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A contour is a piecewise smooth curve together with a direction, and reversing it changes the sign of everything computed along it; the winding number counts how many times a closed contour goes round a point, and it is the quantity the general theorems of this unit are stated with.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to parametrise segments, circles and polygons as contours, compute the length of a contour, say why the direction is part of the contour rather than a detail, decide whether a contour is simple and whether it is closed, and find the winding number of a closed contour about a point, including on a boundary with a hole in it.
Parametrised curves and line integrals from the vector calculus course, and the modulus as a distance. A contour is a parametrised curve in the plane, read as a complex-valued function of a real parameter — which is the only change, and it is enough to make the integral along it a complex number rather than a real one.
A contour is a continuous, piecewise smooth map $z(t)$ from a real interval into the plane, together with the direction in which $t$ increases. It is closed when it ends where it began, and simple when it passes through no point twice. Its length is $\int |z'(t)|\,dt$. The winding number of a closed contour about a point not on it is the number of times it goes round, counted anticlockwise. Positively oriented means the enclosed region stays on the left.
A contour is a function of a real parameter. Write $z(t) = x(t) + iy(t)$ for $t$ in $[\alpha, \beta]$. Three examples cover most of what this course needs:
Length. $L = \int_{\alpha}^{\beta} |z'(t)|\,dt$, which is the same number whatever parametrisation is chosen. A circle of radius $\rho$ has length $2\pi\rho$; a segment has the length of the distance between its ends.
Direction is part of the contour. Reversing $C$ gives a contour usually written $-C$, with the same points and the opposite direction. Every integral along it changes sign. This is the source of essentially every sign in the remaining units, and it is why a problem says anticlockwise.
Winding number. For a closed contour $C$ and a point $p$ not on it, the winding number $n(C, p)$ counts the net number of anticlockwise turns $C$ makes round $p$. A circle traversed once anticlockwise has winding number $1$ about every interior point and $0$ about every exterior one; traversed $k$ times clockwise it has $-k$ and $0$. It is always a whole number, and it is constant on each connected piece of the complement of $C$.
It also has an integral formula, which is the reason it appears here at all:
$$n(C, p) = \frac{1}{2\pi i}\oint_C \frac{dz}{z - p}.$$
The next lesson computes the right-hand side directly for a circle and gets $1$. So how many times round and what a particular integral comes to are the same number, and the residue theorem is in the end a statement about turns.
Another way: picture
Stand at the point $p$ and watch a walker follow the contour, keeping a string taut between you. The winding number is the number of times the string wraps round you by the time the walker gets back to the start — positive for anticlockwise, negative for clockwise, and zero if the walker never got round you at all. Nothing about distance enters: a point just outside a huge circle has winding number zero, and a point just inside a tiny one has winding number one.
Another way: steps
Cauchy's theorem is usually first met as: if $f$ is analytic on and inside a simple closed contour, the integral round it is zero. The word inside is doing a lot of work there, and it is harder to define than it looks — the statement that a simple closed curve has an inside and an outside is the Jordan curve theorem, which is genuinely difficult.
Worse, inside stops meaning anything for a contour that is not simple. A figure eight has two loops running opposite ways; a circle traversed three times has an inside, but a theorem that ignores the repetition will get the answer wrong by a factor of three.
The winding number solves both problems at once. It is defined for any closed contour, simple or not; it is a whole number; it is $0$ for every point of the unbounded piece of the complement; and it counts the repetitions correctly. The general forms of Cauchy's theorem, the integral formula and the residue theorem are all stated with it, and read the integral is $2\pi i$ times the sum of the residues, each weighted by the winding number about it.
For the contours in this course — circles, rectangles, triangles, traversed once the positive way — the winding number is $1$ inside and $0$ outside, and the general statement collapses into the familiar one. It is worth knowing that the general statement is there, because the first time a problem asks about a contour that winds twice, the familiar one has nothing to say.
Two contours through exactly the same points can be different contours, and the difference matters. The unit circle anticlockwise and the unit circle clockwise consist of the same points; every integral along them differs by a sign. The unit circle once and the unit circle three times are again the same points, and their integrals differ by a factor of three.
So a contour carries three things: the points, the direction, and how many times each point is visited. A problem statement that says the circle of radius 2 has not finished specifying the contour, and the convention filling the gap is once, anticlockwise.
A second confusion is between length and distance. The length of a contour depends on the route; the distance between its ends does not. Going from $0$ to $4 + 3i$ along the two sides of a rectangle has length $7$ and displacement $5$. The two agree only for a straight contour, which is exactly the statement that a straight line is the shortest route.
Finally, a winding number is not a measure of nearness. A point may sit a thousandth of a unit outside a contour and have winding number $0$, while a point far away inside has winding number $1$. Enclosure, not proximity.
The boundary of the rectangle with corners $0$, $3$, $3 + 2i$, $2i$, anticlockwise: four segments, each $z(t) = p + t(q - p)$ on $[0, 1]$.
Piecewise smooth, not smooth.
Its length is $3 + 2 + 3 + 2 = 10$, and it is simple and closed. Its winding number is $1$ about any interior point and $0$ about any point outside.
Length adds over the pieces.
$z(t) = e^{it}$ on $[0, 2\pi]$: the unit circle once anticlockwise, winding number $1$ about the origin.
The standard parametrisation.
$z(t) = e^{-it}$ on $[0, 2\pi]$: the same points, traversed clockwise, winding number $-1$.
Reversal negates.
$z(t) = e^{it}$ on $[0, 6\pi]$: three times round, winding number $3$, and no longer a simple contour.
Repetition counts.
Two straight pieces, so add their lengths.
Piecewise.
$4 + 3 = 7$. Note this is not the distance from $0$ to $4 + 3i$, which is $5$: length depends on the path.
Let $C$ be the circle of radius $2$ about the origin, traversed once anticlockwise. Fill in the table.
| Value | |
|---|---|
| The radius of the contour | |
| The length of the contour, as a multiple of pi | |
| The winding number about the origin | |
| The winding number about a point at distance three more than the radius | |
| The winding number about the origin when the circle is traversed n times clockwise |
What is the length of the straight contour from $0$ to $6 + 8i$?
Answer:
Put the steps of describing a curve in the plane as a contour in order.
Number the steps in order (write the number in the box):
A closed contour runs from $0$ to $6$, then to $3i$, then back to $0$. Plot its three vertices, reading the first coordinate as the real part and the second as the imaginary part.
Plot your answer on the grid:
Match each contour to what it is.
| Simple and closed | Simple but not closed | Closed but not simple, crossing itself once | Closed but not simple, retracing every point | |
|---|---|---|---|---|
| A circle traversed once | ||||
| A straight segment between two different points | ||||
| A figure eight | ||||
| A circle traversed $4$ times |
The region between the circles of radius $4$ and $8$ about the origin has a boundary made of both, taken positively: the outer one anticlockwise and the inner one clockwise. What is the total winding number of that boundary about the origin?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Let $C$ be the circle of radius $7$ about the origin, traversed once anticlockwise. Fill in the table.
| Value | |
|---|---|
| The radius of the contour | |
| The length of the contour, as a multiple of pi | |
| The winding number about the origin | |
| The winding number about a point at distance three more than the radius | |
| The winding number about the origin when the circle is traversed n times clockwise |
You can parametrise a contour, find its length and read off its winding numbers. Say in your own words why the winding number is a better idea than the word inside. Next: integrating along a contour, where the direction you just recorded decides every sign.
9. Your turn: the length of the contour from $0$ to $4$ to $4 + 3i$, step 2