§14.2.2 The Factorization of Transcendental Meromorphic Functions

14.2.2 The Factorization of Transcendental Meromorphic Functions

We now give a meromorphic analog of the Hadamard Factorization Theorem:

Theorem 14.2.2.1.

Let $𝑓 : ℂ \to \hat{ℂ}$ be non-constant and meromorphic of finite order $𝜌$ such that $𝑓 (0) \neq 0, \infty$ . If ${𝑎_{𝑛}}_{𝑛 = 1}^{\infty}$ and ${𝑏_{𝑛}}_{𝑛 = 1}^{\infty}$ are its zeros and poles in $ℂ^{*}$ listed in order of non-decreasing moduli, counting multiplicities and zeros, then the products

𝑃_{1} (𝑧) = \prod_{𝑛 = 1}^{\infty} 𝐸_{𝑝} (\frac{𝑧}{𝑎_{𝑛}}), 𝑃_{2} (𝑧) = \prod_{𝑛 = 1}^{\infty} 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}})

locally uniformly converge in

ℂ

for some

𝑝, 𝑞 \leq ⌊ 𝜌 ⌋

. Then

𝑓 (𝑧) = e^{𝜑 (𝑧)} 𝑃_{1} (𝑧) / 𝑃_{2} (𝑧)

, where

𝜑

is a polynomial of degree

\leq 𝜌

Proof.

By Theorem 14.2.1.5 $Theorem 14.2.1.5 Theorem 14.2.1.5$ , letting $𝑞 = ⌊ 𝜌 ⌋$ gives that the two given products converge locally uniformly and the corresponding summation of exponent $𝑞 + 1$ converge. Define $𝑔$ to be the analytic continuation across all removable singularities of $𝑓 \cdot 𝑃_{2}$ such that $𝑔$ is entire and vanishes at each ${𝑎_{𝑛}}$ . Observe that

𝑇 (𝑟, 𝑔) \leq 𝑇 (𝑟, 𝑓) + 𝑇 (𝑟, 𝑃_{2}), 𝑇 (𝑟, 𝑓) = 𝒪︀ (𝑟^{𝜌 + 𝜀})

by Proposition 14.2.5 $Proposition 14.2.5 Proposition 14.2.5$ and Proposition 14.2.1.4 $Proposition 14.2.1.4 Proposition 14.2.1.4$ . For sufficiently large $𝑟$ ,

𝑇 (𝑟, 𝑃_{2}) = 𝑚 (𝑟, 𝑃_{2}) \leq log 𝑀 (𝑟, 𝑃_{2}) .

Define

𝑃_{2} (𝑧) = \prod_{| 𝑏_{𝑛} | \leq 2 | 𝑧 |} 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) \cdot \prod_{| 𝑏_{𝑛} | > 2 | 𝑧 |} 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) = 𝑃_{2}^{\leq} (𝑧) \cdot 𝑃_{2}^{>} (𝑧) .

Since $| 𝑧 / 𝑏_{𝑛} | < 1 / 2$ for $𝑃_{2}^{>}$ ,

| Log 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) | \leq \sum_{𝑘 > 𝑞}^{\infty} \frac{1}{𝑘} {| \frac{𝑧}{𝑏_{𝑛}} |}^{𝑞 + 1} \leq {| \frac{𝑧}{𝑏_{𝑛}} |}^{𝑞 + 1} \sum_{𝑘 > 𝑞}^{\infty} \frac{1}{𝑘} {| \frac{1}{2} |}^{𝑘 - 𝑞 - 1} \leq 𝑐 𝑟^{𝑞 + 1} {| 𝑏_{𝑛} |}^{- 𝑞 - 1},

where $𝑐$ is a constant dependent only on $𝑞$ and $𝑟 = | 𝑧 |$ (see (4.4.3.4) $(4.4.3.4) (4.4.3.4)$ ). Since for sufficiently small $𝜀 > 0$ , $𝑞 + 1 > 𝜌 + 𝜀$ ,

\begin{aligned} log | 𝑃_{2}^{>} (𝑧) | & \leq | Log 𝑃_{2}^{>} (𝑧) | \leq \sum_{| 𝑏_{𝑛} | > 2 𝑟} | Log 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) | \leq 𝑐 𝑟^{𝑞 + 1} \sum_{| 𝑏_{𝑛} | > 2 𝑟} \frac{1}{{| 𝑏_{𝑛} |}^{𝑞 + 1}} \\ \leq 𝑐 𝑟^{𝜌 + 𝜀} \sum_{| 𝑏_{𝑛} | > 2 𝑟} \frac{𝑟^{𝑞 + 1 - (𝜌 + 𝜀)}}{{| 𝑏_{𝑛} |}^{𝜌 + 𝜀} {| 𝑏_{𝑛} |}^{𝑞 + 1 - (𝜌 + 𝜀)}} \leq 2^{𝜌 + 𝜀 - 𝑞 - 1} 𝑐 𝑟^{𝜌 + 𝜀} \sum_{| 𝑏_{𝑛} | > 2 𝑟} {| 𝑏_{𝑛} |}^{- 𝜌 - 𝜀} \\ = 𝒪︀ (𝑟^{𝜌 + 𝜀}) \end{aligned}

