2 Conformal mapping

In Section 26.1 we saw that the real and imaginary parts of an analytic function each satisfies Laplace’s equation. We shall show now that the curves

$u\left(x,y\right)=\text{constant}\text{and}v\left(x,y\right)=\text{constant}$

intersect each other at right angles (i.e. are orthogonal ). To see this we note that along the curve $u\left(x,y\right)=\text{constant}$ we have $du=0$ . Hence

$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=0$ .

Thus, on these curves the gradient at a general point is given by

$\frac{dy}{dx}=-\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}$ .

Similarly along the curve $v\left(x,y\right)=$ constant, we have

$\frac{dy}{dx}=-\frac{\frac{\partial v}{\partial x}}{\frac{\partial v}{\partial y}}$ .

The product of these gradients is

$\frac{\left(\frac{\partial u}{\partial x}\right)\left(\frac{\partial v}{\partial x}\right)}{\left(\frac{\partial u}{\partial y}\right)\left(\frac{\partial v}{\partial y}\right)}=-\frac{\left(\frac{\partial u}{\partial x}\right)\left(\frac{\partial u}{\partial y}\right)}{\left(\frac{\partial u}{\partial y}\right)\left(\frac{\partial u}{\partial x}\right)}=-1$

where we have made use of the Cauchy-Riemann equations. We deduce that the curves are orthogonal.

As an example of the practical application of this work consider two-dimensional electrostatics. If $u=$ constant gives the equipotential curves then the curves $v=$ constant are the electric lines of force . Figure 2 shows some curves from each set in the case of oppositely-charged particles near to each other; the dashed curves are the lines of force and the solid curves are the equipotentials.

Figure 2

In ideal fluid flow the curves $v=$ constant are the streamlines of the flow.

In these situations the function $w=u+\text{i}v$ is the complex potential of the field.

2.1 Function as mapping

A function $w=f\left(z\right)$ can be regarded as a mapping, which maps a point in the $z$ -plane to a point in the $w$ -plane. Curves in the $z$ -plane will be mapped into curves in the $w$ -plane.

Consider aerodynamics where we are interested in the fluid flow in a complicated geometry (say flow past an aerofoil). We first find the flow in a simple geometry that can be mapped to the aerofoil shape (the complex plane with a circular hole works here). Most of the calculations necessary to find physical characteristics such as lift and drag on the aerofoil can be performed in the simple geometry - the resulting integrals being much easier to evaluate than in the complicated geometry.

Consider the mapping

$w=z^2$ .

The point $z=2+\text{i}$ maps to $w={\left(2+\text{i}\right)}^2=3+4\text{i}$ . The point $z=2+\text{i}$ lies on the intersection of the two lines $x=2$ and $y=1$ . To what curves do these map? To answer this question we note that a point on the line $y=1$ can be written as $z=x+\text{i}$ . Then

$w={\left(x+\text{i}\right)}^2=x^2-1+2x\text{i}$

As usual, let $w=u+\text{i}v$ , then

$u=x^2-1\text{and}v=2x$

Eliminating $x$ we obtain:

$4u=4x^2-4=v^2-4$ so $v^2=4+4u$ is the curve to which $y=1$ maps.

Example 5

Onto what curve does the line $x=2$ map?

Solution

A point on the line is $z=2+y\text{i}$ . Then

$w={\left(2+y\text{i}\right)}^2=4-y^2+4y\text{i}$

Hence $u=4-y^2$ and $v=4y$ so that, eliminating $y$ we obtain

$16u=64-v^2$ or $v^2=64-16u$

In Figure 3(a) we sketch the lines $x=2$ and $y=1$ and in Figure 3(b) we sketch the curves into which they map. Note these curves intersect at the point $\left(3,4\right)$ .

Figure 3

The angle between the original lines in (a) is clearly $90^0$ ; what is the angle between the curves in (b) at the point of intersection?

The curve $v^2=4+4u$ has a gradient $\frac{dv}{du}$ . Differentiating the equation implicitly we obtain

$2v\frac{dv}{du}=4\text{or}\frac{dv}{du}=\frac{2}{v}$

At the point $\left(3,4\right)$ $\frac{dv}{du}=\frac{1}{2}$ .

Task!

Find $\frac{dv}{du}$ for the curve $v^2=64-16u$ and evaluate it at the point $\left(3,4\right)$ .

Answer

In general, if two curves in the $z$ -plane intersect at a point $z_0$ , and their image curves under the mapping $w=f\left(z\right)$ intersect at $w_0=f\left(z_0\right)$ and the angle between the two original curves at $z_0$ equals the angle between the image curves at $w_0$ we say that the mapping is conformal at $z_0$ .

