September21 ,2020 FUNCTIONS,SEQUENCESANDSERIES ComplexVariables

MAT334H1-F – LEC0101

Complex Variables

FUNCTIONS, SEQUENCES AND SERIES

September 21^st, 2020

Functions: generalities – 1

Definition (informal): function

Afunction(ormap) is the data of two sets𝐴and𝐵together with a ”process” that associates to each𝑥 ∈ 𝐴a unique𝑓 (𝑥) ∈ 𝐵:

𝑓 ∶{

𝐴 → 𝐵

𝑥 ↦ 𝑓 (𝑥)

Here: 𝑓 is the name of the function,𝐴is thedomain, and𝐵is thecodomain.

Beware

The domain and codomain are part of the definition of a function:

sin∶ ℝ → ℝandsin∶ [−𝜋, 𝜋] → ℝare not the same function.

A function is not simply a ”formula”.

Nonetheless, if you have a question like ”what’s the domain of𝑓 (𝑧) = _1+𝑧¹ ?”, it is implied that you are asked to give the greatest domain such that this formula makes sense (here: ℂ ⧵ {−1}).

Functions: generalities – 2

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• Theimage of𝐸 ⊂ 𝐴by𝑓 is𝑓 (𝐸) ≔ {𝑓 (𝑥) ∶ 𝑥 ∈ 𝐸} ⊂ 𝐵.

• Theimage of f(orrange of𝑓) isRange(𝑓 ) ≔ 𝑓 (𝐴).

• Thepreimage of𝐹 ⊂ 𝐵by𝑓 is𝑓⁻¹(𝐹 ) ≔ {𝑥 ∈ 𝐴 ∶ 𝑓 (𝑥) ∈ 𝐹 }.

• Thegraph of𝑓 is the setΓ_𝑓 ≔ {(𝑥, 𝑦) ∈ 𝐴 × 𝐵 ∶ 𝑦 = 𝑓 (𝑥)}.

The graph of a function𝑓 ∶ ℂ → ℂis quite difficult to visualize: it is a2-dimensional object lying in a4-dimensional space.

Instead, it can be useful to visualize separately the graphs of the real and imaginary parts. Or to study how the function transforms some subsets.

Functions: generalities – 3

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• We say that𝑓 isinjective(orone-to-one) if ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑥₁≠ 𝑥₂ ⟹ 𝑓 (𝑥₁) ≠ 𝑓 (𝑥₂) or equivalently by taking the contrapositive ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑓 (𝑥₁) = 𝑓 (𝑥₂) ⟹ 𝑥₁= 𝑥₂

• We say that𝑓 issurjective(oronto) if ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

• We say that𝑓 isbijectiveif it is injective and surjective, i.e. ∀𝑦 ∈ 𝐵, ∃!𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

Figure:Injective

𝑎 𝑏 𝑐

1 2 3 4

Figure:Not injective

𝑎 𝑏 𝑐

1 2 3 4

Functions: generalities – 3

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• We say that𝑓 isinjective(orone-to-one) if ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑥₁≠ 𝑥₂ ⟹ 𝑓 (𝑥₁) ≠ 𝑓 (𝑥₂) or equivalently by taking the contrapositive ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑓 (𝑥₁) = 𝑓 (𝑥₂) ⟹ 𝑥₁= 𝑥₂

• We say that𝑓 issurjective(oronto) if ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

• We say that𝑓 isbijectiveif it is injective and surjective, i.e. ∀𝑦 ∈ 𝐵, ∃!𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

Figure:Surjective

𝑎 𝑏 𝑐 𝑑

1 2 3

Figure:Not surjective

𝑎 𝑏 𝑐 𝑑

1 2 3 4

Functions: generalities – 3

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• We say that𝑓 isinjective(orone-to-one) if ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑥₁≠ 𝑥₂ ⟹ 𝑓 (𝑥₁) ≠ 𝑓 (𝑥₂) or equivalently by taking the contrapositive ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑓 (𝑥₁) = 𝑓 (𝑥₂) ⟹ 𝑥₁= 𝑥₂

• We say that𝑓 issurjective(oronto) if ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

• We say that𝑓 isbijectiveif it is injective and surjective, i.e. ∀𝑦 ∈ 𝐵, ∃!𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

Figure:Bijective

𝑎 𝑏 𝑐 𝑑

1 2 3 4

Functions: generalities – 3

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• We say that𝑓 isinjective(orone-to-one) if ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑥₁≠ 𝑥₂ ⟹ 𝑓 (𝑥₁) ≠ 𝑓 (𝑥₂) or equivalently by taking the contrapositive ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑓 (𝑥₁) = 𝑓 (𝑥₂) ⟹ 𝑥₁= 𝑥₂

• We say that𝑓 issurjective(oronto) if ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

• We say that𝑓 isbijectiveif it is injective and surjective, i.e. ∀𝑦 ∈ 𝐵, ∃!𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

Proposition

𝑓 ∶ 𝐴 → 𝐵is bijective if and only if there exists𝑔 ∶ 𝐵 → 𝐴such that {

∀𝑥 ∈ 𝐴, 𝑔(𝑓 (𝑥)) = 𝑥

∀𝑦 ∈ 𝐵, 𝑓 (𝑔(𝑦)) = 𝑦 . Then𝑔 is unique, it is called theinverse of𝑓 and denoted by𝑓⁻¹∶ 𝐵 → 𝐴.

