# MATH3401

Lecture 18

### Contours

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 $$t_1\neq t_2 \in [a,b].$$

### Contours

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.

### Contours

Example 2

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

### Contours

Example 3

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

### Contours

Example 4

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

### Contours

Remarks

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

Examples 2 and 3 are Jordan curves.

Example 4 is not a Jordan curve.

### Contours

Differentiability

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.

### Contours

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

### Contours

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.

### Contours

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) $\int_C f\left(z\right) dz = \int_a^b f\left( z\left( t\right)\right)z'( t) dt.$