periodic functions

This entry concerns the periodicity of the meromorphic functions.

Theorem 1.  If $\omega$ is a period of a function $f$, then also $n\omega$, with $n$ an arbitrary integer, is a period of $f$.

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

$$f(z-k\omega )=f((z-k\omega )+k\omega )=f(z)$$

which is true for all $z$’s. Q.E.D.

Note.  If a function has no other periods than  $\pm \omega ,\pm 2\omega ,\pm 3\omega ,\mathrm{\dots }$,  the function is called one-periodic and $\omega$ the prime period or primitive period of the function.  Examples of one-periodic functions are the trigonometric functions sine and cosine (with prime period $2\pi$), tangent and cotangent (prime period $\pi$), the exponential function and the hyperbolic sine and cosine (http://planetmath.org/HyperbolicFunctions) (with prime period $2i\pi )$, hyperbolic tangent and cotangent (http://planetmath.org/HyperbolicFunctions) (prime period $i\pi$).

Theorem 2.  The moduli (http://planetmath.org/Complex) 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 sequence  ${\omega }_1,{\omega }_2,\mathrm{\dots }$  of the periods such that  ${lim}_{n\to \mathrm{\infty }}{\omega }_n=0$.  If $z_0$ is a regularity point of $f$, we have

$$f(z_0)=f(z_0+{\omega }_n)\mathit{\hspace{1em}}\forall n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots },$$

i.e. the function $f(z)-f(z_0)$ has infinitely many zeros  $z_0+{\omega }_n$  ($n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }$)  which have the accumulation point $z_0$.  But then $f(z)-f(z_0)$ vanishes identically (cf. this (http://planetmath.org/IdentityTheoremOfHolomorphicFunctions) entry), i.e. $f(z)$ is a constant function.  This contradicts the assumption, and therefore the antithesis is wrong. Q.E.D.

Figure 1: The argument 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 $z_0$.  Thus we can choose two periods ${\omega }_1$ and ${\omega }_2$ within a disc with center $z_0$ and with radius an arbitrary positive number $\epsilon$.  The difference ${\omega }_1-{\omega }_2$ is also a period.  Because ,  $f(z)$ 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 $m_1$.  Let ${\omega }_1$ be such a period that  $|{\omega }_1|=m_1$.  Then each multiple $n{\omega }_1$  ($n=\pm 1,\pm 2,\mathrm{\dots }$)  is a period.  The points of the complex plane corresponding these periods lie all on the same line

$\arg z=\arg {\omega }_1$ (1)

and are situated at .  The line does not contain points corresponding other periods, since if there were a period $\omega$ on the line between the points $\nu {\omega }_1$ and $(\nu +1){\omega }_1$, then the period $\omega -\nu {\omega }_1$ would have the modulus .

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 $m_2$.  Suppose that ${\omega }_2$ is such a period giving the minimum distance $m_2$.  Then also all numbers  $\omega =n_1{\omega }_1+n_2{\omega }_2$,  with  $n_1,n_2\in \mathbb{Z}$,  are periods of $f$.  The corresponding points of the complex plane form the vertices of a lattice of congruent (http://planetmath.org/Congruence) parallelograms.  Conversely, one can infer that all the periods of $f$ are of the form

$\omega =n_1{\omega }_1+n_2{\omega }_2\mathit{\hspace{1em}}(n_1,n_2\in \mathbb{Z}).$ (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 $0,{\omega }_1,{\omega }_2,{\omega }_1+{\omega }_2$.  This however would contradict the minimality of ${\omega }_1$ and ${\omega }_2$.

Figure 3: The basic period parallelogram

The numbers ${\omega }_1$ and ${\omega }_2$ 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 functions.

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

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