17 - Conformal mappings

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welcome to our second lecture in the

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fourth week of our course analysis of a

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complex kind today we'll learn about

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conformal mappings what is a conformal

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mapping intuitively it's simply a

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mapping that preserves angles between

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curves what I mean by that is the

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following

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look at these two curves gamma 1 and

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let's give it an orientation and gamma 2

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they intersect at this point that I drew

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right here and if you look from this

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point in a direction of gamma 1 and then

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look over to where gamma 2 is that gives

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an orientation to the angle between

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those two curves that's the angle

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between the curves I'm talking about now

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let's imagine a function f Maps this

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picture to a separate picture so here is

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f of gamma 1 and here's F of gamma 2 and

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again if I stand at the point of

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intersection and look from F of gamma 1

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to F of gamma 2 I get another oriented

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angle if these two angles are the same

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then we say the mapping F preserved that

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angle so it can form a mapping is a

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mapping that preserves all angles

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between all curves you could possibly

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draw in order to make this precise we're

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going to have to define exactly what we

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mean by a curve and also what we mean by

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the angle between curves how exactly do

01:26

you measure an angle between curvy

01:29

segments that's what we're going to do

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right now let's start with the notion of

01:35

a path a path in the complex plane from

01:37

a point A to a point B is a continuous

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function gamma that lives on an interval

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this is an interval on the real line so

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you know an inter from A to B so

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continuous function from this interval

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into the complex plane such that gamma

01:58

of a is a and gamma of b is b it's a

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continuous function and so between a and

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b you'll find the image of this integral

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as some kind of continue is not an

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interrupted path so here I drew a point

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gamma of T for example let's look at

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some examples gamma of T is the function

02:25

two plus I plus e to the I T for T

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between 0 and pi what does that look

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like let me draw some axes let's try to

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figure this out e to the I T we know

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describes a circle of radius 1 if T

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one's from 0 to 2 pi then we're running

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from 0 to pi so it's going to be a half

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a circle but we're also shifting that

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circle because we're adding 2 plus I to

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all those numbers 2 plus I say right

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here and now we're adding e to the I T

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to that so this is going to be a curve

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like this 1/2 a circle centered at 2

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plus I so this is our function gamma of

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T this is the point gam of 0 and this is

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the point gamma of PI so in our

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definition up here a is 0 B is PI and

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gamma of t is this function that you see

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down here we could break this function

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up into its real part X of T and its

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imaginary part y of T by breaking up e

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to the I T as cosine T plus I sine T so

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just like we did for complex valued

03:47

function of a complex variable we can

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break up this complex valued function of

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a real variable into its real part and

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imaginary part the only difference is

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that T is a real variable here and not a

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complex variable and that's where we get

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a path bar it says something

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two-dimensional asked the image look at

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another example gamma of T equals 2 plus

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I plus T times -3 minus 5 I and this

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time T goes from 0 to 1 so a 0 and B is

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1 let's first of all find out from where

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to where in the complex plane this path

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works when T is equal to 0

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gamma of zero equals two plus I plus

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nothing so just two plus I in other

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words this path starts at two plus I

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this is my point gamma of zero where

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does it end

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what end said gamma of 1 what is gamma

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of 1 that's 2 plus I plus 1 times minus

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3 minus 5 I and that is equal to 2 minus

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3 which is minus 1 plus I times 1 minus

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5 so minus 4 so minus 1 minus 4i that is

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somewhere down here what is the path to

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in between keep subtracting more and

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more from gamma of 0 it is a linear

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mapping from gamma of 0 to Kim of 1 and

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again we could break this up into the

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real part and the imaginary part if we

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wanted to

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use another example gamma of T equals T

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times e to the I T again let's try to

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draw this this time relating T run from

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zero to three PI gamma of zero

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what's zero times whatever is equal to

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zero gamma of three pi is equal to three

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pi times e to the I times 3 pi but

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either the items 3 pi is the same is e

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to the I pi because the function e to

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the I something is periodic with period

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2pi but either i pi is negative 1 so

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that is minus 3 PI so the starts at 0

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and an end somewhere way out here is

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minus 3 pi when T is equal to PI over 2

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normally that would be at the 90-degree

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turning point of the circle the value is

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e to the I PI over 2 which is I times pi

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over 2 so we're going to be right here

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so the curve does something like this

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it's like a circle but instead of being

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a circle of radius 1 the radius keeps

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increasing what does it do from there 2

