Residue Staircase
y = (p · ∫₀^(2 · π) (x² − x · cos(s))/(x² − 2x · cos(s) + 1) ds + q · ∫₀^(2 · π) (x² − 2x · cos(s))/(x² − 4x · cos(s) + 4) ds)/(2 · π)
Open in the app The dials and keys named below are the app's.
What it draws
The function is f(z) = p/(z − 1) + q/(z − 2), with a pole at 1 and a pole at 2. The graph takes a circle |z| = x about the origin, integrates f once round it, and divides by 2πi. The height is that number, plotted against the circle's radius x.
On the circle z = x e^{is}, so dz = iz ds and the i cancels. Each pole a leaves (1/2π)∫ z/(z − a) ds. Its imaginary part cancels round the circle, and its real part is the fraction inside each integral. Its denominator is |z − a|².
Why it steps
Cauchy's residue theorem says the integral round a closed curve is 2πi times the sum of the residues of the poles inside it. The residue of p/(z − 1) at 1 is p, and of q/(z − 2) at 2 it is q.
So the height is 0 while the circle is smaller than 1, and nothing is inside. It is p once the circle takes in the pole at 1, and p + q once it takes in both. At the opening p = q = 1 that is 0, 1 and 2: the staircase.
A negative x is the same circle started from the other side, which is why the left half mirrors the right.
At the poles
When the circle runs through a pole the integral is not defined, since |z − a|² reaches 0. At exactly x = 1 and x = 2 the formula gives no value, and the graph has a gap.
Close by, the integrand is a sharp peak that the numerical integral cannot resolve. Within about 0.02 of each pole it swings past the step: 1.14 at x = 1.01 where it should be 1.
History
Augustin-Louis Cauchy developed the calculus of residues in the 1820s, from his integral theorem for a function with no poles inside the curve, whose integral is then 0.
Try
- Drag p to −1. This is f(z) = 1/((z − 1)(z − 2)): the first step goes down to −1, and outside both poles the two residues cancel back to 0.
- Drag p to 0. The pole at 1 is gone, and so is its step: only the step at 2 is left.
- Set p to 2 and q to 0.5. The steps are 2 high and then 0.5 more, each the size of its residue.
- Press Front in the deck to see the staircase side on.