Functions: generalities – 3

We fix a function𝑓 ∶ 𝐴 → 𝐵.

• We say that𝑓 isinjective(orone-to-one) if ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑥₁≠ 𝑥₂ ⟹ 𝑓 (𝑥₁) ≠ 𝑓 (𝑥₂) or equivalently by taking the contrapositive ∀𝑥₁, 𝑥₂∈ 𝐴, 𝑓 (𝑥₁) = 𝑓 (𝑥₂) ⟹ 𝑥₁= 𝑥₂

• We say that𝑓 issurjective(oronto) if ∀𝑦 ∈ 𝐵, ∃𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

• We say that𝑓 isbijectiveif it is injective and surjective, i.e. ∀𝑦 ∈ 𝐵, ∃!𝑥 ∈ 𝐴, 𝑦 = 𝑓 (𝑥)

Figure:Bijective function

𝑎 𝑏 𝑐 𝑑

1 2 3 4

Figure:Its inverse

𝑎 𝑏 𝑐 𝑑

1 2 3 4

Limits of complex functions – 1

Definition: limit point

Let𝑆 ⊂ ℂ. We say that𝑧₀∈ ℂis alimit point of𝑆if

∀𝛿 > 0, ∃𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 or equivalently

∀𝛿 > 0, (𝐷𝛿(𝑧₀) ∩ 𝑆) ⧵ {𝑧0} ≠ ∅

Theorem

𝑧₀is a limit point of𝑆if and only if𝑧₀∈ 𝑆 ⧵ {𝑧₀}.

Definition: limit at 𝑧

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ,𝑧₀be a limit point of𝑆andℓ ∈ ℂ. We say that𝑓 tends toℓas𝑧tends to𝑧₀, denoted bylim

𝑧→𝑧₀𝑓 (𝑧) = ℓ, if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − ℓ| < 𝜀.

Limits of complex functions – 1

Definition: limit point

Let𝑆 ⊂ ℂ. We say that𝑧₀∈ ℂis alimit point of𝑆if

∀𝛿 > 0, ∃𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 or equivalently

∀𝛿 > 0, (𝐷𝛿(𝑧₀) ∩ 𝑆) ⧵ {𝑧0} ≠ ∅

Theorem

𝑧₀is a limit point of𝑆if and only if𝑧₀∈ 𝑆 ⧵ {𝑧₀}.

Definition: limit at 𝑧

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ,𝑧₀be a limit point of𝑆andℓ ∈ ℂ. We say that𝑓 tends toℓas𝑧tends to𝑧₀, denoted bylim

𝑧→𝑧₀𝑓 (𝑧) = ℓ, if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − ℓ| < 𝜀.

Limits of complex functions – 1

Definition: limit point

Let𝑆 ⊂ ℂ. We say that𝑧₀∈ ℂis alimit point of𝑆if

∀𝛿 > 0, ∃𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 or equivalently

∀𝛿 > 0, (𝐷𝛿(𝑧₀) ∩ 𝑆) ⧵ {𝑧0} ≠ ∅

Theorem

𝑧₀is a limit point of𝑆if and only if𝑧₀∈ 𝑆 ⧵ {𝑧₀}.

Definition: limit at 𝑧

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ,𝑧₀be a limit point of𝑆andℓ ∈ ℂ.

We say that𝑓 tends toℓas𝑧tends to𝑧₀, denoted bylim

𝑧→𝑧₀𝑓 (𝑧) = ℓ, if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, 0 < |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − ℓ| < 𝜀.

