Entire functions are complex functions that are analytic everywhere in the complex plane. The Hadamard factorization theorem provides a way to represent these functions as products involving their zeros, offering insights into their growth and behavior.

This theorem connects an entire function's zeros to its growth rate, revealing how the distribution of zeros relates to the function's order and type. It's a powerful tool for understanding the structure and properties of entire functions in complex analysis.

Hadamard Factorization Theorem

Statement and Implications

• The Hadamard factorization theorem asserts that any entire function of finite order can be expressed as a product involving its zeros
• Provides a canonical representation of entire functions in terms of their zeros, order, and type
• The factorization includes a polynomial factor, a product over the function's non-zero roots, and an exponential factor
• Establishes a connection between the growth of an entire function and the distribution of its zeros
  • Functions with similar growth properties often have similar zero distributions
  • Rapid growth of an entire function typically implies a high density of zeros

Components of the Factorization

• The polynomial factor $z^m$ accounts for any zero at the origin
  • $m$ is the multiplicity of the zero at $z = 0$
• The infinite product $\Pi_k (1 - z/a_k)$ incorporates the non-zero roots $a_k$ of the function
  • Each factor corresponds to a single non-zero root
  • The product converges due to the exponential factors $e^{z/a_k + (z/a_k)^2/2 + ... + (z/a_k)^{p-1}/(p-1)}$
• The exponential factor $e^{P(z)}$ involves a polynomial $P(z)$ of degree $\leq p$
  • The degree of $P(z)$ is determined by the order and type of the entire function
  • $P(z)$ captures the growth behavior not accounted for by the zeros

Entire Functions as Products

Hadamard Factorization Formula

• The Hadamard factorization expresses an entire function $f(z)$ as:

$f(z) = z^m \cdot e^{P(z)} \cdot \Pi_k \left(1 - \frac{z}{a_k}\right) \cdot e^{\frac{z}{a_k} + \frac{(z/a_k)^2}{2} + ... + \frac{(z/a_k)^{p-1}}{p-1}}$ - $m$ is a non-negative integer - $P(z)$ is a polynomial of degree $\leq p$ - $a_k$ are the non-zero roots of $f(z)$ - $p$ is the genus of $f(z)$, related to its order

• The genus $p$ is the smallest integer such that the series $\sum_k \frac{1}{|a_k|^{p+1}}$ converges
  • Convergence of this series is necessary for the infinite product to converge
• The exponential factors in the product ensure convergence of the infinite product
  • Without these factors, the product might diverge even if the series $\sum_k \frac{1}{|a_k|^{p+1}}$ converges

Examples and Applications

• The Hadamard factorization of the sine function $\sin(\pi z)$ is:

$\sin(\pi z) = \pi z \cdot \Pi_{k=1}^\infty \left(1 - \frac{z}{k}\right) \cdot \left(1 + \frac{z}{k}\right)$ - Roots at non-zero integers, genus $p = 1$, order $\rho = 2$

• The Gamma function $\Gamma(z)$ has Hadamard factorization:

$\Gamma(z) = \frac{e^{-\gamma z}}{z} \cdot \Pi_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} \cdot e^{z/k}$ - Roots at non-positive integers, genus $p = 1$, order $\rho = 1$

• Hadamard factorizations are used in the study of zeta functions and L-functions in number theory

Order and Type of Entire Functions

Determining Order and Type

• The order $\rho$ of an entire function $f(z)$ is related to the genus $p$ in its Hadamard factorization: $\rho = p + 1$
• The genus $p$ is the smallest integer such that the series $\sum_k \frac{1}{|a_k|^{p+1}}$ converges, where $a_k$ are the non-zero roots of $f(z)$
  • Larger values of $p$ correspond to faster growth of $f(z)$
• The type $\sigma$ of an entire function is determined by the degree of the polynomial $P(z)$ in its Hadamard factorization
  • If $\deg(P) < p$, then $f(z)$ is of minimal type
  • If $\deg(P) = p$, then $f(z)$ is of normal type, and $\sigma = \frac{\text{leading coefficient of } P}{p!}$
  • If $\deg(P) > p$, then $f(z)$ is of maximal type
• The order and type provide information about the growth rate of the entire function
  • Higher order implies faster growth as $|z| \to \infty$
  • Within an order, higher type implies faster growth

Examples and Implications

• The exponential function $e^z$ has order $\rho = 1$ and type $\sigma = 1$
  • Grows faster than any polynomial but slower than $e^{z^2}$
• The function $e^{z^2}$ has order $\rho = 2$ and type $\sigma = 1$
  • Grows faster than $e^z$ and any function of order $< 2$
• Functions of the same order can have different types, e.g., $e^z$ and $e^{2z}$
  • Both have order $\rho = 1$, but $e^{2z}$ has type $\sigma = 2$ and grows faster
• The order and type can help classify entire functions and understand their asymptotic behavior

Analyzing Growth and Zeros of Entire Functions

Relationship between Growth and Zeros

• The Hadamard factorization relates the growth of an entire function to the distribution of its zeros
• The order $\rho$ of an entire function determines the rate of growth of the function as $|z| \to \infty$
  • Functions of order $\rho$ grow roughly like $e^{|z|^\rho}$
• The type $\sigma$ provides a more precise characterization of growth within the order
  • Functions of order $\rho$ and type $\sigma$ grow roughly like $e^{\sigma |z|^\rho}$
• The distribution of zeros is encoded in the infinite product and the convergence of the series $\sum_k \frac{1}{|a_k|^{p+1}}$
  • Faster growth typically implies a higher density of zeros
  • The convergence rate of the series is related to the density of zeros

Applications and Examples

• The Hadamard factorization can be used to estimate the number of zeros in a given region using the Jensen formula
  • Relates the number of zeros in a disk to the growth of the function on the boundary
• The factorization helps analyze the asymptotic behavior of entire functions and their derivatives
  • Differentiation of the Hadamard product can provide information about the growth and zeros of derivatives
• Comparing the Hadamard factorizations of two entire functions can provide insights into their relative growth and zero distributions
  • Functions with similar factorizations often have similar growth and zero properties
• Example: The Riemann zeta function $\zeta(s)$ is related to the prime numbers and has zeros at the negative even integers and non-trivial zeros in the critical strip $0 < \Re(s) < 1$
  • The Hadamard factorization of $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ encodes information about the distribution of these zeros