Definition: order of a pole

MAT334H1-F – LEC0101

Complex Variables

ZEROES OF ANALYTIC FUNCTIONS – 2

November 16^th, 2020

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 1 / 9

Reviews from Oct 23 – Poles

Theorem

Let𝑈 ⊂ ℂbe open and𝑧₀∈ 𝑈. Assume that𝑓 ∶ 𝑈 ⧵ {𝑧₀} → ℂis holomorphic/analytic. Then TFAE:

1 𝑧₀is a pole of𝑓, i.e. lim

𝑧→𝑧0

|𝑓 (𝑧)| = +∞.

2 There exist𝑛 ∈ ℕ_>0and𝑔 ∶ 𝑈 → ℂanalytic such that𝑔(𝑧₀) ≠ 0and𝑓 (𝑧) = 𝑔(𝑧)

(𝑧 − 𝑧₀)^𝑛 on𝑈 ⧵ {𝑧₀}.

3 𝑧₀is not a removable singularity of𝑓 and there exists𝑛 ∈ ℕ_>0such thatlim

𝑧→𝑧₀(𝑧 − 𝑧₀)^𝑛+1𝑓 (𝑧) = 0.

Definition: order of a pole

The integer𝑛 ∈ ℕ_>0in 2 is uniquely defined and we say that𝑓 admits apole of order𝑛at𝑧₀.

Proposition

The order of the pole𝑧₀is also:

• The order of vanishing of1/𝑓 at𝑧₀.

• The smallest𝑛such thatlim

𝑧→𝑧₀(𝑧 − 𝑧₀)^𝑛+1𝑓 (𝑧) = 0.

Logarithmic residue

Lemma

• If𝑧₀is an isolated zero of𝑓 thenRes(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀.

• If𝑧₀is an isolated pole of𝑓 then−Res(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀.

Proof.

• Assume that𝑓 (𝑧) = (𝑧 − 𝑧₀)^𝑚𝑔(𝑧)in a neighborhood of𝑧₀where𝑔is analytic and𝑔(𝑧₀) ≠ 0. Then ^𝑓_{𝑓 (𝑧)}^′(𝑧) = 𝑚(𝑧 − 𝑧₀)⁻¹+ ^𝑔_𝑔(𝑧)^′(𝑧).

We conclude using that ^𝑔_𝑔^′ is holomorphic in a neighborhood of𝑧₀.

• 𝑧₀is a pole of order𝑚of𝑓 if and only if it is a zero of order𝑚of _𝑓¹. We conclude using that

Res( (1/𝑓 )^′

(1/𝑓 ), 𝑧₀

)= −Res (

𝑓^′ 𝑓 , 𝑧₀

)

■

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 3 / 9

Logarithmic residue

Lemma

• If𝑧₀is an isolated zero of𝑓 thenRes(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀.

• If𝑧₀is an isolated pole of𝑓 then−Res(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀. Proof.

• Assume that𝑓 (𝑧) = (𝑧 − 𝑧₀)^𝑚𝑔(𝑧)in a neighborhood of𝑧₀where𝑔is analytic and𝑔(𝑧₀) ≠ 0.

Then ^𝑓_{𝑓 (𝑧)}^′(𝑧) = 𝑚(𝑧 − 𝑧₀)⁻¹+ ^𝑔_𝑔(𝑧)^′(𝑧). We conclude using that ^𝑔′

𝑔 is holomorphic in a neighborhood of𝑧₀.

• 𝑧₀is a pole of order𝑚of𝑓 if and only if it is a zero of order𝑚of _𝑓¹. We conclude using that

Res( (1/𝑓 )^′

(1/𝑓 ), 𝑧₀

)= −Res (

𝑓^′ 𝑓 , 𝑧₀

)

■

Logarithmic residue

Lemma

• If𝑧₀is an isolated zero of𝑓 thenRes(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀.

• If𝑧₀is an isolated pole of𝑓 then−Res(^𝑓

′

𝑓 , 𝑧₀)is the order of𝑧₀. Proof.

• Assume that𝑓 (𝑧) = (𝑧 − 𝑧₀)^𝑚𝑔(𝑧)in a neighborhood of𝑧₀where𝑔is analytic and𝑔(𝑧₀) ≠ 0.

Then ^𝑓_{𝑓 (𝑧)}^′(𝑧) = 𝑚(𝑧 − 𝑧₀)⁻¹+ ^𝑔_𝑔(𝑧)^′(𝑧). We conclude using that ^𝑔′

𝑔 is holomorphic in a neighborhood of𝑧₀.

