Weierstrass factorization theorem

It is clear that any finite set $\backslash \{c\_n\backslash \}$ of points in the complex plane has an associated polynomial $p(z)\; =\; \backslash prod\_n\; (z-c\_n)$ whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function $p(z)$ in the complex plane has a factorization

$p(z)\; =\; a\backslash prod\_n(z-c\_n),$

where {{math|a}} is a non-zero constant and $\backslash \{c\_n\backslash \}$ is the set of zeroes of $p(z)$.{{citation |last=Knopp |first=K. |title=Theory of Functions, Part II |pages=1–7 |year=1996 |contribution=Weierstrass's Factor-Theorem |location=New York |publisher=Dover}}.

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 $\backslash prod\_n\; (z-c\_n)$ where the sequence $\backslash \{c\_n\backslash \}$ 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 $z$, the factors replacing $(z-c\_n)$ must approach 1 as $n\backslash to\backslash infty$. 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 $(z-c\_n)$ above.

Consider the functions of the form $\backslash exp\backslash left(-\backslash tfrac\{z^\{n+1\}\}\{n+1\}\backslash right)$ for $n\; \backslash in\; \backslash mathbb\{N\}$. At $z=0$, they evaluate to $1$ and have a flat slope at order up to $n$. Right after $z=1$, they sharply fall to some small positive value. In contrast, consider the function $1-z$ which has no flat slope but, at $z=1$, evaluates to exactly zero. Also note that for {{math|{{abs|z}} < 1}},

:$(1-z)\; =\; \backslash exp(\backslash ln(1-z))\; =\; \backslash exp\; \backslash left(\; -\backslash tfrac\{z^1\}\{1\}\; -\; \backslash tfrac\{z^2\}\{2\}\; -\; \backslash tfrac\{z^3\}\{3\}\; +\; \backslash cdots\; \backslash right).$

File:First_5_Weierstrass_factors_on_the_unit_interval.svg

The elementary factors,{{citation|last=Rudin|first=W.|title=Real and Complex Analysis|edition=3rd|url=https://perso.telecom-paristech.fr/decreuse/_downloads/c22155fef582344beb326c1f44f437d2/rudin.pdf|publisher=McGraw Hill|location=Boston|pages=299–304|year=1987|isbn=0-07-054234-1|oclc=13093736}}

also referred to as primary factors,{{citation|last=Boas|first=R. P.|title=Entire Functions|publisher=Academic Press Inc.|location=New York|year=1954|isbn=0-8218-4505-5|oclc=6487790}}, chapter 2.

are functions that combine the properties of zero slope and zero value (see graphic):

:

For {{math|{{abs|z}} < 1}} and $n>0$, one may express it as

$E\_n(z)=\backslash exp\backslash left(-\backslash tfrac\{z^\{n+1\}\}\{n+1\}\backslash sum\_\{k=0\}^\backslash infty\backslash tfrac\{z^k\}\{1+k/(n+1)\}\backslash right)$ and one can read off how those properties are enforced.

The utility of the elementary factors $E\_n(z)$ lies in the following lemma:

Lemma (15.8, Rudin) for {{math|{{abs|z}} ≤ 1}}, $n\; \backslash in\; \backslash mathbb\{N\}$

:$\backslash vert\; 1\; -\; E\_n(z)\; \backslash vert\; \backslash leq\; \backslash vert\; z\; \backslash vert^\{n+1\}.$

Let $\backslash \{a\_n\backslash \}$ be a sequence of non-zero complex numbers such that $|a\_n|\backslash to\backslash infty$.

If $\backslash \{p\_n\backslash \}$ is any sequence of nonnegative integers such that for all $r>0$,

:

then the function

: $E(z)\; =\; \backslash prod\_\{n=1\}^\backslash infty\; E\_\{p\_n\}(z/a\_n)$

is entire with zeros only at points $a\_n$. If a number $z\_0$ occurs in the sequence $\backslash \{a\_n\backslash \}$ exactly {{math|m}} times, then the function {{math|E}} has a zero at $z=z\_0$ of multiplicity {{math|m}}.

• The sequence $\backslash \{p\_n\backslash \}$ in the statement of the theorem always exists. For example, we could always take $p\_n=n$ and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence {{math|p′_n ≥ p_n}}, 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.

