periodic functions

This entry concerns the periodicity of the meromorphic functions.

TheoremMathworldPlanetmath 1.  If ω is a period of a functionMathworldPlanetmath f, then also nω, with n an arbitrary integer, is a period of f.

Proof.  For the positive values of n the theorem is easily proved by inductionMathworldPlanetmath.  If n then is any negative integer -k, we can write


which is true for all z’s. Q.E.D.

Note.  If a function has no other periods than  ±ω,±2ω,±3ω,,  the function is called one-periodic and ω the prime period or primitive period of the function.  Examples of one-periodic functions are the trigonometric functionsDlmfMathworldPlanetmath sine and cosine (with prime period 2π), tangentPlanetmathPlanetmath and cotangent (prime period π), the exponential functionDlmfDlmfMathworldPlanetmath and the hyperbolic sineMathworldPlanetmath and cosine ( (with prime period 2iπ), hyperbolic tangent and cotangent ( (prime period iπ).

Theorem 2.  The moduli ( of all periods of a non-constant meromorphic function f have a positive lower bound.

Proof.  Antithesis:  there are periods of f with arbitrarily little modulus.  Thus we could choose a sequenceω1,ω2,  of the periods such that  limnωn=0.  If z0 is a regularity point of f, we have

f(z0)=f(z0+ωn)n=1, 2,,

i.e. the function f(z)-f(z0) has infinitely many zeros  z0+ωn  (n=1, 2,)  which have the accumulation pointMathworldPlanetmathPlanetmath z0.  But then f(z)-f(z0) vanishes identically (cf. this ( entry), i.e. f(z) is a constant functionMathworldPlanetmath.  This contradicts the assumptionPlanetmathPlanetmath, and therefore the antithesis is wrong. Q.E.D.

Figure 1: The argumentMathworldPlanetmath of Theorem 2

Theorem 3.  The periods of a non-constant meromorphic function f do not accumulate to a finite point.

Proof.  We make the antithesis, that the periods of f have a finite accumulation point z0.  Thus we can choose two periods ω1 and ω2 within a disc with center z0 and with radius an arbitrary positive number ε.  The differencePlanetmathPlanetmath ω1-ω2 is also a period.  Because |ω1-ω2|<2ε,  f(z) seems to have periods with arbitrarily little modulus.  This contradicts the theorem 2, and so the antithesis is wrong.

Figure 2: The argument of Theorem 3

The theorems 2 and 3 imply, that the moduli of all periods of the function f have a positive minimum m1.  Let ω1 be such a period that  |ω1|=m1.  Then each multiple nω1  (n=±1,±2,)  is a period.  The points of the complex plane corresponding these periods lie all on the same line

argz=argω1 (1)

and are situated at .  The line does not contain points corresponding other periods, since if there were a period ω on the line between the points νω1 and (ν+1)ω1, then the period ω-νω1 would have the modulus <|ω1|=m1.

Can a function have other periods than those on the line (1)?  If there are such ones, then it’s rather easy to prove, using the theorem 3, that their distances from this line have a positive minimum m2.  Suppose that ω2 is such a period giving the minimum distance m2.  Then also all numbers  ω=n1ω1+n2ω2,  with  n1,n2,  are periods of f.  The corresponding points of the complex plane form the vertices of a latticeMathworldPlanetmath of congruent ( parallelograms.  Conversely, one can infer that all the periods of f are of the form

ω=n1ω1+n2ω2(n1,n2). (2)

In fact, if f had some period point other than (2), then one such would be also in the basic parallelogram with the vertices 0,ω1,ω2,ω1+ω2.  This however would contradict the minimality of ω1 and ω2.

Figure 3: The basic period parallelogram

The numbers ω1 and ω2 are called the prime periods of the function.  We have the

Theorem 4.  A non-constant meromorphic function has at most two prime periods.  Their ratio is not real.

The functions, which have two prime periods, are called two-periodic, doubly periodic or elliptic functionsMathworldPlanetmath.

Figure 4: Lattice generated by the prime periods as the basis


  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
Title periodic functions
Canonical name PeriodicFunctions
Date of creation 2013-03-22 16:51:30
Last modified on 2013-03-22 16:51:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Topic
Classification msc 30D20
Classification msc 30D05
Classification msc 30A99
Synonym periodic function
Related topic EllipticFunction
Related topic PossibleDegreesOfEllipticFunctions
Related topic PeriodicityOfExponentialFunction
Related topic PeriodicExtension
Related topic CounterperiodicFunction
Related topic RolfNevanlinna
Related topic ExamplesOfPeriodicFunctions
Defines one-periodic
Defines two-periodic
Defines doubly periodic
Defines prime period
Defines primitive period