• 𝑧₀is a pole of order𝑚of𝑓 if and only if it is a zero of order𝑚of _𝑓¹. We conclude using that

Res( (1/𝑓 )^′

(1/𝑓 ), 𝑧₀

)= −Res (

𝑓^′ 𝑓 , 𝑧₀

)

■

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 3 / 9

The argument principle – Statement

Theorem: the argument principle

Let𝑈 ⊂ ℂbe open. Let𝑆 ⊂ 𝑈 be finite. Let𝑓 ∶ 𝑈 ⧵ 𝑆 → ℂbe holomorphic/analytic.

Let𝛾 ∶ [𝑎, 𝑏] → ℂbe piecewise smooth positively oriented simple closed curve on𝑈which doesn’t pass through a zero or a pole of𝑓 and such that its inside is entirely included in𝑈.

Then

1 2𝑖𝜋 ∫_𝛾

𝑓^′(𝑧)

𝑓 (𝑧)d𝑧 = 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾} where

• 𝑍_{𝑓 ,𝛾}is the number of zeroes of𝑓 enclosed in𝛾counted with their multiplicites/orders,

• 𝑃_{𝑓 ,𝛾} is the number of poles of𝑓 enclosed in𝛾counted with their multiplicites/orders.

Proof.We apply Cauchy’s residue theorem to ^𝑓′

𝑓 and then we use the above lemma: 1

2𝑖𝜋 ∫_𝛾 𝑓^′(𝑧)

𝑓 (𝑧)d𝑧 =

𝑧∈Inside(𝛾)∑ Res(

𝑓^′

𝑓 , 𝑧)= ∑

𝑧zero of𝑓

Res( 𝑓^′

𝑓 , 𝑧₀)+ ∑

𝑧pole of𝑓

Res( 𝑓^′

𝑓 , 𝑧₀)= 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾} ■

The argument principle – Statement

Theorem: the argument principle

Let𝑈 ⊂ ℂbe open. Let𝑆 ⊂ 𝑈 be finite. Let𝑓 ∶ 𝑈 ⧵ 𝑆 → ℂbe holomorphic/analytic.

Let𝛾 ∶ [𝑎, 𝑏] → ℂbe piecewise smooth positively oriented simple closed curve on𝑈which doesn’t pass through a zero or a pole of𝑓 and such that its inside is entirely included in𝑈.

Then

1 2𝑖𝜋 ∫_𝛾

𝑓^′(𝑧)

𝑓 (𝑧)d𝑧 = 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾} where

• 𝑍_{𝑓 ,𝛾}is the number of zeroes of𝑓 enclosed in𝛾counted with their multiplicites/orders,

• 𝑃_{𝑓 ,𝛾} is the number of poles of𝑓 enclosed in𝛾counted with their multiplicites/orders.

Proof.We apply Cauchy’s residue theorem to ^𝑓′

𝑓 and then we use the above lemma:

1 2𝑖𝜋 ∫_𝛾

𝑓^′(𝑧) 𝑓 (𝑧)d𝑧 =

𝑧∈Inside(𝛾)∑ Res(

𝑓^′

𝑓 , 𝑧)= ∑

𝑧zero of𝑓

Res( 𝑓^′

𝑓 , 𝑧₀)+ ∑

𝑧pole of𝑓

Res( 𝑓^′

𝑓 , 𝑧₀)= 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾} ■

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 4 / 9

The argument principle – Interpretation

The value 1 2𝑖𝜋 ∫_𝛾

𝑓^′(𝑧)

𝑓 (𝑧)d𝑧involved in the previous slide is equal to the number of counterclockwise turns made by𝑓 (𝑧)as𝑧goes through𝛾.

Indeed, if we set𝛾(𝑡) = 𝑓 ∘ 𝛾̃ then

∫_𝛾 𝑓^′(𝑧)

𝑓 (𝑧)d𝑧 =

∫_𝛾̃ 1 𝑤d𝑤.

Assume for instance that𝛾 ∶ [0, 1] → ℂ̃ is defined by𝛾(𝑡) = 𝑧̃ ₀+ 𝑟𝑒^{2𝑖𝜋𝑛𝑡}where𝑛 ∈ ℤ.

Then 1 2𝑖𝜋 ∫_𝛾̃

1

𝑤d𝑤 = 𝑛which is the number of counterclockwise turns made by𝛾̃around𝑧₀. Then the conclusion of the previous statement can be rewritten as

changes ofarg(𝑓 (𝑧))as𝑧goes through𝛾

2𝜋 = 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾}

That’s why it is calledthe argument principle.

Generalization at infinity

The previous lemma holds at∞:

Lemma

• If∞is an isolated zero of𝑓 thenRes(^𝑓

′

𝑓 , ∞)is the order of∞.

• If∞is an isolated pole of𝑓 then−Res(^𝑓

′

𝑓 , ∞)is the order of∞.

Proof.∞is an isolated zero (resp. pole) of order𝑚of𝑓 if and only if0is an isolated zero (resp.

pole) of order𝑚of𝑔(𝑧) = 𝑓 (1/𝑧).