Limits of complex functions – 2

Definition: limit at ∞

Let𝑆 ⊂ ℂbe unbounded,𝑓 ∶ 𝑆 → ℂandℓ ∈ ℂ. We say that𝑓 tends toℓas𝑧tends to∞, denoted by lim

𝑧→∞𝑓 (𝑧) = ℓ, if

∀𝜀 > 0, ∃𝑀 > 0, ∀𝑧 ∈ 𝑆, |𝑧| > 𝑀 ⟹ |𝑓 (𝑧) − ℓ| < 𝜀

Examples

𝑧→∞lim 1 𝑧 = 0 lim𝑧→0

𝑧 𝑧 DNE

𝑧→∞lim 𝑧 𝑧 DNE

Limits of complex functions – 3

Proposition (here 𝑧

∈ ℂ ⊔ {∞})

𝑧→𝑧lim₀𝑓 (𝑧) = 𝐿 ⇔⎧⎪

⎨⎪

⎩

𝑧→𝑧lim₀ℜ(𝑓 ) = ℜ(𝐿)

𝑧→𝑧lim₀ℑ(𝑓 ) = ℑ(𝐿)

Proposition

Assume that lim

𝑧→𝑧₀𝑓 (𝑧) = 𝐿, lim

𝑧→𝑧₀𝑔(𝑧) = 𝑀,𝛼, 𝛽 ∈ ℂ(here𝑧₀∈ ℂ ⊔ {∞}). Then

• lim

𝑧→𝑧₀(𝛼𝑓 + 𝛽𝑔) = 𝛼𝐿 + 𝛽𝑀

• lim

𝑧→𝑧₀𝑓 𝑔 = 𝐿𝑀

• lim

𝑧→𝑧₀𝑓 = 𝐿

• lim

𝑧→𝑧₀|𝑓 | = |𝐿|

• If𝑀 ≠ 0, lim

𝑧→𝑧

𝑓 𝑔 = 𝐿

𝑀

Continuity – 1

Definition

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂand𝑧₀∈ 𝑆.

We say that𝑓 is continuous at𝑧₀if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − 𝑓 (𝑧₀)| < 𝜀.

We say that𝑓 iscontinuousif𝑓 is continuous at every𝑧₀∈ 𝑆.

If𝑧₀∈ 𝑆is a limit point then𝑓 is continuous at𝑧₀if and only if lim

𝑧→𝑧₀𝑓 (𝑧) = 𝑓 (𝑧₀)

Theorem

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ. The following are equivalent:

1 𝑓 is continuous,

2 ∀𝒰 ⊂ ℂopen, ∃𝒱 ⊂ ℂopen, 𝑓⁻¹(𝒰) = 𝒱 ∩ 𝑆,

3 ∀𝒞 ⊂ ℂclosed, ∃𝒟 ⊂ ℂclosed, 𝑓⁻¹(𝒞 ) = 𝒟 ∩ 𝑆.

Continuity – 1

Definition

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂand𝑧₀∈ 𝑆.

We say that𝑓 is continuous at𝑧₀if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − 𝑓 (𝑧₀)| < 𝜀.

We say that𝑓 iscontinuousif𝑓 is continuous at every𝑧₀∈ 𝑆.

If𝑧₀∈ 𝑆is a limit point then𝑓 is continuous at𝑧₀if and only if lim

𝑧→𝑧₀𝑓 (𝑧) = 𝑓 (𝑧₀)

Theorem

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ. The following are equivalent:

1 𝑓 is continuous,

2 ∀𝒰 ⊂ ℂopen, ∃𝒱 ⊂ ℂopen, 𝑓⁻¹(𝒰) = 𝒱 ∩ 𝑆,

3 ∀𝒞 ⊂ ℂclosed, ∃𝒟 ⊂ ℂclosed, 𝑓⁻¹(𝒞 ) = 𝒟 ∩ 𝑆.

Continuity – 1

Definition

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂand𝑧₀∈ 𝑆.

We say that𝑓 is continuous at𝑧₀if

∀𝜀 > 0, ∃𝛿 > 0, ∀𝑧 ∈ 𝑆, |𝑧 − 𝑧₀| < 𝛿 ⟹ |𝑓 (𝑧) − 𝑓 (𝑧₀)| < 𝜀.

We say that𝑓 iscontinuousif𝑓 is continuous at every𝑧₀∈ 𝑆.

If𝑧₀∈ 𝑆is a limit point then𝑓 is continuous at𝑧₀if and only if lim

𝑧→𝑧₀𝑓 (𝑧) = 𝑓 (𝑧₀)

Theorem

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂ. The following are equivalent:

1 𝑓 is continuous,

2 ∀𝒰 ⊂ ℂopen, ∃𝒱 ⊂ ℂopen, 𝑓⁻¹(𝒰) = 𝒱 ∩ 𝑆,

3 ∀𝒞 ⊂ ℂclosed, ∃𝒟 ⊂ ℂclosed, 𝑓⁻¹(𝒞 ) = 𝒟 ∩ 𝑆.

Continuity – 2

Proposition

𝑓 is continuous at𝑧₀if and only ifℜ(𝑓 )andℑ(𝑓 )are.

