>s=\tfrac{1}{2}</math>.Template:Citation needed
Hadamard factorization theoremEdit
A special case of the Weierstraß factorization theorem occurs for entire functions of finite order. In this case the <math>p_n</math> can be taken independent of <math>n</math> and the function <math>g(z)</math> is a polynomial. Thus <math display="block">f(z)=z^me^{P(z)}\prod_{k=1}^\infty E_p(z/a_k)</math>where <math>a_k</math> are those roots of <math>f</math> that are not zero (<math>a_k \neq 0</math>), <math>m</math> is the order of the zero of <math>f</math> at <math>z = 0</math> (the case <math>m = 0</math> being taken to mean <math>f(0) \neq 0</math>), <math>P</math> a polynomial (whose degree we shall call <math>q</math>), and <math>p</math> is the smallest non-negative integer such that the series<math display="block">\sum_{n=1}^\infty\frac{1}{|a_n|^{p+1}}</math>converges. This is called Hadamard's canonical representation.<ref name="conway" /> The non-negative integer <math>g=\max\{p,q\}</math> is called the genus of the entire function <math>f</math>. The order <math>\rho</math> of <math>f</math> satisfies <math display="block">g \leq \rho \leq g + 1</math> In other words: If the order <math>\rho</math> is not an integer, then <math>g = [ \rho ]</math> is the integer part of <math>\rho</math>. If the order is a positive integer, then there are two possibilities: <math>g = \rho-1</math> or <math>g = \rho </math>.
For example, <math>\sin</math>, <math>\cos</math> and <math>\exp</math> are entire functions of genus <math>g = \rho = 1</math>.
See alsoEdit
- Mittag-Leffler's theorem
- Wallis product, which can be derived from this theorem applied to the sine function
- Blaschke product
NotesEdit
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