§4.4 Further Properties of Meromorphic and Entire Functions

4.4 Further Properties of Meromorphic and Entire Functions

Theorem 4.4.1.

Let $𝑈 \subseteq ℂ$ be a region and $𝑓 : 𝑈 \to ℂ$ be meromorphic. Let $𝛾 \subset 𝑈$ be a positively oriented Jordan curve that is null-homotopic in $𝑈$ . If $𝑓$ has no zeros on $𝛾$ , then $𝑓$ has finitely many zeros and poles in the region bounded by $𝛾$ . Denote the zeros of $𝑓$ in the bounded region by $𝑎_{1}, \dots, 𝑎_{𝑘}$ with respective multiplicities $𝛼_{1}, \dots, 𝛼_{𝑘}$ , and the poles by $𝑏_{1}, \dots, 𝑏_{𝑚}$ with respective orders $𝛽_{1}, \dots, 𝛽_{𝑚}$ . Let $𝜓$ be any function holomorphic on a neighborhood of the closure of the bounded region. Then

\frac{1}{2 π i} \oint_{𝛾} \frac{𝜓 (𝑧) 𝑓^{'} (𝑧)}{𝑓 (𝑧)} d 𝑧 = \sum_{𝑖 = 1}^{𝑘} 𝛼_{𝑖} 𝜓 (𝑎_{𝑖}) - \sum_{𝑗 = 1}^{𝑚} 𝛽_{𝑗} 𝜓 (𝑏_{𝑗}) .

Proof.

Choose disks $𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})$ with pairwise disjoint closures around each zero $𝑎_{𝑖}$ and $𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})$ around each pole $𝑏_{𝑗}$ , with $𝜀_{𝑖}, 𝜀_{𝑗}^{'} > 0$ sufficiently small so that these disks are contained in $int (𝛾)$ , disjoint from $𝛾$ , and contained in the neighborhood where $𝜓$ is holomorphic. The function

𝑔 (𝑧) = \frac{𝜓 (𝑧) 𝑓^{'} (𝑧)}{𝑓 (𝑧)}

is holomorphic on

int (𝛾) ∖ (⋃_{𝑖 = 1}^{𝑘} 𝐷 (𝑎_{𝑖}, 𝜀_{𝑖}) \cup ⋃_{𝑗 = 1}^{𝑚} 𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})),

since $𝜓$ is holomorphic there, $𝑓$ is meromorphic with no other singularities, and $𝑓 \neq 0$ on $𝛾$ . The oriented boundary of this punctured domain is $𝛾^{+} \cup ⋃_{𝑖 = 1}^{𝑘} 𝜕 {𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})}^{-} \cup ⋃_{𝑗 = 1}^{𝑚} 𝜕 {𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})}^{-}$ . By Cauchy–Goursat (Theorem 3.1.9 $Theorem 3.1.9 Theorem 3.1.9$ ),

\oint_{𝛾^{+}} 𝑔 (𝑧) d 𝑧 + \sum_{𝑖 = 1}^{𝑘} \oint_{𝜕 {𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})}^{-}} 𝑔 (𝑧) d 𝑧 + \sum_{𝑗 = 1}^{𝑚} \oint_{𝜕 {𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})}^{-}} 𝑔 (𝑧) d 𝑧 = 0 .

Thus,

- \oint_{𝛾} 𝑔 (𝑧) d 𝑧 = \sum_{𝑖 = 1}^{𝑘} \oint_{𝜕 {𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})}^{+}} 𝑔 (𝑧) d 𝑧 + \sum_{𝑗 = 1}^{𝑚} \oint_{𝜕 {𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})}^{+}} 𝑔 (𝑧) d 𝑧 .

Near each zero $𝑎_{𝑖}$ , write $𝑓 (𝑧) = {(𝑧 - 𝑎_{𝑖})}^{𝛼_{𝑖}} ℎ (𝑧)$ where $ℎ$ is holomorphic at $𝑎_{𝑖}$ with $ℎ (𝑎_{𝑖}) \neq 0$ . Then

\frac{𝑓^{'} (𝑧)}{𝑓 (𝑧)} = \frac{𝛼_{𝑖}}{𝑧 - 𝑎_{𝑖}} + \frac{ℎ^{'} (𝑧)}{ℎ (𝑧)},

𝑔 (𝑧) = 𝜓 (𝑧) (\frac{𝛼_{𝑖}}{𝑧 - 𝑎_{𝑖}} + \frac{ℎ^{'} (𝑧)}{ℎ (𝑧)}) .