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pi here if the radius keeps growing 2 pi

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2 pi it's not a good picture but you get

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the idea it's a bad picture really you

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can do much better than that but this is

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a spiral this curve spirals from a

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radius of 0 and the radius keeps

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increasing and the curve keeps going

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around and around and around here's

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another example gamma of T is defined in

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three different ways depending on

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whether T is between 0 and 1 between 1

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and 2 or between 2 and 3 let's try to

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draw this curve as well

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40 between 0 and 1 we're looking at the

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function T times 1 plus I when T is

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equal to 0 that's isn't 0 when T is

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equal to 1 we're at 1 plus I which is

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say here if this is 1 and this is I so

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from 0 to 1 that's what the curve does

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next between 1 and 2 we're looking at T

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plus I in other words the real part

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changes from 1 to 2 but the imaginary

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part stays fixed at 1 so in other words

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this is what happens the imaginary part

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stays fixed and the real part increases

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from 1 to 2 between 2 & 3 we're looking

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at 2 plus I times 3 minus T so now the

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real part remains fixed and the

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imaginary part changes and it starts

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with T equals 2 and 3 minus 2 is 1 and I

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so it starts right here and decreases

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from there to 0 in other words this is

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what the curve does so altogether this

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is my curve gamma of T we say that the

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path is smooth if these functions X of T

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and Y of T that make up gamma of T's if

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you split up gamma of T into is real

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part X of T and it's imaginary of y of T

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if these two functions are smooth and by

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that we mean that they have as many

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derivatives as we need to let me quickly

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remind you of the example that we just

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looked at example number one was this

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semicircle example number two was the

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straight line segment example number

09:58

three was our spiral

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and example number four was this path

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that was put together from line segments

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like this so in these examples this

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first one is nice and smooth you could

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have found the derivatives in fact of X

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of T and Y of T next one is just this

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nice and smooth this one is just as nice

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and smooth but this one has some corners

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in the derivative these functions X of T

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and Y of T would have a hard time

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existing at these corners so 1 2 & 3 are

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smooth

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whereas 4 is called piecewise smooth

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because away from the corners everything

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is good but at the corners the

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derivatives don't exist so this is

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called piecewise smooth which means it's

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put together or concatenated from

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finally many smooth paths the term curve

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is typically used for a smooth or a

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piecewise smooth path so all these

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examples from above are allowable for

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being curves now if gamma written as X

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plus iy is a smooth curve and T 0 is a

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point in that interval where the

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function gamma is defined then we say

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the derivative gamma Prime at T 0 is X

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prime of to 0 plus iy prime of to 0 and

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that is a tangent vector to the curve

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gamma at T 0 and let me demonstrate to

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you why that is the case

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so suppose here's this curve gamma and

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suppose here is gamma of T 0 that's the

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point we're looking at now let's look at

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this derivative gamma prime of T 0 what

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is the derivative the derivative is

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obtained by looking at gamma of T 0 plus

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h which is maybe this point here and

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looking at the difference gamma of T 0

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plus h minus gamma of to 0 and dividing

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that by H but gamma of T 0 plus h minus

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gamma of T 0 is this

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and we're dividing that by age which is

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a real number so it changes the length

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but not the direction of this vector as

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H moves closer and closer to zero the

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second point most closer and closer to

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the first point there by this vector

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becomes more and more a tangent vector

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in the limit we get the derivative which

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is the tangent vector to this curve at

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the point gamma of T zero so the

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derivative represents a tangent vector

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to the curve at two zero

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now if you have two smooth curves gamma

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1 and gamma 2 and they intersect at a

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point Z 0 then we say the angle between

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these two curves is simply the angle

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between the 2 tangent vectors so you

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find a tangent vector to gamma 1 you

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find a tangent vector to gamma 2 and you

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take the angle between the two of them

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from gamma 1 to gamma 2 that is defined

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as the angle between two curves because

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we know how to find the angle between

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two lines the angle between the curves

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is simply defined us the angle between

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the tangent vectors to the two curves

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let's look at an example suppose gamma 1

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is given by gamma 1 of T's e to the I

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t's again that's a circle and we're

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letting T run from 0 to PI so you get

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this curve right here this curve gamma 1

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right there gamma 2 is also of the form

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2 plus I plus 2 e to the I T so it's

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also a circle this circle is centered at

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2 plus I however and it's radius is 2

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and we're starting at PI over 2 so we're

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starting at this point up here and like