An analytic function is conformal everywhere except where $f^{\prime }\left(z\right)=0$ .

Task!

At which points is $w=e^z$ not conformal?

Answer

2.2 Inversion

The mapping $w=f\left(z\right)=\frac{1}{z}$ is called an inversion . It maps the interior of the unit circle in the $z$ -plane to the exterior of the unit circle in the $w$ -plane, and vice-versa. Note that

$w=u+\text{i}v=\frac{x}{x^2+y^2}-\frac{y}{x^2+y^2}\text{i}$ and similarly $z=x+\text{i}y=\frac{u}{u^2+v^2}-\frac{v}{u^2+v^2}\text{i}$

so that

$u=\frac{x}{x^2+y^2}\text{and}v=-\frac{y}{x^2+y^2}$ .

A line through the origin in the $z$ -plane will be mapped into a line through the origin in the $w$ -plane. To see this, consider the line $y=mx$ , for $m$ constant. Then

$u=\frac{x}{x^2+m^2{x}^2}\text{and}v=-\frac{mx}{x^2+m^2{x}^2}$

so that $v=-mu$ , which is a line through the origin in the $w$ -plane.

Task!

Consider the line $ax+by+c=0$ where $c\ne 0$ . This represents a line in the $z$ -plane which does not pass through the origin. To what type of curve does it map in the $w$ -plane?

Answer

Similarly, it can be shown that a circle in the $z$ -plane passing through the origin maps to a line in the $w$ -plane which does not pass through the origin. Also a circle in the $z$ -plane which does not pass through the origin maps to a circle in the $w$ -plane which does pass through the origin. The inversion mapping is an example of the bilinear transformation :

$w=f\left(z\right)=\frac{az+b}{cz+d}\text{where we demand that}ad-bc\ne 0$

(If $ad-bc=0$ the mapping reduces to $f\left(z\right)=$ constant.)

Task!

Find the set of bilinear transformations $w=f\left(z\right)=\frac{az+b}{cz+d}$ which map $z=2$ to $w=1$ .

Answer

Find the bilinear transformations for which $z=-1$ is mapped to $w=3$ .

Answer

Example 6

Find the bilinear transformation which maps

1. $z=2$ to $w=1$ , and
2. $z=-1$ to $w=3$ , and
3. $z=0$ to $w=-5$

Solution

We have the answers to 1. and 2. from the previous two Tasks:

If $z=0$ is mapped to $w=-5$ then $-5=\frac{b}{d}$ so that $b=-5d$ . Substituting this last relation into the first two obtained we obtain

Solving these two in terms of $d$ we find $2c=11d$ and $2a=17d$ . Hence the transformation is:

$w=\frac{17z-10}{11z+2}$ (note that the $d$ ’s cancel in the numerator and denominator).

Some other mappings are shown in Figure 4.

Figure 4

As an engineering application we consider the Joukowski transformation

$w=z-\frac{{\ell }^2}{z}$ where $\ell$ is a constant.

It is used to map circles which contain $z=1$ as an interior point and which pass through $z=-1$ into shapes resembling aerofoils. Figure 5 shows an example:

Figure 5

This creates a cusp at which the associated fluid velocity can be infinite. This can be avoided by adjusting the fluid flow in the $z$ -plane. Eventually, this can be used to find the lift generated by such an aerofoil in terms of physical characteristics such as aerofoil shape and air density and speed.

Exercise

Find a bilinear transformation $w=\frac{az+b}{cz+d}$ which maps

1. $z=0$ into  $w=\text{i}$
2. $z=-1$  into  $w=0$
3. $z=-\text{i}$  into  $w=1$

Answer

1. $z=0,w=\text{i}$ gives  $\text{i}=\frac{b}{d}$ so that $b=d\text{i}$
2. $z=-1,w=0$ gives $0=\frac{-a+b}{-c+d}$ so $-a+b=0$ so $a=b$ .
3. $z=-\text{i},w=1$ gives $1=\frac{-a\text{i}+b}{-c\text{i}+d}$ so that  $-c\text{i}+d=-a\text{i}+b=d+d\text{i}$ (using 1. and 2.)

We conclude from 3. that $-c=d$ . We also know that $a=b=d\text{i}$ .

Hence $w=\frac{d\text{i}z+d\text{i}}{-dz+d}=\frac{\text{i}z+\text{i}}{-z+1}$