which is true for all ’s. Q.E.D.
Note. If a function has no other periods than , the function is called one-periodic and the prime period or primitive period of the function. Examples of one-periodic functions are the trigonometric functions sine and cosine (with prime period ), tangent and cotangent (prime period ), the exponential function and the hyperbolic sine and cosine (http://planetmath.org/HyperbolicFunctions) (with prime period , hyperbolic tangent and cotangent (http://planetmath.org/HyperbolicFunctions) (prime period ).
i.e. the function has infinitely many zeros () which have the accumulation point . But then vanishes identically (cf. this (http://planetmath.org/IdentityTheoremOfHolomorphicFunctions) entry), i.e. is a constant function. This contradicts the assumption, and therefore the antithesis is wrong. Q.E.D.
Theorem 3. The periods of a non-constant meromorphic function do not accumulate to a finite point.
Proof. We make the antithesis, that the periods of have a finite accumulation point . Thus we can choose two periods and within a disc with center and with radius an arbitrary positive number . The difference is also a period. Because , seems to have periods with arbitrarily little modulus. This contradicts the theorem 2, and so the antithesis is wrong.
The theorems 2 and 3 imply, that the moduli of all periods of the function have a positive minimum . Let be such a period that . Then each multiple () is a period. The points of the complex plane corresponding these periods lie all on the same line
and are situated at . The line does not contain points corresponding other periods, since if there were a period on the line between the points and , then the period 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 . Suppose that is such a period giving the minimum distance . Then also all numbers , with , are periods of . 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 are of the form
In fact, if had some period point other than (2), then one such would be also in the basic parallelogram with the vertices . This however would contradict the minimality of and .
The numbers and 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.
- 1 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).
|Date of creation||2013-03-22 16:51:30|
|Last modified on||2013-03-22 16:51:30|
|Last modified by||pahio (2872)|