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What is a conformal mapping?

Intuitively, it's simply a mapping that preserves angles between curves.

What I mean by that is the following.

Look at these two curves, gamma one, and it's given an orientation, and gamma two.

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They intersect at this point that I drew right here.

And if you look from this point in a direction of gamma one and

then look over to where gamma two is

that gives the orientation to the angle between those two curves.

That's the angle between the curves I am talking about.

Now let's imagine a function f maps this picture to a separate picture.

So here is f of gamma one, and here is f of gamma two.

And again, if I stand at the point of intersection and

look from f of gamma one to f of gamma two, I get another oriented angle.

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In order to make this precise, we're going to have to define

exactly what we mean by a curve, and also what we mean by the angle between curves.

How exactly do you measure an angle between curvy segments?

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It's a continuous function, and so between A and B, you'll find the image of this

interval, as some kind of continuous, non interrupted path.

So here I drew a point gamma of t for example.

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Let's try to figure this out.

E to the it we know describes the circle of radius 1 if t runs from zero to 2 pi.

Now, we're running from zero to pi so it's going to be a half a circle, but

we're also shifting that circle because we're adding 2 + i to all those numbers.

2 + I is say right here, and

now we're adding e to the i t to that.

This is going to be a curve like this,

a half a circle centered at 2 + i.

So this is our function, gamma of t, this is the point gamma of 0,

and this is the point gamma of pi.

So in our definition up here, a is 0, b is pi, and

gamma of t is this function that you see down here.

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So just like we did for

complex valued function of a complex variable, we can break up this complex

valued function of a real variable into its real part and imaginary part.

The only difference is that t is a real variable here and not a complex variable.

And that's where we get a path versus something two dimensional as the image.

Let's look at another example.

Gamma of (t) = (2 + i) + t (- 3- 5 i) and this time.

And this time t goes from zero to one.

So a is zero and b is one.

Let's first of all find out from where to

where in the complex plane in this path works.

When t = 0, gamma of 0

= 2 + i + nothing.

So, just 2 + i.

In other words, this path starts at 2 + i.

This is my point gamma of 0.

Where does it end?

Well, it ends at gamma of 1.

What is gamma of 1?

That's 2 + i +1 (- 3- 5 i).

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You keep subtracting and more and more from gamma of (0).

It is a linear mapping from gamma (0) to gamma (1).

And again, we could break this up into the real part and

the imaginary part if we wanted to.

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But e to the i x 3 pi is the same as e to the i pi because

the function e to the i something is periodic with

period 2 pi but e to the i pi is -1 so that is- 3 pi.

So this starts at zero and it ends somewhere way out here at- 3 pi.

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So, the curve does something like this.

It's like a circle but instead of being a circle radius 1,

the radius keeps increasing.

What does it do from there to pi?

From here the radius keeps growing to pi.

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2 pi.

It's not a good picture, but you get the idea.

It's a bad picture, really.

You can do better much than that.

But this is a spiral, this curve spirals from a radius of 0.

And the radius keeps increasing.

And th curve keeps going around, and around, and around.

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Gamma of T is defined in 3 different ways, depending on whether T is between 0 and 1,

between 1 and 2, or between 2 and 3.

Let's try to draw this curve, as well.

For t between zero and one we're looking at the function t ( 1 + i ).

When t=0, that's just 0.

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When t=1, we're at 1+i, which is say here.

If this is 1 and this is i.

So from 0 to 1, that's what the curve does.

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Next, between 1 and 2, well looking at t + i,

in other words the real part changes from one to two,

but the imaginary part stays fixed at 1.

So in other words this is what happens.

The imaginary part stays fixed and the real part increases from 1 to 2.

Between 2 and 3, we're looking at 2+i(3-t).

Now the real part remains fixed and the imaginary part changes.

And it starts with equals 2, at 3-2 is 1 at i, so it starts right here.

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We say that the path is smooth if these functions x(t) and

y(t) that make up gamma of t.

So if you split up gamma of t into it's real part x(t) and it's imaginary y(t).

If these two functions are smooth, and

by that we mean that they have as many derivatives as we need to.

Let me quickly remind you of the examples that we just looked at.

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So in these examples, this first one is nice and smooth.

You could have found the derivatives in fact of x(t) and y(t).

