Quasi-elliptic functions Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 9. № 4 (2017). С. 129-136.

УДК 517.53

QUASI-ELLIPTIC FUNCTIONS

A.YA. KHRYSTIYANYN, DZ.V. LUKIVSKA

Abstract. We study certain generalizations of elliptic functions, namely quasi-elliptic functions.

Let p = eia, q = ellS, a, ft e R. A meromorphic in € function g is called quasi-elliptic if there exist e €*, Im ^ > 0, such that g(u + wi) = pg(u), g(u + w2) = qg(u) for each

u e € In the case a = ft = 0 mod this is a classical theory of elliptic functions. A class of quasi-elliptic functions is denoted bv Q£. We show that the class Q£ is nontrivial. For this class of functions we construct analogues pap, (ap of p and ( Weierstrass functions. Moreover, these analogues are in fact the generalizations of the classical p and ( functions in such a way that the latter can be found among the former by letting a = 0 and ft = 0. We also study an analogue of the Weierstrass a function and establish connections between this function and pap as well as (ap.

Let q,p e €*, |q| < 1. A meromorphic in C* function f is said to be p-loxodromic of multiplicator q if for each z e €* f (qz) = pf (z). We obtain telations between quasi-elliptic and p-loxodromic functions.

Keywords: quasi-elliptic function, the Weierstrass p-function, the Weierstrass (-function, the Weierstrass ct-function, p-loxodromic function.

Mathematics subject classification: 30D30

1. Introduction

Denote C* = C\{0}. A meromorphic in C function g is called elliptic [1] if there exist u1,u2 e C* such that Im ^ > 0 and for each u e C

Ш1

g(u + ui) = g(u), g(u + ^2) = g(u).

The theory of elliptic functions was developed by K. Jacobi, N. Abel, A. Legendre, K. Weierstrass. The following definition was introduced by A. Kondratyuk.

Definition 1. [2] A meromorphic in C function f is said to be modulo-elliptic if there exist u1,u2 e C* such th at Im ^ > 0 and for ea ch u e C

If (u + Wi )| = If (u)|, If (u + = If («)|. Consider the first of these identities

^(U + ^i)| = ^(u)|, u e C. If f (u) = 0 and f (u) = to, we can divide (1) bv ^(w)| to obtain

f (u + ui)

(1)

f Ы

1.

A.YA. KHRYSTIYANYN, dz.v. LUKIVSKA, QUASI-ELLIPTIC FUNCTIONS. ©KHRYSTIYANYN a.YA., LUKIVSKA dz.v. 2017. Поступила 21 сентября 2016 г.

The function g(u) = ^(u + ) is meromorphic in C, It follows from (2) that the function g is

f W

holomorphie and bounded in C except for a set of the zeros an d poles of /.Since g is bounded, these points are removable, and relation (2) implies

Vu e C : lg(u)l = 1.

By the Liouville theorem g is constant and the latter identity implies the existence of a e R such that g(u) = e%a. This means that

Vu e C : f (u + Wi) = ezaf (u).

In the same way as above, we conclude that there exists ft e R such that

Vu e C : f (u + u2) = ezl3 f (u).

We consider separately the following cases:

(i) a = ft = 0 mod

(ii) a = 0 mod ft = 0 mod 2-n (or a = 0 mod ft = 0 mod 2n);

(iii) a = 0 mod 2n, ft = 0 mod 2-k.

In the first case we obtain the classical theory of elliptic functions including the famous Weierstrass ^-function

p(u) = -1 + ^(7—- —, u = mui + nu2, m,n e Z. (3)

u2 \(u — u)2 u2 I

The Weierstrass ^-function is elliptic [1] with periods u2. The representations for classical Weierstrass (and a functions are well-known [1], [3]:

((u) = —+ --1---+ u = mu1 + nu2, m,n e Z. (4)

u \ u — U U U)2 )

T-r / U\ U I u2

a(u) = «11 (1--)e^ + 2w2, u = mu1 + nu2, m,n e Z. (5)

We also observe that the following identities

p(u) = —C(u), ((u) = , p(u) = — (^l) .

a(u) \ a(u) )

hold true. We note that each elliptic function can be representedbv using (3), (4), (5) (see [3]). In other words, these functions play an important role in representations of elliptic functions. In the second case we obtain so-called p-elliptic functions.

