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mapping the real line to a polygon

If integrals are used instead of interpolating polynomials, increments in the integrand can be used to guide the curve followed by an indefinite integral. A readily visualized process is to use increments of constant direction to create a straight line, varying the direction from time to time to outline a polygon. Doing the same by interpolation would mean assigning vertices to the data points, without the assurance of their being connected adequately by intermediate values.

Increments of constant direction are best gotten by taking real increments and multiplying them by an overall phase factor. The real axis could contribute the increments, relying upon roots to contribute the phase factors; the minus in a square root for a reversal of direction, a cube root for a $120^\circ$ shift, a fourth root to make a square corner, and so on. The roots should be negative fractional powers to give finite integrals, leaving the question of their placement along the real axis as one to be resolved. It is desirable that the terms vanish strongly enough at infinity that an excursion along the whole real axis produces a closed polygon.

Consider the derivative defined by

$\displaystyle \frac{dw}{dz}$ = $\displaystyle (z-z_1)^{-\mu_1}(z-z_2)^{-\mu_2}\cdots(z-z_n)^{-\mu_n}$ (214)

as it appears on the real line. Writing each singularity in the form
$\displaystyle (z-z_p)^{-\mu_p}$ = $\displaystyle \vert z-z_p\vert^{\mu_p}e^{i\mu_p\arg(z-z_p)},$ (215)

we can examine the effect of moving z from left to right along the real axis. The only effect will be to change the argument of z-zp from $\pi$ to 0, and thus change the argument of the derivative by $\pi\mu_p$, which sends the integral off in a new direction. The condition for closure of the polygon is
$\displaystyle \mu_1 + \mu_2 + \cdots + \mu_n$ = 2, (216)

restoring the increments to their original direction. Putting angles in the exponents is easy enough, but deciding where to place the zp to get desired edge lengths, and thus the wp, requires more careful planning.

This whole process is called a Schwartz-Christoffel transformation. To map one polygon to another, consider inverting the procedure, sending the first polygon to the real axis. Composing the inverse with the mapping of the real axis to the second polygon makes the transition directly.

next up previous contents
Next: functions with boundary values Up: Mapping theorems Previous: interpolation