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runs through T equals 3 PI over 2 so all

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the way down here so gamma 2 is this

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green curve over here these two curves

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intersect at I in fact gamma 1 of PI

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over 2 is the same as gamma 2 of PI

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that's where they intersect at i let's

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find the tangent vectors to these two

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curves gamma 1 prime the derivative can

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be found in two ways either you just use

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the chain rule and you get its ie to the

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I T or if you wanted to write gamma 1 of

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T in real part and imaginary part that

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would be cosine T plus I sine T and then

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you could find the derivatives

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separately so gamma 1 prime of T would

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then be the derivative of cosine which

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is minus sine T plus I times the

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derivative of sine T which is cosine T

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if you factor out an i you notice this

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cosine T is left by itself and

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factor out an eye from minus sign T

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that's the same as writing plus I times

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sine T in these prep please write here

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because if you were to multiply this I

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back in there the I squared gives you

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this negative sign but cosine T plus I

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sine T is simply e to the I T so this is

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I times e to the I T or you could have

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just as I said apply the chain rule the

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derivative of e to the I T can be found

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as I times e to the I T gamma 1 Prime at

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the point where these two curves

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intersect namely PI over two is then I

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times e to the I PI over two but e to

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the I PI over two is another I so we get

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I squared which is negative one so the

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tangent vector to the curve gamma 1 at T

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equals PI over two or at the point I is

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minus one I move this vector and had it

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start right at the point I and it points

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simply in the negative direction it has

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length negative one so that is this

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vector right here it's the tangent

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vector to the curve gamma one at the

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point I we can do the same thing for

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gamma 2 gamma two prime of T 2 is a

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constant

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Isak constant the derivatives is 0 the

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derivative of 2 e to the I T is 2i e to

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the I T if you plug in PI which is the

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parameter that yields the point I for

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gamma 2 you get gamma 2 prime is 2i e to

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the I T e to the I T is negative 1 so

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you find 2i times negative 1 which is

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minus 2i if I attach a factor the vector

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minus 2i at this point I I get this

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vector right here in other words the

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angle between these two curves is this

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angle right there so this angle points

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counterclockwise and that is what we

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call the angle PI over 2 the angle

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between those two curves equals PI over

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2

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finally we're able to really define what

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it means for a function to be conformal

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we say your function is conformal if it

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preserves angles between Kirk's so that

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means precisely a smooth complex valued

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function Chi is conformal at a point Z

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zero if whenever gamma 1 and gamma 2 are

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two curves that intersect at that point

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Z zero so we need to encourage gamma 1

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and gamma 2 that intersect there with

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nonzero tangents then they're images

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under G namely do you compose with gamma

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1 and G composed gamma to have also

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nonzero tangents they are and intersect

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at the same angle so you know suppose

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this is G of gamma 1 and this is G of

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gamma 2 the requirement would be that

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the angle from gamma 1 to gamma 2 is the

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same as the angle from G of gamma 1 to G

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of gamma 2 when you're standing at this

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intersection point this is the point G

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of C 0 and the angle is measured in

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terms of their tangents a conformal

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mapping of a domain D onto some V is a

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continuously differentiable mapping that

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is conformal at each point in D and maps

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T 1 to 1 onto B it is that's not enough

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to simply be conformal at each point in

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the domain D moreover we do require that

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the mapping math D in a one-to-one

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manner onto V then we say we're dealing

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with a conformal mapping otherwise we

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can say the mapping is conformal at a

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point but in order for it to be a

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conformal mapping we require that is

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conformal at each point in addition to

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being one-to-one and onto B

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what does all this have to do with

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analytic functions what analytic

19:38

functions kind of brink one formality

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with them for free under very minimal

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assumptions if F is an analytic function

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in a set U and C 0 is a point in this

19:52

set u such that it's derivative is non

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zero at C 0 then FS automatically

19:59

control at that point analytic functions

20:03

are confirmed almost everywhere except

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where the derivative is equal to 0 as

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soon as the derivative of an analytic

20:09

function is non zero the function is

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conformal at that point it preserves

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angles between curves let me show you

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the reason why that is true so both

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gamma is a curve that goes through this

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point Z 0 and then what is the

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derivative of F of gamma the derivative

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of f of gamma at t 0 by the chain rule

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is f prime of gamma of t 0 times gamma

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prime of t 0 but F prime of gamma of T 0

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gamma of T 0 is just our Z 0 so that's F