Definition 2. [4] Let p = e^. A meromorphic in C function g is called p-elliptic if there exist u1,u2 e C* such that Im ^ > 0 and for each u e C

g(u + ui) = g(u), g(u + ^2) = pg(u). This case was studied in [6].

The aim of this article is to consider the third case. This is a generalization of elliptic functions in some sense as the following definition says.

Definition 3. Let p = eza, q = e^. A meromorph ic in C functio n g is called quasi-elliptic if there exist u1,u2 e C*, Im ^ > 0, such that for each u e C

g(u + U1) = pg(u), g(u + ^2) = qg(u).

We denote the class of quasi-elliptic functions by Q£.

Let u = mu\ + nu2, m,n E Z. If f E Q£, Definition 3 implies

g(u + u) = pmqng{u).

If p = ^d q = 1 in Definition 3, we obtain classic elliptic function. If p = 1 or q = 1 in Definition 3, we obtain p-elliptic function.

Remark 1. There is one special case when Definition 3 still gives an elliptic function. Namely, if p = eta, q = e1^, where a, ft E 2nQ, then

f {u + lux ) = f (u), f {u + IU2) = f {u),

a ft

where I is the least common denominator of — and —.

J 2n 2n

Indeed, if a = 2n~, using Definition 3, we have b

f (u + lu,) = f (u + (l — 1)^y2*f = ■ ■ ■ = f (u)e^f = f (u).

The same conclusion can be made for ft.

Remark 2. The class Q£ of quasi-elliptic functions is not trivial. For example, consider the function

_ ^ima^inß

f (u)=y --—, u = mui + nu2, m,n E Z. (6)

(u — u)3

Consider a compact subset K from C. Since f|l], [3]J

E ¿3 < m

1 1

we obtain that the series in the right hand side of (6), or at least its remainder, is uniformly convergent on K. Therefore f is meromorphic in C, and we have for each, u E C

pi(m— l)apinß

f (u + u, ) = e^Y. 1-(-n-^ = el№f (U).

mj^ez (u - (m - 1)ui - nu2)3

In the same way, for each u E C we obtain f (u + u2) = e%li f (u).

Our main aim is to construct a quasi-elliptic function paß being an analogue of p(u) and also to construct corresponding analogues of ^d a functions,

2. Generalization of the Weierstrass ^-function Let p = eza, q = ezß. Consider the function

Gaß(u) = 1 + £ - -1) (8)

u2 \(u — u)2 UJ2 I

where E C, Im ^ > 0 w = mw\ + nu2, m,n E Z, Similarly, in view of (7), as in the case

of the series from (6), we obtain that Gaß is meromorph ic in C,

It is obvious that, G00 coincides with the classical Weierstrass function p. Consider the case a = 0 mod 2-n and ft = 0 mod 2k, that is, p = 1 and q =1.

Theorem 1. A function of the form

paß (u) = Gaß (u) + Caß,

where

_ Gaß( y) - ¿"Goß (-f) _ Gaß( f) - ^ Gaß ^ f)

Uaß e™ - 1 e*ß - 1

belongs to Q£ with p _ eia, q _ eiß.

Proof. Consider the function Gaß. We shall show that there exists a unique constant Caß such that (Gaß (u) + Caß) E Q£, that is

Gaß (u + + Caß — Gia(Gaß + Caß),

Gaß (u + U2t) + Caß _ eiß (Gaß + Gaß).

These properties are called multi ^periodicity with the period u1 and multi ^-periodicity with the period u2, respectively.

Let us consider the derivative of Gaß\

gi(ma+nß)

G'aß («)_ -2

(u - u)3 We have:

_^ gi(ma+nß) _^ gi(ma+nß)

G'aß(u + _ - 2 7 7-r^ _ -2 V^ ----—-—

' (u + — mu1 — nu2)3 (u — (m — 1)u1 — nu2)3

pi((m- 1)a+raß)

_ - 2eia V -?---- _ eiaG'aß(u).

