MATH3401
Complex Analysis
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):
- \(z(b)= z(a)\)
- Otherwise \(z(t_1)\neq z(t_2)\) for \(t_1\neq t_2 \in [a,b].\)
Contours
Example 1
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\(z = \left\{ \begin{array}{lr} t+it & 0\leq t\leq 1\\ t+i & 1 \lt t \leq 2 \end{array} \right. \) is a simple arc. |
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Contours
Example 2
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\(z = z_0 + R e ^{i \theta}\), \(0\leq \theta \leq 2 \pi\) |
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Contours
Example 3
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\(z = z_0 + R e ^{-i \theta}\), \(0\leq \theta \leq 2 \pi\) |
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Contours
Example 4
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\(z = z_0 + R e ^{2 i \theta}\), \(0\leq \theta \leq 2 \pi\) |
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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:
- \(z\) is continuous;
- \(z\) 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:
- on the curve;
- inside the curve;
- 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) \[ \int_C f\left(z\right) dz = \int_a^b f\left( z\left( t\right)\right)z'( t) dt. \]