(the summation of converges by Theorem 14.2.1.5 $Theorem 14.2.1.5 Theorem 14.2.1.5$ ). Assume now that $| 𝑏_{𝑛} | \leq 2 𝑟$ , or $2 | 𝑧 / 𝑏_{𝑛} | \geq 1$ . Then,

log | 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) | = log | 1 - \frac{𝑧}{𝑏_{𝑛}} | + ℜ𝔢 (\sum_{𝑘 = 1}^{𝑞} \frac{1}{𝑘} \frac{𝑧^{𝑘}}{𝑏_{𝑛}^{𝑘}}) \leq log (1 + | \frac{𝑧}{𝑏_{𝑛}} |) + 𝑐 {| \frac{𝑧}{𝑏_{𝑛}} |}^{𝑞}

for some $𝑐$ such that $ℜ𝔢 (\sum_{𝑘 = 1}^{𝑞} \frac{1}{𝑘} 𝑤^{𝑘}) \leq 𝑐 {| 𝑤 |}^{𝑞}$ for all $| 𝑤 | \geq 1 / 2$ , which depends only on $𝑞$ . First assume that $𝑞 \geq 1$ , and it then follows that

\begin{aligned} log | 𝑃_{2}^{\leq} (𝑧) | & \leq \sum_{| 𝑏_{𝑛} | \leq 2 𝑟} [𝑐 {| \frac{𝑟}{𝑏_{𝑛}} |}^{𝑞} + log (1 + | \frac{𝑟}{𝑏_{𝑛}} |)] \leq \sum_{| 𝑏_{𝑛} | \leq 2 𝑟} [𝑐 {| \frac{𝑟}{𝑏_{𝑛}} |}^{𝑞} + | \frac{𝑟}{𝑏_{𝑛}} |] \\ \leq \sum_{| 𝑏_{𝑛} | \leq 2 𝑟} (𝑐 + 2^{𝑞 - 1}) {| \frac{𝑧}{𝑏_{𝑛}} |}^{𝑞} = \sum_{| 𝑏_{𝑛} | \leq 2 𝑟} 𝑐^{'} \frac{𝑟^{𝜌 + 𝜀}}{{| 𝑏_{𝑛} |}^{𝜌 + 𝜀}} {(\frac{| 𝑏_{𝑛} |}{𝑟})}^{𝜌 + 𝜀 - 𝑞} \\ \leq \sum_{| 𝑏_{𝑛} | \leq 2 𝑟} 2^{𝜌 + 𝜀 - 𝑞} 𝑐^{'} \frac{𝑟^{𝜌 + 𝜀}}{{| 𝑏_{𝑛} |}^{𝜌 + 𝜀}} = 𝒪︀ (𝑟^{𝜌 + 𝜀}) . \end{aligned}

It follows that

𝑇 (𝑟, 𝑔) \leq log | 𝑃_{2} (𝑧) | = 𝒪︀ (𝑟^{𝜌 + 𝜀}),

and thus $𝑔$ is of order $𝜌^{'} \leq 𝜌$ . For any $𝜀$ , there is some $𝑐$ such that $log (1 + | 𝑤 |) \leq 𝑐 {| 𝑤 |}^{𝜀}$ ; thus for $𝑞 = 0$ , the assertion is direct from Proposition 14.2.1.4 $Proposition 14.2.1.4 Proposition 14.2.1.4$ :

\begin{matrix} log | 𝐸_{𝑞} (\frac{𝑧}{𝑏_{𝑛}}) | \leq log (1 + | \frac{𝑧}{𝑏_{𝑛}} |) \leq 𝑐 \frac{𝑟^{𝜀}}{{| 𝑏_{𝑛} |}^{𝜀}} \leq 𝑐 \frac{𝑟^{𝜀}}{{| 𝑏_{1} |}^{𝜀}} \\ log | 𝑃_{2}^{\leq} | \leq 𝑛 (2 𝑟, 0, 𝑓) \frac{𝑟^{𝜀}}{{| 𝑏_{1} |}^{𝜀}} = 𝒪︀ (𝑟^{𝜌 + 2 𝜀}) . \end{matrix}

Similar logic then derives that

𝑔

is of order

\leq 𝜌

. There then exists an integer

𝑝 \leq 𝜌^{'}

such that

𝑃_{1}

converges and

𝑔 (𝑧) = e^{𝜑 (𝑧)} 𝑃_{1} (𝑧)

has the property that

𝜑

is a polynomial of degree

𝑔 \leq 𝜌^{'}

by the Hadamard Factorization Theorem (Theorem 4.4.5.3

Theorem 4.4.5.3 Theorem 4.4.5.3

) for entire functions. Writing

𝑓 = 𝑔 / 𝑃_{2}

then yields the desired representation. ⁠

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