September28 ,2020andSeptember30 ,2020 LINEINTEGRALS http://uoft.me/MAT334-LEC0101 ComplexVariables

http://uoft.me/MAT334-LEC0101

LINE INTEGRALS

September 28^th, 2020 and September 30^th, 2020

Curves – 1

• A (complex)curveis a continuous function𝛾 ∶ [𝑎, 𝑏] → ℂwhere𝑎 < 𝑏. We say that𝛾(𝑎)is thestart-pointof𝛾and that𝛾(𝑏)is itsend-point.

Careful: it is quite common to designate by ”curve” either the function𝛾or its range𝛾([𝑎, 𝑏]).

• We say that a curve𝛾 ∶ [𝑎, 𝑏] → ℂissimpleif it doesn’t admit double points (i.e.

self-intersections), except maybe𝛾(𝑎) = 𝛾(𝑏), formally

(𝑎 ≤ 𝑡¹< 𝑡₂≤ 𝑏and𝛾(𝑡₁) = 𝛾(𝑡₂)) ⟹ (𝑡¹= 𝑎and𝑡₂= 𝑏)

• We say that a curve𝛾 ∶ [𝑎, 𝑏] → ℂisclosedif𝛾(𝑎) = 𝛾(𝑏).

Closed, simple. Not closed, simple.

Closed, not simple. Not closed, not simple.

Curves – 3

Jordan curve theorem – /ʒɔʀdã/

If𝛾 ∶ [𝑎, 𝑏] → ℂis a simple closed curve thenℂ ⧵ 𝛾 ([𝑎, 𝑏])consists of two disjoints open connected sets, one bounded (the inside) and one unbounded (the outside).

Outside Inside

This result seemsobvious, nonetheless it is quite difficult to prove (note that we only assume that 𝛾is continuous, so the curve can be quite nasty, e.g. Koch snowflake).

• We say that a curve𝛾 ∶ [𝑎, 𝑏] → ℂissmoothif𝛾is𝒞¹.

Here𝛾is a function of a real variable, so by𝒞¹, I mean thatℜ(𝛾)andℑ(𝛾)are𝒞¹as real functions.

• We say that a curve𝛾 ∶ [𝑎, 𝑏] → ℂispiecewise-smoothif there exist𝑡₀, … , 𝑡_𝑛∈ [𝑎, 𝑏]such that𝑎 = 𝑡₀< 𝑡₁< ⋯ < 𝑡_𝑛= 𝑏and𝛾is𝒞¹on(𝑡_𝑘, 𝑡_𝑘+1)for𝑘 = 0, … , 𝑛 − 1.

Figure:A piecewise-smooth curve.

Curves – 5

Definition: concatenation/sum

Given two curves𝛾₁∶ [𝑎₁, 𝑏₁] → ℂand𝛾₂∶ [𝑎₂, 𝑏₂] → ℂsuch that𝛾₁(𝑏₁) = 𝛾₂(𝑎₂), we define the concatenation(orsum) of𝛾₁and𝛾₂by(𝛾₁+ 𝛾₂) ∶ [𝑎₁, 𝑏₁+ 𝑏₂− 𝑎₂] → ℂwhere

(𝛾₁+ 𝛾₂)(𝑡) ≔ {

𝛾₁(𝑡) if𝑡 ∈ [𝑎₁, 𝑏₁]

𝛾₂(𝑡 + 𝑎₂− 𝑏₁) if𝑡 ∈ [𝑏₁, 𝑏₁+ 𝑏₂− 𝑎₂] It is obviously a curve, i.e. it is continuous.

𝛾₁(𝑎₁)

𝛾₁(𝑏₁) = 𝛾₂(𝑎₂)

𝛾₂(𝑏₂)

𝛾₁

𝛾₂

𝛾₁(𝑏₁) = 𝛾₂(𝑎₂)

𝛾₁(𝑎₁) = 𝛾₂(𝑏₂)

𝛾₁

𝛾₂

Proposition

If𝛾₁and𝛾₂are piecewise-smooth then so is(𝛾₁+ 𝛾₂).

CAREFUL

It is possible for𝛾₁and𝛾₂to be smooth but for(𝛾₁+ 𝛾₂)not to be smooth, but only piecewise-smooth, since it may not be smooth at thegluing point.

𝛾₁(𝑎₁)

𝛾₁(𝑏₁) = 𝛾₂(𝑎₂)

𝛾₂(𝑏₂)

Curves – 7

Curves are naturally oriented by the usual orientation of[𝑎, 𝑏]:𝛾goes from𝛾(𝑎)to𝛾(𝑏).

Definition: reversed curve

For a curve𝛾 ∶ [𝑎, 𝑏] → ℂwe define(−𝛾) ∶ [𝑎, 𝑏] → ℂby(−𝛾)(𝑡) ≔ 𝛾(𝑎 + 𝑏 − 𝑡).

It is the same curve but with reversed orientation: it is traversed from𝛾(𝑏)to𝛾(𝑎).

