Complex Analysis

Lecture 18


Jordan arc (a.k.a simple arc):

\(z(t_1) \neq z(t_2)\) for \(t_1\neq t_2\)

Here \(z(t) = x(t) + i y(t)\), \(t\in [a,b]\).

Jordan curve (a.k.a imple cosed curve):

  1. \(z(b)= z(a)\)
  2. Otherwise \(z(t_1)\neq z(t_2)\) for \(t_1\neq t_2 \in [a,b].\)


Example 1

\(z = \left\{ \begin{array}{lr} t+it & 0\leq t\leq 1\\ t+i & 1 \lt t \leq 2 \end{array} \right. \)

is a simple arc.


Example 2

\(z = z_0 + R e ^{i \theta}\), \(0\leq \theta \leq 2 \pi\)


Example 3

\(z = z_0 + R e ^{-i \theta}\), \(0\leq \theta \leq 2 \pi\)


Example 4

\(z = z_0 + R e ^{2 i \theta}\), \(0\leq \theta \leq 2 \pi\)



Examples 2, 3, and 4 have the same trace.

Examples 2 and 3 are Jordan curves.

Example 4 is not a Jordan curve.



An arc/curve is called differentiable if \(z'(t)\) exists

(\( \forall t\in (a,b)\) for an arc, \( \forall t\in [a,b]\) for a closed curve).

If \(z'\) is also continuous, then \(\ds \int_a^b \left|z'(t)\right| dt\) exists, and defines the arc length.


Note: If \(z(t)\) is a parametrisation of the image arc, we can defined another one by \(t = \Phi(\tau)\) with \[ \Phi (\alpha ) = a \quad \text{and} \quad \Phi (\beta) = b \] such that \[ \Phi \in \mathcal C \big( \left[\alpha,\beta\right] \big) \quad \text{and} \quad \Phi' \in \mathcal C \big( \left(\alpha,\beta\right) \big). \]

So \(z(t) = Z(\tau) = z\left(\Phi \left(\tau\right)\right)\).


Note (cont): Assume \(\Phi' (\tau)>0\; \forall \tau\), then

\( \ds \int_a^b \left|z'(t)\right| dt = \int_{\alpha}^{\beta} \left|z'\left(\Phi \left(\tau\right)\right)\right| \Phi' \left(\tau\right) d\tau \)

\( \qquad \qquad \;\; = \ds \int_{\alpha}^{\beta} \left|Z'\left(\tau \right)\right| d\tau. \)

i.e. arc-length is independent of parametrisation.


A contour is an arc/curve/simple closed curve such that:

  1. \(z\) is continuous;
  2. \(z\) is piecewise differentiable.

If initial and final values of \(z\) coincide and no other self intersection, we have a simple closed contour.

Jordan curve theorem

Any simple closed contour divides \(\C\) into three parts:

  1. on the curve;
  2. inside the curve;
  3. outside the curve.

Remark: Statement still holds if we remove 2, CARE!

Contour integral

\( \ds \int_C f\left(z\right) dz, \)      or \(\ds \quad \int_{z_1}^{z_2} f\left(z\right) dz\)

If we know:

a) integral is independent of path \(z_1\) to \(z_2\);

b) path is understood.

Contour integral

Suppose the contour \(C\) is specified by \(z(t)\) where \[ z_1=z(a), \; z_2=z(b), \;\text{ with }\; a\leq t\leq b, \]

and suppse \(f\) is piece-wise continuous (pwc) on \(C\).

Then (cf. line integrals) \[ \int_C f\left(z\right) dz = \int_a^b f\left( z\left( t\right)\right)z'( t) dt. \]