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prime of Z 0 times gamma prime of T 0

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this was the tangent vector to the curve

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gamma hat c0 this is the tangent vector

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to the curve f of gamma at F of C 0 and

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how do they differ from each other they

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differ by being multiplied with F prime

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of C 0 which by assumption is a nonzero

21:10

number what does it mean to be

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multiplied by a complex number

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multiplying by a complex number means

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we're rotating and possibly stretching

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so the tangent vector to the original

21:22

curve maybe gets rotated and possibly

21:24

stretched but it's rotated and stretched

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by this number F prime of C 0 so now I

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have two curves gamma 1 and gamma 2

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through this point Z 0 and with their

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respective tangent vectors then the

21:38

image curves the tangent vectors to

21:41

those image curves are also obtained

21:44

from the original tangent

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just by simply rotating and stretching

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with the exact same rotation and

21:50

stretching stretching doesn't change an

21:52

angle at all rotation is a rigid

21:56

Euclidean motion and if you rotate two

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vectors by the same amount then in their

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image the angle hasn't changed the angle

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doesn't change by a rotation and that's

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why the angle is preserved under

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analytic functions as long as the

22:14

derivative is nonzero

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let's look at some examples as look at

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the function f of Z equals Z squared

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let's restrict it to the right half

22:24

plane so U is this right half plane over

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here if we restrict after the right half

22:31

plane it's a one-to-one function on to

22:35

the whole complex plane minus the

22:37

negative real axis and furthermore in

22:41

the right half plane the derivative of

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this function is 2 Z which is never

22:47

equal to zero in you because we're

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seeing away from the origin you're

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requiring the real part to be actually

22:53

positive so the origin itself is not in

22:57

there which means the function is

22:59

one-to-one has a non zero derivative and

23:01

Maps it on to the whole plane minus the

23:06

negative real axis so Z squared is a

23:08

conformal mapping from the right half

23:11

plane onto the whole plane minus the

23:13

negative real axis

23:21

the function e to the Z is conformal at

23:24

each point in C because it's derivative

23:28

is never equal to 0 f prime of Z is also

23:31

equal to e to the C that's never equal

23:33

to 0 no matter what you plug in and so e

23:36

to the Z is conformal at each point in

23:38

the complex plane but f is not a

23:43

one-to-one mapping in C because we know

23:45

it is periodic with period 2pi keeps

23:49

repeating so if we look at the function

23:51

on the whole complex plane is never

23:53

going to be a conformal mapping however

23:56

if you restrict e to the Z 2 for example

23:59

this strip with the imaginary part is

24:02

between 0 & 2 pi the function does not

24:07

repeat in there because for any point it

24:10

repeats 2 pi later but you can't have

24:13

two points in this strip that if you buy

24:15

from each other so if you restrict after

24:18

this strip then it's derivative is still

24:21

nonzero everywhere it's 1 to 1 in there

24:24

and it maps that strip conformally onto

24:27

its image which happens to be the whole

24:30

plane minus the positive real axis we're

24:35

not including the real axis in this

24:38

domain or the line 2 pi which means

24:43

we're not really hitting every point in

24:46

the complex plane as an image we're

24:48

omitting those points that are images

24:50

under F of the real line or the line 2

24:52

pi or the line 4 pi so that's why it's

24:55

the plane minus the positive real axis

24:58

that were mapped onto finally let's look

25:02

at F of Z equals the complex conjugate

25:05

of C that's clearly a one-to-one and

25:08

onto function from C to C however angles

25:12

between curves are reversed in

25:14

orientation let's look at this curve

25:17

gamma 1 for example and a curve gamma 2

25:20

if you find the complex conjugates all

25:24

you have to do is you'll reflect this

25:26

over the x-axis that's what the

25:31

conjugate really does so

25:32

the image looks something like this

25:34

here's F F gamma one then there's F of

25:35

gamma two but where is the angle from

25:38

gamma 1 to gamma 2 was counterclockwise

25:42

the angle from F of gamma 1 to F of

25:46

gamma 2 goes clockwise so even though

25:49

the magnitude of the Engel hasn't

25:51

changed its orientation has changed that

25:54

is not allow for a conformal mapping and

25:56

therefore this mapping is not conformal

26:00

anywhere in the next lecture we'll look

26:03

at Mobius transformations these are

26:05

special conformal mappings with many

26:08

special properties that are very

26:10

exciting would spend two lectures

26:12

learning about these Mobius

26:14

transformations