Proposition

If𝑓 and𝑔are continuous at𝑧₀, so are:

• 𝛼𝑓 + 𝛽𝑔(here𝛼, 𝛽 ∈ ℂ)

• 𝑓 𝑔

• 𝑓

• |𝑓 |

• 𝑓

𝑔 (assuming that𝑔(𝑧₀) ≠ 0).

Sequences – 1

Definition

We say that a sequence of complex numbers(𝑧_𝑘)_𝑘∈ℕtends toℓ ∈ ℂas𝑛tends to+∞, denoted by lim

𝑘→+∞𝑧_𝑘= ℓ, if

∀𝜀 > 0, ∃𝐾 ∈ ℕ, ∀𝑘 ∈ ℕ, 𝑘 ≥ 𝐾 ⟹ |𝑧_𝑘− ℓ| < 𝜀

Proposition

𝑘→+∞lim 𝑧_𝑘= ℓ ⇔ lim

𝑘→+∞|𝑧_𝑘− ℓ| = 0

Proposition

𝑘→+∞lim 𝑧_𝑘= ℓ ⇔⎧⎪

⎨⎪

⎩

𝑘→+∞lim ℜ(𝑧_𝑘) = ℜ(ℓ)

𝑘→+∞lim ℑ(𝑧_𝑘) = ℑ(ℓ)

Sequences – 2

Proposition

Assume that lim

𝑘→+∞𝑧_𝑘= 𝐿and lim

𝑘→+∞𝑤_𝑘= 𝑀then

• lim

𝑘→+∞(𝑧_𝑘+ 𝑤_𝑘) = 𝐿 + 𝑀

• lim

𝑘→+∞𝑧_𝑘𝑤_𝑘= 𝐿𝑀

• lim

𝑘→+∞

𝑧_𝑘 𝑤_𝑘 = 𝐿

𝑀 (assuming𝑀 ≠ 0)

• lim

𝑘→+∞𝑧_𝑘= 𝐿

• lim

𝑘→+∞|𝑧_𝑘| = |𝐿|

Series – 1

Definitions

Given a sequence(𝑧_𝑘)_𝑘∈ℕ:

• We define the𝑛-th partial sum associated to(𝑧_𝑘)_𝑘∈ℕby𝑆_𝑛=

𝑛

∑𝑘=0

𝑧_𝑘,

• We say that the series

+∞

∑𝑘=0

𝑧_𝑘isconvergentif lim

𝑛→+∞𝑆_𝑛exists then we set

+∞

∑𝑘=0

𝑧_𝑘≔ lim

𝑛→+∞

𝑛

∑𝑘=0

𝑧_𝑘,

• Otherwise, we say that the series

+∞

∑𝑘=0

𝑧_𝑘isdivergent.

Convergence doesn’t depend on the starting index:

+∞

∑𝑘=0

𝑧_𝑘is convergent⇔

+∞

𝑘=𝑘∑₀

𝑧_𝑘is convergent.

But the value of the series DOES depend on the starting index.

Series – 2

Proposition

If

+∞

∑𝑘=0

𝑧_𝑘is convergent then lim

𝑘→+∞𝑧_𝑘= 0.

• The contrapositive can be very useful: if lim

𝑘→+∞𝑧_𝑘≠ 0then

+∞

∑𝑘=0

𝑧_𝑘is divergent.

• The converse is false:

+∞

∑𝑘=1

1

𝑘 is divergent.

Homework: geometric series

Study the series

+∞

𝑘=𝑘∑₀ 𝑧^𝑘.

Series – 3

Definition: absolute convergence

We say that the series

+∞

∑𝑘=0

𝑧_𝑘isabsolutely convergentif

+∞

∑𝑘=0

|𝑧_𝑘|is convergent.

Theorem

If

+∞

∑𝑘=0

𝑧_𝑘isabsolutely convergentthen

+∞

∑𝑘=0

𝑧_𝑘isconvergent.

Beware

The converse is false:

+∞

∑𝑘=1

(−1)^𝑘

𝑘 is convergent but not absolutely convergent.

Series – 4

Theorem: d’Alembert’s ratio test

We consider a series

+∞

𝑘=0∑

𝑧_𝑘. We assume thatℓ ≔ lim

𝑘→+∞| 𝑧_𝑘+1

𝑧_𝑘 |exists, then

• Ifℓ < 1then

+∞

∑𝑘=0

𝑧_𝑘is absolutely convergent.

• Ifℓ > 1then

+∞

∑𝑘=0

𝑧_𝑘is divergent.

We can’t conclude whenℓ = 1:

•

+∞

∑𝑘=0

1is divergent. •

+∞

∑𝑘=1

(−1)^𝑘

𝑘² is absolutely convergent.

•

+∞

∑𝑘=1

(−1)^𝑘

𝑘 is convergent but not absolutely convergent.