Then,

\oint_{𝜕 𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})} 𝑔 (𝑧) d 𝑧 = \oint_{𝜕 𝐷 (𝑎_{𝑖}, 𝜀_{𝑖})} 𝜓 (𝑧) (\frac{𝛼_{𝑖}}{𝑧 - 𝑎_{𝑖}} + \frac{ℎ^{'} (𝑧)}{ℎ (𝑧)}) d 𝑧 = 2 π i 𝛼_{𝑖} 𝜓 (𝑎_{𝑖}),

where the first term has been reduced by the Cauchy–Goursat Formula (Theorem 3.1.10 $Theorem 3.1.10 Theorem 3.1.10$ ) and the second integral vanishes by the Cauchy–Goursat Theorem (Theorem 3.1.9 $Theorem 3.1.9 Theorem 3.1.9$ ).

Near a pole $𝑏_{𝑗}$ , write $𝑓 (𝑧) = {(𝑧 - 𝑏_{𝑗})}^{- 𝛽_{𝑗}} 𝑘 (𝑧)$ where $𝑘$ is holomorphic at $𝑏_{𝑗}$ with $𝑘 (𝑏_{𝑗}) \neq 0$ . Then

\frac{𝑓^{'} (𝑧)}{𝑓 (𝑧)} = - \frac{𝛽_{𝑗}}{𝑧 - 𝑏_{𝑗}} + \frac{𝑘^{'} (𝑧)}{𝑘 (𝑧)},

𝑔 (𝑧) = 𝜓 (𝑧) (- \frac{𝛽_{𝑗}}{𝑧 - 𝑏_{𝑗}} + \frac{𝑘^{'} (𝑧)}{𝑘 (𝑧)}) .

A similar calculation yields that

\oint_{𝜕 𝐷 (𝑏_{𝑗}, 𝜀_{𝑗}^{'})} 𝑔 (𝑧) d 𝑧 = - 2 π i 𝛽_{𝑗} 𝜓 (𝑏_{𝑗}) .

Combining these,

\begin{aligned} \oint_{𝛾} \frac{𝜓 (𝑧) 𝑓^{'} (𝑧)}{𝑓 (𝑧)} d 𝑧 & = \sum_{𝑖 = 1}^{𝑘} 2 π i 𝛼_{𝑖} 𝜓 (𝑎_{𝑖}) - \sum_{𝑗 = 1}^{𝑚} 2 π i 𝛽_{𝑗} 𝜓 (𝑏_{𝑗}) \\ = 2 π i (\sum_{𝑖 = 1}^{𝑘} 𝛼_{𝑖} 𝜓 (𝑎_{𝑖}) - \sum_{𝑗 = 1}^{𝑚} 𝛽_{𝑗} 𝜓 (𝑏_{𝑗})) . □ \end{aligned}

Theorem 4.4.2 (Argument Principle).

Let $𝑈 \subseteq ℂ$ be a region and $𝑓 : 𝑈 \to ℂ$ be meromorphic. Let $𝛾 \subset 𝑈$ be a simple, closed, positively oriented curve that is null-homotopic in $𝑈$ . If $𝑓$ has no zeros or poles on $𝛾$ , then $𝑓$ has finitely many zeros and poles in the region bounded by $𝛾$ , and the number of zeros, $𝑘$ , minus the number of poles, $𝑘^{'}$ , counting multiplicities and orders, is given by

𝑘 - 𝑘^{'} = \frac{1}{2 π i} \oint_{𝛾} \frac{𝑓^{'} (𝑧)}{𝑓 (𝑧)} d 𝑧 .

Let

Γ

be the image of

𝛾

under the map

𝑤 = 𝑓 (𝑧)

. Then

𝑘 - 𝑘^{'} = {Ind}_{Γ} (0)

Proof.

By Theorem 4.4.1 $Theorem 4.4.1 Theorem 4.4.1$ for $𝜓 \equiv 1$ ,

\frac{1}{2 π i} \oint_{𝛾} \frac{𝑓^{'} (𝑧)}{𝑓 (𝑧)} d 𝑧 = 𝑘 - 𝑘^{'} .

Parametrize $Γ$ by $𝑤 = 𝑓 (𝑧)$ . Then $d 𝑤 = 𝑓^{'} (𝑧) d 𝑧$ , and

𝑘 - 𝑘^{'} = \frac{1}{2 π i} \oint_{Γ} \frac{d 𝑤}{𝑤} = {Ind}_{Γ} (0) . □