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restraints on periodic functions

The fact that there are many $\wp $ functions is somewhat hidden from view, especially when the two periods $\omega$ and $\omega'$ are not given explicitly. There could easily be still more doubly periodic functions, even for the same lattice, gotten by just averaging something else besides the inverse powers, just as periodic real functions can be built up from sines and cosines of different frequencies.

Cauchy's integral formula and the various consequences derived from it, such as the the constancy of bounded functions or the use of the logarithm to count zeroes and poles, imposes corresponding constraints on periodic functions.

To start, should a periodic function be bounded within its unit cell, repetition bounds it throughout the whole plane, making it constant by Liouville's theorem. Constancy is trivially periodic for whatever period, but any other alternative requires some singularity within the unit cell. The simplest possibility would be a finite number of poles, but even these are subject to some further restrictions.

For example, counting the difference between poles and zeroes by integrating the logarithm finds that integrals along opposite sides of the unit cell cancel because they traverse the same values in opposite directions. So there are just as many poles (counting multiplicity) as zeroes (again counting multiplicity).

If there were just a simple pole with residue, that residue would have to be zero because the boundary integral evaluating it would would always vanish by symmetry. So single isolated poles in periodic functions can't be found. In fact, the residues of whatever collection of poles would have to sum to zero; in particular, a pair of poles would have to have residues of opposite sign.

The Weierstrass function has a pole of order 2 and residue 0 at the origin (and indeed, at every lattice point), requiring every unit cell to have exactly two zeroes. In fact, the number of occurrences of any particular value within a unit cell must be the same as the number of zeroes. The reason is that subtracting a constant from a function does not change the location of its poles, but of course shifts the values and locations of its zeroes. So including infinity, there is a uniform multiplicity of values within each unit cell of a periodic function.

Moreover, all possible values must appear, although this is an observation which ought to be discussed in its own right. Isolated intervals are excluded because otherwise a Móbius transformation could be used to put the center of the interval at infinity while excluding a neighborhood of infinity. Then the composite function would have to be bounded, hence constant by Liouville's theorem, with only trivial periodicity. Once it is known that there are no missing intervals, continuity precludes excluding discrete values.

next up previous contents
Next: the differential equation for Up: Periodic functions Previous: Weierstrass function