Then𝑚 =Res(^𝑔

′

𝑔, 0) =Res(⁻¹𝑧² 𝑓^′(1/𝑧)

𝑓 (1/𝑧), 0) =Res(^𝑓

′

𝑓 , ∞). ■

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 6 / 9

Rouché’s theorem – 1

Rouché’s theorem – version 1

Let𝑈 ⊂ ℂbe open,𝑓 , 𝑔 ∶ 𝑈 → ℂbe two holomorphic/analytic functions on𝑈, and𝛾 ∶ [𝑎, 𝑏] → ℂ be a piecewise smooth simple closed curve on𝑈 whose inside is also included in𝑈.

Assume that

∀𝑡 ∈ [𝑎, 𝑏], |𝑔(𝛾(𝑡))| < |𝑓 (𝛾(𝑡))|

Then𝑓 and𝑓 + 𝑔have the same number of zeroes inside𝛾, counted with multiplicities.

Proof.For𝑡 ∈ [0, 1], set𝜑_𝑡(𝑧) = 𝑓 (𝑧) + (1 − 𝑡)𝑔(𝑧)andℎ(𝑡) = 1 2𝑖𝜋 ∫_𝛾

𝜑^′_𝑡(𝑧) 𝜑_𝑡(𝑧)d𝑧.

The functionℎis continuous since𝜑_𝑡doesn’t vanish on𝛾, indeed for𝑧 ∈ 𝛾

|𝜑_𝑡(𝑧)| ≥ |𝑓 (𝑧)| + (1 − 𝑡)|𝑔(𝑧)| ≥ |𝑓 (𝑧)| − |𝑔(𝑧)| > 0

Henceℎis a continuous function taking values inℤ(by the principle argument), so it is constant.

Henceℎ(0) = ℎ(1), i.e. 𝑍_{𝑓 +𝑔,𝛾}− 𝑃_{𝑓 +𝑔,𝛾} = 𝑍_{𝑓 ,𝛾}− 𝑃_{𝑓 ,𝛾} by the principle argument.

But these functions have no poles in the inside of𝛾, hence𝑍_{𝑓 +𝑔,𝛾}= 𝑍_{𝑓 ,𝛾}. ■

Rouché’s theorem – 2

Rouché’s theorem – version 2

Let𝑈 ⊂ ℂbe open,𝑓 , 𝑔 ∶ 𝑈 → ℂbe two holomorphic/analytic functions on𝑈, and𝛾 ∶ [𝑎, 𝑏] → ℂbe a piecewise smooth simple closed curve on𝑈 whose inside is also included in𝑈.

Assume that

∀𝑧 ∈ 𝛾, |𝑓 (𝑧) − 𝑔(𝑧)| < |𝑓 (𝑧)|

Then𝑓and𝑔have the same number of zeroes inside𝛾, counted with multiplicities.

Proof.That’s an immediate consequence of the previous version since𝑧₀is a zero of order𝑛of𝑔iff it is a

zero of order𝑛of−𝑔. ■

Rouché’s theorem – version 3

Let𝑈 ⊂ ℂbe open,𝑓 , 𝑔 ∶ 𝑈 → ℂbe two holomorphic/analytic functions on𝑈, and𝛾 ∶ [𝑎, 𝑏] → ℂbe a piecewise smooth simple closed curve on𝑈 whose inside is also included in𝑈.

Assume that

∀𝑧 ∈ 𝛾, |𝑓 (𝑧) + 𝑔(𝑧)| < |𝑓 (𝑧)|

Then𝑓and𝑔have the same number of zeroes inside𝛾, counted with multiplicities.

Proof.Since𝑧₀is a zero of order𝑛of𝑔iff it is a zero of order𝑛of−𝑔. ■

Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Nov 16, 2020 8 / 9

The fundamental theorem of algebra – bis repetita placent

We already proved the FTA (or d’Alembert–Gauss theorem) using Liouville’s theorem (Oct 21): a non-constant complex polynomial admits at least one root.

Here is another proof using Rouché’s theorem.

Theorem

A complex polynomial of degree𝑛has exactly𝑛complex roots (counted with multiplicity).

Proof.Assume that𝑃 (𝑧) = 𝑎_𝑛𝑧^𝑛+ 𝑄(𝑧)where𝑄is a polynomial of degree< 𝑛and𝑎_𝑛≠ 0.

If we take𝑅 > 0big enough then|𝑄(𝑧)| < |𝑎_𝑛𝑧^𝑛|on𝛾 ∶ [0, 1] → ℂdefined by𝛾(𝑡) = 𝑅𝑒^{2𝑖𝜋𝑡}. By Rouché’s theorem,𝑃 (𝑧) = 𝑎_𝑛𝑧^𝑛+ 𝑄(𝑧)and𝑎_𝑛𝑧^𝑛have the same number of zeroes counted with

multiplicity. ■