Complex Analysis

Lecture 23

Conformal mapping


  • \(f: z\mapsto w\),
  • \(f\) 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.  \(\Delta U = 0\).