𝛾

𝛾(𝑎) 𝛾(𝑏)

−𝛾

𝛾(𝑎) 𝛾(𝑏)

Definition: positive orientation (closed simple curves)

We say that a simple closed curve𝛾ispositively orientedif𝛾keeps its inside on its left.

Figure:A positively oriented simple closed curve Figure:A simple closed curve which is NOT positively oriented

Curves – 9

Examples

• 𝛾 ∶ [0, 1] → ℂ,𝛾(𝑡) = (1 − 𝑡)𝑧₀+ 𝑡𝑧₁.

𝑧₀

𝑧₁

• 𝛾 ∶ [0, 1] → ℂ,𝛾(𝑡) = 𝑒^𝑖𝜋𝑡.

Example

Pick𝜀 ∈ (0, 1)and set𝛾 = 𝛾₁+ 𝛾₂+ 𝛾₃+ 𝛾₄where

• 𝛾₁∶ [0, 𝜋] → ℂ,𝛾₁(𝑡) = 𝑒^𝑖𝑡.

• 𝛾₂∶ [−1, −𝜀] → ℂ,𝛾₂(𝑡) = 𝑡.

• 𝛾₃∶ [0, 𝜋] → ℂ,𝛾₃(𝑡) = 𝜀𝑒^{𝑖(𝜋−𝑡)}.

• 𝛾₄∶ [𝜀, 1] → ℂ,𝛾₄(𝑡) = 𝑡.

ℜ ℑ

Integrals

Definition

For𝑓 ∶ [𝑎, 𝑏] → ℂcontinuous we set

∫

𝑏 𝑎

𝑓 (𝑡)d𝑡 ≔

∫

𝑏 𝑎

ℜ(𝑓 (𝑡))d𝑡 + 𝑖

∫

𝑏 𝑎

ℑ(𝑓 (𝑡))d𝑡

Propositions

For𝑓 ∶ [𝑎, 𝑏] → ℂcontinuous,

ℜ(∫

𝑏 𝑎

𝑓 (𝑡)d𝑡 )=

∫

𝑏 𝑎

ℜ(𝑓 (𝑡))d𝑡

ℑ(∫

𝑏 𝑎

𝑓 (𝑡)d𝑡 )=

∫

𝑏 𝑎

ℑ(𝑓 (𝑡))d𝑡

Definition: line integral

Let𝑆 ⊂ ℂand𝑓 ∶ 𝑆 → ℂbe continuous. Let𝛾 ∶ [𝑎, 𝑏] → ℂbe a piecewise-smooth curve such that𝛾([𝑎, 𝑏]) ⊂ 𝑆. Pick a subdivision𝑎 = 𝑡₀< 𝑡₁< ⋯ < 𝑡_𝑛= 𝑏of[𝑎, 𝑏]such that𝛾is𝐶¹on(𝑡_𝑘, 𝑡_𝑘+1) then theline integral of𝑓 along𝛾is defined by

∫_𝛾𝑓 (𝑧)d𝑧 ≔

𝑘−1

∑𝑘=0∫

𝑡_𝑘+1 𝑡_𝑘

𝑓 (𝛾(𝑡))𝛾^′(𝑡)d𝑡

It doesn’t depend on the choice of the subdivision: any subdivision as above gives the same value for∫_𝛾𝑓.

If𝛾is smooth, we can simply write

∫_𝛾𝑓 (𝑧)d𝑧 =

∫

𝑏 𝑎

𝑓 (𝛾(𝑡))𝛾^′(𝑡)d𝑡

Line integrals – 2

Proposition

• ∫_−𝛾𝑓 = −

∫_𝛾𝑓

• ∫_𝛾

1+𝛾₂

𝑓 =∫_𝛾

1

𝑓 +∫_𝛾

2

𝑓

• ∫_𝛾𝛼𝑓 + 𝛽𝑔 = 𝛼

∫_𝛾𝑓 + 𝛽

∫_𝛾𝑔, 𝛼, 𝛽 ∈ ℂ

Homework

Prove the above properties.

Theorem: reparametrization

Let𝛾₁∶ [𝑎₁, 𝑏₁] → ℂand𝛾₂∶ [𝑎₂, 𝑏₂] → ℂbe two curves.

Assume that𝛾₁= 𝛾₂∘ 𝜑where𝜑 ∶ [𝑎₁, 𝑏₁] → [𝑎₂, 𝑏₂]is a𝒞¹-diffeomorphism, then

• If𝜑is increasing then

∫_𝛾

1

𝑓 =∫_𝛾

2

𝑓,and we say that𝜑preserves the orientation.

• If𝜑is decreasing then

∫_𝛾

1

𝑓 = −

∫_𝛾

2

𝑓,and we say that𝜑reverses the orientation.

Line integrals – 4

Definition: length (or arclength) of a curve

The(arc)lengthof a piecewise smooth¹curve𝛾 ∶ [𝑎, 𝑏] → ℂislength(𝛾) ≔

𝑛−1

∑𝑘=0∫

𝑡_𝑘+1

𝑡_𝑘 |𝛾^′(𝑡)|d𝑡.