Let {{math|ƒ}} be an entire function, and let $\backslash \{a\_n\backslash \}$ be the non-zero zeros of {{math|ƒ}} repeated according to multiplicity; suppose also that {{math|ƒ}} has a zero at {{math|1=z = 0}} of order {{math|m ≥ 0}}.{{efn|A zero of order {{math|1=m = 0}} at {{math|1=z = 0}} is taken to mean {{math|ƒ(0) ≠ 0}} — that is, $f$ does not have a zero at $0$.}}

Then there exists an entire function {{math|g}} and a sequence of integers $\backslash \{p\_n\backslash \}$ such that

: $f(z)=z^m\; e^\{g(z)\}\; \backslash prod\_\{n=1\}^\backslash infty\; E\_\{p\_n\}\backslash !\backslash !\backslash left(\backslash frac\{z\}\{a\_n\}\backslash right).${{citation|last=Conway|first=J. B.|title=Functions of One Complex Variable I, 2nd ed.|publisher=Springer|location=springer.com|year=1995|isbn=0-387-90328-3}}

The case given by the fundamental theorem of algebra is incorporated here. If the sequence $\backslash \{a\_n\backslash \}$ is finite then we can take $p\_n\; =\; 0$, $m=0$ and $e^\{g(z)\}=c$ to obtain $\backslash ,\; f(z)\; =\; c\backslash ,\{\backslash displaystyle\backslash prod\}\_n\; (z-a\_n)$.

The trigonometric functions sine and cosine have the factorizations

$$\backslash sin\; \backslash pi\; z\; =\; \backslash pi\; z\; \backslash prod\_\{n\backslash neq\; 0\}\; \backslash left(1-\backslash frac\{z\}\{n\}\backslash right)e^\{z/n\}\; =\; \backslash pi\; z\backslash prod\_\{n=1\}^\backslash infty\; \backslash left(1-\backslash left(\backslash frac\{z\}\{n\}\backslash right)^2\backslash right)$$

$$\backslash cos\; \backslash pi\; z\; =\; \backslash prod\_\{q\; \backslash in\; \backslash mathbb\{Z\},\; \backslash ,\; q\; \backslash ;\; \backslash text\{odd\}\; \}\; \backslash left(1-\backslash frac\{2z\}\{q\}\backslash right)e^\{2z/q\}\; =\; \backslash prod\_\{n=0\}^\backslash infty\; \backslash left(\; 1\; -\; \backslash left(\backslash frac\{z\}\{n+\backslash tfrac\{1\}\{2\}\}\; \backslash right)^2\; \backslash right)$$

while the gamma function $\backslash Gamma$ has factorization

$$\backslash frac\{1\}\{\backslash Gamma\; (z)\}=e^\{\backslash gamma\; z\}z\backslash prod\_\{n=1\}^\{\backslash infty\; \}\backslash left\; (\; 1+\backslash frac\{z\}\{n\}\; \backslash right\; )e^\{-z/n\},$$

where $\backslash gamma$ is the Euler–Mascheroni constant.{{citation needed|date=April 2019}} The cosine identity can be seen as special case of

$$\backslash frac\{1\}\{\backslash Gamma(s-z)\backslash Gamma(s+z)\}\; =\; \backslash frac\{1\}\{\backslash Gamma(s)^2\}\backslash prod\_\{n=0\}^\backslash infty\; \backslash left(\; 1\; -\; \backslash left(\backslash frac\{z\}\{n+s\}\; \backslash right)^2\; \backslash right)$$

for $s=\backslash tfrac\{1\}\{2\}$.{{citation needed|date=April 2019}}

{{Main page|Hadamard factorization theorem}}

A special case of the Weierstraß factorization theorem occurs for entire functions of finite order. In this case the $p\_n$ can be taken independent of $n$ and the function $g(z)$ is a polynomial. Thus $$f(z)=z^me^\{P(z)\}\backslash prod\_\{k=1\}^\backslash infty\; E\_p(z/a\_k)$$where $a\_k$ are those roots of $f$ that are not zero ($a\_k\; \backslash neq\; 0$), $m$ is the order of the zero of $f$ at $z\; =\; 0$ (the case $m\; =\; 0$ being taken to mean $f(0)\; \backslash neq\; 0$), $P$ a polynomial (whose degree we shall call $q$), and $p$ is the smallest non-negative integer such that the series$$\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{1\}\{|a\_n|^\{p+1\}\}$$converges. This is called Hadamard's canonical representation. The non-negative integer $g=\backslash max\backslash \{p,q\backslash \}$ is called the genus of the entire function $f$. The order $\backslash rho$ of $f$ satisfies $$g\; \backslash leq\; \backslash rho\; \backslash leq\; g\; +\; 1$$

In other words: If the order $\backslash rho$ is not an integer, then $g\; =\; [\; \backslash rho\; ]$ is the integer part of $\backslash rho$. If the order is a positive integer, then there are two possibilities: $g\; =\; \backslash rho-1$ or $g\; =\; \backslash rho$.

For example, $\backslash sin$, $\backslash cos$ and $\backslash exp$ are entire functions of genus $g\; =\; \backslash rho\; =\; 1$.

• Mittag-Leffler's theorem
• Wallis product, which can be derived from this theorem applied to the sine function
• Blaschke product

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• {{springer|title=Weierstrass theorem|id=p/w097510}}

• {{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}}

Category:Theorems in complex analysis