periodic functions

periodic functions

This entry concerns the

periodicity

of the

meromorphic functions

.

Theorem

1.

If

ω

is a period of a

function

f

, then also

n

⁢

ω

, 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

⁢

ω

)

=

f

⁢

(

(

z

-

k

⁢

ω

)

+

k

⁢

ω

)

=

f

⁢

(

z

)

which is true for all

z

’s. Q.E.D.

Note.

If a function has no other periods than

±

ω

,

±

2

⁢

ω

,

±

3

⁢

ω

,

…

,  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 functions

sine and cosine (with prime period

2

⁢

π

),

tangent

and

cotangent

(prime period

π

), the

exponential function

and the

hyperbolic sine

and cosine (

http://planetmath.org/HyperbolicFunctions

) (with prime period

2

i

π

)

,

hyperbolic tangent

and cotangent (

http://planetmath.org/HyperbolicFunctions

) (prime period

i

⁢

π

).

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

ω

1

,

ω

2

,

…

of the periods such that

lim

n

→

∞

⁡

ω

n

=

0

.  If

z

0

is a

regularity

point of

f

, we have

f

⁢

(

z

0

)

=

f

⁢

(

z

0

+

ω

n

)

∀

n

=

1

,

2

,

…

,

i.e. the function

f

⁢

(

z

)

-

f

⁢

(

z

0

)

has infinitely many zeros

z

0

+

ω

n

(

n

=

1

,

2

,

…

)  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

ω

1

and

ω

2

within a disc with center

z

0

and with radius an arbitrary positive number

ε

.  The

difference

ω

1

-

ω

2

is also a period.  Because

|

ω

1

-

ω

2

|

<

2

⁢

ε

,

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

ω

1

be such a period that

|

ω

1

|

=

m

1

.  Then each

multiple

n

⁢

ω

1

(

n

=

±

1

,

±

2

,

…

)  is a period.  The points of the complex plane corresponding these periods lie all on the same line

arg

⁡

z

=

arg

⁡

ω

1

(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

ν

⁢

ω

1

and

(

ν

+

1

)

⁢

ω

1

, then the period

ω

-

ν

⁢

ω

1

would have the modulus

<

|

ω

1

|

=

m

1

.

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

ω

2

is such a period giving the minimum distance

m

2

.  Then also all numbers

ω

=

n

1

⁢

ω

1

+

n

2

⁢

ω

2

,  with

n

1

,

n

2

∈

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

ω

=

n

1

ω

1

+

n

2

ω

2

(

n

1

,

n

2

∈

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

,

ω

1

,

ω

2

,

ω

1

+

ω

2

.  This however would contradict the minimality of

ω

1

and

ω

2

.

Figure 3:

The basic period parallelogram

The numbers

ω

1

and

ω

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

References

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

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