Next one's just as nice and smooth.

This one is just as nice and smooth, but this one has some corners.

And the derivative of these functions x(t) and

y(t) would have a hard time existing at these corners.

So 1, 2, and 3 are smooth.

Whereas 4 is called piece wise smooth

because away from the corners everything is good.

But at the corners the derivatives don't exist.

So this is called piece wise smooth, which means it's put together or

concatenated from finitely many smooth paths.

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Now if gamma written as x plus i y is a smooth curve and

t zero is point in that interval where the function gamma is defined.

Then we say the derivative gamma prime at t zero is

x prime up to zero plus y prime of t zero.

And that is a tangent vector to the curve gamma at t zero.

And let me demonstrate to you why that is the case.

So, suppose here's this curve gamma.

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And suppose here is gamma of 2, 0.

That's the point we're looking at.

Now let's look at this derivative, gamma prime of 2,0.

What is the derivative?

The derivative is obtained by looked of t 0 plus h, which is maybe this one here.

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And we're dividing that by h, which is a real number,

so that changes the length but not the direction of this vector.

As h moves closer and closer to zero,

the second point moves closer and closer to the first point.

Thereby this vector becomes more and more a tangent vector.

In the limit, we get the derivative,

which is the tangent vector to this curve at the point gamma of t zero.

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Now, if you have two smooth curves gamma 1 and gamma 2.

And the inner second of point C zero then we see the angle between these two curves

is simply the angle between the two tangent vectors.

So you find the tangent vector to gamma one.

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And you take the angle between the two of them, from gamma 1 to gamma 2.

That is defined as the angle between two curves.

because we know how to find the angle between two lines.

The angle between the curves is simply defined as the angle between the tangent

vectors to the two curves.

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Let's look an example.

Suppose gamma 1 is given of gamma 1 of t's eve it.

So, again, that's a circle.

And we're letting t run from zero to pi.

So you get this curve right here.

This curve gamma 1 right there.

Gamma 2 is also of the form 2 plus i plus 2 e to the i t, so it's also a circle.

This circle is centered at 2 plus i however, and it's radius is 2.

And we're starting at pi / 2.

So we're starting at this point up here, and

that runs through t = 3 pi / 2, so all the way down here.

So gamma 2 is this green curve over here.

These two curves intersect at i.

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That's where they intersect at i.

Let's find the tangent factor to these 2 curves.

Gamma 1 prime, the derivative can be found in two ways.

Either you just you the sheng rule, and you get it's i equal to i t.

Or if you wanted to write gamma 1 of t in a real part and an imaginary part,

that would be cosine t + i sine t, and

then you could find the derivatives separately.

So gamma 1 prime of t would then be the derivative of cosine which is minus sin t

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That's the same as right + i sine t.

In these parenthesis right here.

Because if you were to multiply this i back in there,

the i squared gives you this negative sign.

But cos t + i sin t is simply e to the i t.

So this is i times e to the i t.

Or you could have chose, as I said, apply the chain rule to the derivative,

e to the i t can be found as i times e to the i t.

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Gamma 1 prime at the point where these two curves intersect

namely pi over 2 is then I times E to the I pi over 2 but

either the I power of 2 is another I, so we can I square which is negative 1.

So, the tangent vector to the curve gamma 1

at t = pi / 2, or at the point i, is -1.

I move this vector and had it start right at the point i.

And it points simply in the negative direction, it has length -1.

So that is

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this vector right here is the tangent vector to the curve gamma1 at the point i.

We can do the same thing for gamma2.

Gamma2 prime of t.

2 is a constant, i is a constant, the derivatives are 0.

The derivative of 2 e to the i t is 2 i e to the i t.

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If you plug in pi, which is the parameter that yields the point i for

gamma2, you get gamma2 prime = 2 i e to the i t.

e to the i t = -1, so you find 2 i x -1, which is -2 i.

If I attach a vector, the vector -2 i at this point i,

I get this vector right here.

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In other words, the angle between these two curves is this angle right there.

So this angle points counter clockwise, and

that is what we call the angle pi over 2.

The angle between those two curves equals pi over 2.

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Finally we're able to really define what it means for a function to be conformal.

We say a function is conformal if it preserves angles between curves.

So that means precisely.

