10.3 Weierstrass factorization theorem

The Weierstrass factorization theorem is a powerful tool in complex analysis. It states that every entire function can be expressed as a product involving its zeros, providing insight into the function's behavior and structure.

This theorem connects the zeros of an entire function to its overall properties. It's crucial for understanding how entire functions behave and how their zeros influence their characteristics. The theorem also relates to other important results in complex analysis.

Definition of Weierstrass factorization theorem

• States that every entire function can be represented as a product involving its zeros
• Provides a way to express entire functions as an infinite product of factors, each involving a zero of the function
• Fundamental result in complex analysis connects the zeros of an entire function to its overall behavior

Entire functions vs meromorphic functions

• Entire functions are complex-valued functions that are holomorphic (differentiable) on the entire complex plane
• Meromorphic functions are complex-valued functions that are holomorphic on the complex plane except for a set of isolated points called poles
• Weierstrass factorization theorem applies specifically to entire functions, while meromorphic functions have a similar representation called the Mittag-Leffler theorem

Zeros of entire functions

• Points in the complex plane where an entire function takes the value zero
• Can be isolated or have accumulation points
• Play a crucial role in the behavior and properties of entire functions

Order of a zero

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• Measures the multiplicity or degree of a zero of an entire function
• Determined by the number of derivatives of the function that vanish at the zero
• Higher-order zeros contribute more to the product representation in Weierstrass factorization

Multiplicity of zeros

• Counts the number of times a zero appears in the product representation
• Zeros with multiplicity greater than one are called multiple zeros
• Affects the convergence and form of the canonical products used in Weierstrass factorization

Hadamard factorization theorem

• Refinement of the Weierstrass factorization theorem for entire functions of finite order
• States that an entire function of finite order can be represented as a product of its zeros, a polynomial, and an exponential factor
• Provides additional information about the growth and distribution of zeros

Relationship to Weierstrass factorization

• Hadamard factorization is a special case of Weierstrass factorization for entire functions of finite order
• Incorporates the order of the entire function into the factorization
• Allows for more precise statements about the convergence and growth of the product representation

Canonical products

• Building blocks used in the construction of the Weierstrass factorization
• Infinite products involving the zeros of an entire function and certain exponential factors
• Ensure the convergence of the infinite product representation

Definition of canonical products

• Canonical products are of the form $\prod_{n=1}^{\infty} E_{p_n}(z/a_n)$, where $a_n$ are the zeros of the entire function and $E_{p_n}$ are the primary factors
• Primary factors are chosen based on the order and multiplicity of the zeros to ensure convergence
• Examples of primary factors include $(1-z)$, $(1-z)e^z$, and $(1-z)e^{z+z^2/2}$

Convergence of canonical products

• Convergence of the canonical product is crucial for the validity of the Weierstrass factorization
• Depends on the choice of primary factors and the distribution of zeros
• Sufficient conditions for convergence involve the order and multiplicity of the zeros

Types of canonical products

• Canonical products of genus 0, 1, or 2, depending on the growth of the zeros
• Genus determines the choice of primary factors and the form of the exponential factor in the Hadamard factorization
• Examples: $\sin(z)$ has a canonical product of genus 1, while $\cos(z)$ has a canonical product of genus 0

Proof of Weierstrass factorization theorem

• Involves constructing a canonical product with the same zeros as the given entire function
• Shows that the ratio of the entire function to the canonical product is an entire function without zeros
• Concludes that the ratio must be an exponential function, leading to the desired factorization

Key steps in the proof

• Enumerate the zeros of the entire function and construct the canonical product
• Prove that the canonical product converges uniformly on compact subsets of the complex plane
• Show that the ratio of the entire function to the canonical product is an entire function without zeros
• Use Liouville's theorem to conclude that the ratio is an exponential function

Role of canonical products in the proof

• Canonical products provide a way to construct an infinite product with the same zeros as the entire function
• Ensure the convergence of the infinite product and the existence of the ratio function
• Allow for the factorization of the entire function into a product of factors involving its zeros and an exponential term

Applications of Weierstrass factorization

• Provides a powerful tool for studying the properties and behavior of entire and meromorphic functions
• Allows for the representation of functions in terms of their zeros, which can simplify certain problems
• Useful in various areas of complex analysis, such as the study of distribution of zeros, growth estimates, and interpolation problems

Representation of entire functions

• Weierstrass factorization gives a canonical way to represent entire functions as infinite products
• Allows for the study of the relationship between the zeros of an entire function and its growth and behavior
• Examples: $\sin(z)=z\prod_{n=1}^{\infty}(1-\frac{z^2}{n^2\pi^2})$, $\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}(1+\frac{z}{n})^{-1}e^{z/n}$

Representation of meromorphic functions

• Meromorphic functions can be represented as ratios of entire functions using Weierstrass factorization
• Poles of the meromorphic function correspond to zeros of the denominator in the ratio representation
• Allows for the study of the distribution of poles and the behavior of meromorphic functions

Use in complex analysis proofs

• Weierstrass factorization is often used in proofs of important results in complex analysis
• Examples: Proofs of the fundamental theorem of algebra, the Mittag-Leffler theorem, and the Riemann mapping theorem
• Provides a way to construct functions with desired properties or to derive contradictions

Examples of Weierstrass factorization

• Concrete examples help to illustrate the application of the theorem and the form of the factorization
• Showcase the relationship between the zeros of a function and its Weierstrass factorization
• Provide insight into the role of canonical products and the exponential factor in the factorization

Factorization of simple entire functions

• $e^z=e^z\prod_{n=1}^{\infty}(1-\frac{z}{2\pi in})e^{z/(2\pi in)}$, where the zeros are at $2\pi in$ for $n\in\mathbb{Z}$
• $\cos(z)=\prod_{n=1}^{\infty}(1-\frac{z^2}{(n-1/2)^2\pi^2})$, where the zeros are at $(n-1/2)\pi$ for $n\in\mathbb{Z}$
• These examples demonstrate the factorization of common entire functions and the distribution of their zeros

Factorization of meromorphic functions

• $\tan(z)=\frac{\sin(z)}{\cos(z)}=z\prod_{n=1}^{\infty}(1-\frac{z^2}{n^2\pi^2})(1-\frac{z^2}{(n-1/2)^2\pi^2})^{-1}$, where the zeros are at $n\pi$ and the poles are at $(n-1/2)\pi$ for $n\in\mathbb{Z}$
• $\Gamma(z)^{-1}=z\prod_{n=1}^{\infty}(1+\frac{z}{n})e^{-z/n}$, where the zeros are at the negative integers and the pole is at $z=0$
• These examples show how meromorphic functions can be represented using Weierstrass factorization and the relationship between zeros and poles

Limitations of Weierstrass factorization

• While powerful, Weierstrass factorization has certain limitations and does not apply to all complex-valued functions
• Understanding these limitations helps to clarify the scope and applicability of the theorem
• Highlights the need for additional techniques and results in complex analysis

Functions with essential singularities

• Weierstrass factorization does not apply to functions with essential singularities
• Essential singularities cannot be represented as poles or removable singularities
• Examples: $e^{1/z}$ has an essential singularity at $z=0$ and cannot be factored using Weierstrass factorization

Functions with branch points

• Functions with branch points or multi-valued functions are not covered by Weierstrass factorization
• Branch points introduce additional complexity and require different techniques, such as Riemann surfaces
• Examples: $\sqrt{z}$ has a branch point at $z=0$ and is not an entire function