# MATH3401

Lecture 23

### Conformal mapping

Consider

• $$f: z\mapsto w$$,
• $$f$$ analytic,
• $$f'(z_0) \neq 0.$$

Then locally (near $$z_0$$) $$\,f$$ preserves

• angle;
• orientation;
• shape.

We say that $$f$$ is conformal.

### Conformal mapping

$$\Gamma_1 = f(C_1)$$,   $$\Gamma_2 = f(C_2)$$.

Conformality $$\Rightarrow$$ $$\beta = \alpha$$.

### Conformal mapping

If orientation (i. e., sense) is not necessarily preserved, but angle magnitude is, the map is called isogonal.

Consider $$f$$ an analytic function with $$f'(z_0) = 0$$:
$$z_0$$ is a critical point of $$f$$.

Angle will not be preserved. Can show: angle will be multiplied by $$m$$, where $$m$$ is the smallest integer such that $$\,f^{(m)}(z_0) \neq 0.$$

### Local inverses

Conformality $$\Rightarrow$$ locally one-to-one and onto,
i. e.   $$f$$ has a local inverse.

### Local inverses

This follows from MATH2400/2401: Inverse function theorem.

$$f:(x,y) \mapsto (u,v)$$ sufficiently smooth is locally invertible if $\text{det}\left(J_f\right) = \left| \begin{array}{cc} u_x & u_y\\ v_x & v_y \end{array} \right| \neq 0.$

### Local inverses

In our case, $$f$$ is analytic $$\Rightarrow u_x, u_y, v_x, v_y$$ are continuous and

$$\text{det}\left(J_f\right)$$ $$= u_x v_y - u_y v_x$$ $$= u_x^2 + v_x^2$$

$$\quad \qquad = |u_x+iu_y|^2$$ $$= \left|\,f'\right|^2$$

$$\quad \qquad \neq 0$$ at $$z_0$$.

### Harmonic functions

##### Precise definition

$$u: \Omega\subset \R^n \ra \R^n$$ harmonic says:

1. $$u$$ and partials of order 1 and 2 are continuous;
2. $$\Delta u = 0$$.

### Harmonic functions

$$\Omega \subseteq \R^2$$,

$$U: \Omega \ra \R$$ such that

$$\Delta U = 0$$.

### Harmonic functions

Laplacian or Laplace operator

$$\Delta U = U_{xx} + U_{yy}$$

In $$\R^3$$ $$\Delta U = U_{xx} + U_{yy} + U_{zz}$$

In $$\R^n$$ $$\Delta U = \sum_{j=1}^{n} U_{jj}$$

Models many physical situations in "steady state"

### Harmonic functions

$$\Omega \subseteq \R^2$$ or $$\R^3$$

$$\Lambda =$$ sufficiently smooth subdomain of $$\Omega$$ with external normal $$\mathbf v(\mathbf x)$$ on $$\partial \Lambda$$,

$$\mathbf v'$$ unit exterior normal

### Harmonic functions

$$U =$$ density of something on equilibrium.

$$\mathbf F$$: flux density of $$U$$ in $$\Omega$$ in equilibrium.

$\int_{\partial \Lambda} \mathbf F \cdot \mathbf v' dS = 0$ $$dS=$$ surface measure on $$\partial \Lambda$$.

### Harmonic functions

Gauss's divergence theorem:

$\int_{\Lambda} \text{div} \, \mathbf F \, \text{d} \mathbf x = 0$

$$\text{d}\mathbf x= dx \,dy$$ in 2D $$\qquad$$

$$= dx \,dy\, dz$$ in 3D

$$\text{d}\mathbf x= dx_1 \cdots dx_n$$ in $$n$$-D

### Harmonic functions

Since $$\Lambda$$ is (essentially) arbitrary, there holds:

$\text{div} \, \mathbf F = 0 \;\text{ in } \;\Omega,$

i. e. $$\ds \sum_{j=1}^{n} \partial_j F_j = 0\;$$ in $$\;\Omega\quad (*)$$

In many physical situations, $\mathbf F = c \nabla U, \text{ with } c \text{ usually negative}$

$$(*) \Rightarrow$$ $$c\left( \text{div}\, \nabla U\right) =0$$   i. e.  $$\Delta U = 0$$.