A smooth complex value function g is conformal at a point z 0 if

whenever gamma 1 and gamma 2 are two curves that intersect at that point z 0.

So I need two curves, gamma 1.

And gamma 2, that intersect there.

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G(gamma 2).

The requirement would be that the angle from gamma 1

to gamma 2 is the same as angle from g of gamma 1 to g of gamma 2 when you're

standing at this intersection point.

This is the point g(z0).

And the angle is measured in terms of their tangents.

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Moreover, we require that for mapping, map D in a 1 to 1 manner onto V.

Then we say we're dealing with a conformal mapping,

otherwise we can say the mapping is conformal at a point.

But in order for it to be a conformal mapping we require that it's conformal

at each point in addition to being one to one and onto V.

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Analytic functions are confirmed almost everywhere except

where the derivative is equal to 0.

As soon as the derivative of an analytic function is non zero,

the function is conformal at that point.

It preserves angles between curves.

Let me show you the reason why that is true.

Suppose gamma is a curve that goes through this point z 0.

And then what is the derivative of f of gamma?

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The derivative of f of gamma at t 0,

by the chain rule is f prime of gamma of 2 0, + gamma prime of 2 0.

But f prime of gamma of 2 0, or gamma of 2 0 is just our z 0.

So that's f prime of z 0 times gamma prime of t 0.

This was the tangent vector to the curve gamma at z 0.

This is the tangent vector

to the curve f of gamma at f of z 0.

And how do they differ from each other?

They differ by being multiplied with f-prime of z0,

which by assumption is a non-zero number.

What does it mean to be multiplied by a complex number?

Multiplying by a complex number means we're rotating and possibly stretching.

So the tangent vector to the original curve, maybe gets rotated and possibly

stretched, but it's rotated and stretched by this number f prime of z zero.

So, now, I have two curves, gamma one and gamma two,

through this point z zero, and with their respective tangent vectors.

Then the image curves.

The tangent vectors to those image curves are also obtained

from the original tangent vectors by simply rotating and stretching.

With the exact same rotation and stretching.

Stretching doesn't change an angle at all, rotation is a rigid Euclidian motion and

if you rotate two vectors by the same amount, then

in their image the angle hasn't changed, the angle doesn't change by a rotation.

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So, the origin itself is not in there.

Which means the function is one to one, has a non-zero derivative and

maps it onto the whole plane minus the negative real axis.

So z's work is a conformal mapping from the right-half plane onto the whole

plane minus the negative real axis.

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The function e to the z is conformal at each point in C

because its derivative is never equal to 0.

f prime of z is also equal to e to the z.

Does never equal to 0 no matter what you plug in.

And so even the 0s conform like each point in the complex plane.

But, f is not a 1 to 1 mapping in C because we know

it is periodic with periodic 2 pi i, it keeps repeating.

So if we look at the function on the whole complex plane it's

never going to be a conformal mapping.

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However, if you restrict E to the Z, to for example,

the strip, with the imaginary part is between the zero and 2 pi.

The function does not repeat in there,

because for any point, it repeats 2 pi I later,

which can't have two points in the strip 2piI from each other.

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So, if you restrict after the strip,

then its derivative is still non-zero everywhere, it's one-to-one in there and

it mounts that strip conformally onto its image,

which happens to be the whole plane minus the positive real axis.

We're not including the real axis in this domain or the line 2 pi i, which

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means we're not really hitting every point in the complex plane as an image.

We're omitting those points that are images under f of the real line or

the line 2 pi i or the line 4 pi i.

So that's why it's the plane minus the positive real

axis that we're mapped on to.

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Finally, let's look with F of Z equals the complex conjugate of Z.

That's clearly a one to one and onto function from C to C, however,

Angles between curves are reversed in orientation.

Let's look at this curve, gamma 1 for example and then the curve gamma 2.

If you find the complex conjugates all you

have to do is you reflect this over the x axis.

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The angle from f of gamma 1 to f of gamma 2 goes clockwise.

So even though the magnitude of the angle hasn't changed it's orientation

has changed.

That is not allowable for a mapping.

And therefore this mapping is not conformal anywhere.

In the next lecture we'll look at mobius transformations.

These are special conformal mappings with many special

properties that are very excite.

[INAUDIBLE] We'll spend two lectures learning about these Mobius

transformations.