Proposition

If𝛾 ∶ [𝑎, 𝑏] → ℂis a smooth curve thenlength(𝛾) =

∫

𝑏

𝑎 |𝛾^′(𝑡)|d𝑡.

Proposition

Let𝑆 ⊂ ℂ,𝑓 ∶ 𝑆 → ℂcontinuous. Let𝛾 ∶ [𝑎, 𝑏] → ℂbe a piecewise-smooth curve such that 𝛾([𝑎, 𝑏]) ⊂ 𝑆 then

|∫_𝛾𝑓 (𝑧)d𝑧

|≤(max

𝛾([𝑎,𝑏])|𝑓 |)length(𝛾)

The max is achieved since𝛾([𝑎, 𝑏])is compact and|𝑓 |continuous.

1A continuous curve may not be rectifiable, e.g. Koch snowflake.

Green’s theorem (real version)

Let𝑆 ⊂ ℝ²be a regular region with a piecewise smooth boundary𝜕𝑆which is assumed to be positively oriented.

LetF∶ 𝑈 → ℝ²be a𝒞¹-vector field where𝑈 ⊂ ℝ²is open and𝑆 ⊂ 𝑈.

Then

∫_𝜕𝑆F⋅dx=

∬_𝑆(

𝜕𝐹₂

𝜕𝑥 −𝜕𝐹₁

𝜕𝑦 )d𝑥d𝑦 or using the other notation for the line integral of a vector field

∫_𝜕𝑆𝑃d𝑥 + 𝑄d𝑦 =

∬_𝑆(

𝜕𝑄

𝜕𝑥 −𝜕𝑃

𝜕𝑦 )d𝑥d𝑦

I suggest you to review the details of Green’s theorem from your multivariable calculus course.

See for instance:

http://www.math.toronto.edu/campesat/ens/1920/0310-notes.pdf

The real Green’s theorem (from multivariable calculus) – 2

• 𝑆regular region means that𝑆is bounded and that𝑆 = 𝑆,̊

or equivalently that𝑆is bounded and that∀𝑢 ∈ 𝜕𝑆, ∀𝑟 > 0, 𝐷_𝑟(𝑢) ∩ ̊𝑆 ≠ ∅.

• 𝜕𝑆positively oriented means that each piece of𝜕𝑆is parametrized by a closed simple curve such that the interior𝑆̊is on its left.

𝛾₁

𝛾₂

𝛾₃

• If𝜕𝑆 = 𝛾₁∪ 𝛾₂∪ ⋯ ∪ 𝛾_ℓthen

∫_𝜕𝑆 =

∫_𝛾

1

+ ⋯ +

∫_𝛾

ℓ

.

Green’s theorem (complex version) - 1

Let𝑆 ⊂ ℂbe a regular region with a piecewise smooth boundary𝜕𝑆which is assumed to be positively oriented.

Let𝑈 ⊂ ℂopen such that𝑆 ⊂ 𝑈.

Let𝑓 ∶ 𝑈 → ℂbe such that(𝑥, 𝑦) ↦ (ℜ(𝑓 ), ℑ(𝑓 ))is𝒞¹in the real sense (i.e. as a real multivariable function).

Then,

∫_𝜕𝑆𝑓 (𝑧)d𝑧 = 𝑖

∬_𝑆(

𝜕𝑓

𝜕𝑥 + 𝑖𝜕𝑓

𝜕𝑦 )d𝑥d𝑦 where

𝜕𝑓

𝜕𝑥 ≔ 𝜕ℜ(𝑓 )

𝜕𝑥 + 𝑖𝜕ℑ(𝑓 )

𝜕𝑥

and 𝜕𝑓

𝜕𝑦 ≔ 𝜕ℜ(𝑓 )

𝜕𝑦 + 𝑖𝜕ℑ(𝑓 )

𝜕𝑦 Homework: check it!

The complex Green’s theorem – version 2

Green’s theorem (complex version) - 2

Let𝑆 ⊂ ℂbe a regular region with a piecewise smooth boundary𝜕𝑆which is assumed to be positively oriented.

Let𝑈 ⊂ ℂopen such that𝑆 ⊂ 𝑈.

Let𝑓 ∶ 𝑈 → ℂbe such that(𝑥, 𝑦) ↦ (ℜ(𝑓 ), ℑ(𝑓 ))is𝒞¹in the real sense (i.e. as a real multivariable function).

Then,

∫_𝜕𝑆𝑓 (𝑧)d𝑧 = 2𝑖

∬_𝑆

𝜕𝑓

𝜕𝑧 d𝑥d𝑦 where

𝜕𝑓

𝜕𝑧 ≔ 1 2 (

𝜕𝑓

𝜕𝑥 + 𝑖𝜕𝑓

𝜕𝑦 ) and

𝜕𝑓

𝜕𝑧 ≔ 1 2 (

𝜕𝑓

𝜕𝑥 − 𝑖𝜕𝑓

𝜕𝑦 )

a continuous nowhere differentiable simple closed curve