^ (u - (m - 1)w1 - nu2)3 aß

Hence, we obtain

G'aß (u + W1) - eiaG'aß (u)_0. (9)

We note that for each C E C, the function (Gaß + C) satisfies (9), Let

Gaß (tT ) - e%a Gaß (-Tt) C _ Caß _ -V27 ^ia - 1 V 2) ■ (10)

Then relation (9) implies

Gaß (u + Wi) + Caß - eia (Gaß + Gaß) _ A, where A is a constant. If we let u _ -it is easy to obtain that

Gaß (y) - eiaGaß (-y) + (1 - eia)Caß _ A. Taking into consideration the choice of Caß by formula (10), we get A _ 0. Therefore, we have

Gaß (u + Ui) + Caß _ eia (Gaß + Caß) , (11)

that is, we have shown that the function (Gaß + Caß) is multi ^periodic of period

It remains to prove the uniqueness of Caß. Suppose that there exists a constant C diiferent from Caß such that the function (Gaß + C) is multi p-periodic of period too. Then we get

Gaß (u + ui) + C _ eia (Gaß (u) + C).

Deducting this identity from (11), we obtain

C - Caß _ C - Caß).

Since a _ 0 mod 2^, we get C _ Caß.

In the same way, for the period u2 we have

Gaß (u + ^2) + Caß _ eiß (Gaß (u) + Caß) + B, (12)

where B is some constant.

Let us find B. Using identities (11) and (12), we obtain

Gaß (u + U! + u2) + Caß =eiß (Gaß (u + ^ ) + Caß ) + B

=eiß (eia(Gaß (u) + Gaß )) + B

=j(a+ß\Gaß (U) + Gaß)+ B

and

Gaß (U + Ui + U2) + Caß =eia(Gaß (u + U2) + Caß )

=eia(eiß (Gaß (u) + Caß ) + B ) =ei(a+ß)(Gaß (u) + Caß ) + Beia.

Comparing the right hand sides of these relations, we get B = Beia. Since a = 0 mod the previous identity implies that B = 0, Therefore,

Gaß (u + U2 ) + Gaß = ejß (Gaß (u) + Gaß) .

Hence, the function Gaß is multi ^periodic with the period and is multi g-periodic with period u2, respectively.

It is easy to see that Caß can be also expressed as

c Gaß (f) - Gaß (-f )

°aß = eiß - l .

□

Definition 4. A function of the form

Paß (U) = Gaß (U) + Caß = 1 + W, 1 ,2 — -1) ¿(ma+nß) + Caß,

u2 \(u — u)2 UJ2 I

where

= Ggß[ y) — elaG»ß (— = Gaß( f) — ^ Gaß (— f)

aß e- — 1 eiß — 1

is called the generalized Weierstrass p-function.

Remark 3. For the sake of completeness, in the case p = q = 1, in other words, as a = ft = 0 mod 2n, we efine C00 = 0. Then p00 = p.

3. Generalization of Weierstrass (and a functions Now we consider the function

(«/3(u) = - + W — + - +

u \ u — U U UJ2

where u\,u2 E C, Im^ > 0 u = mu\ + nu2l m2 + n2 = 0, m,n E Z, Differentiating (aß, we obtain Gaß (u) = —('aß (u). Hence,

paß (U) = Caß (u) + C»ß.

We denote

Xmn(u) = (—1--+ 4) , m2 + n2 = 0,

\u — U U UJ2

and

Xoo(u) _ -.

u

Then (aß can be rewritten as

Caß(u) _ £ ei(ma+nß)Xmn(u). (13)

00

By A* we denote the plane C with radial slits from ^ to <x. Integrating Xmn and x00 along a path in A* connecting the points ^d u, we obtain

Xmn(t)dt _ log (1 - + - + , m2 + n2 _0 (14)

V u) u 2u2

o

and

J Xoo(t)dt _ logu. (15)

o

We consider entire functions

vmn(u) _ (1 - e™ + , m2 + n2 _ 0

u,

and we let

<oo(u) _ u.

Employing these functions, we can rewrite (14) as

u

J Xmn(t)dt _ log <mn(u), m,n E Z. o

Differentiating this identity and using the definitions of Xm and <oo, we get

Vm,n E Z : xmn(u)_ .

<mn \Uj)

Taking into consideration this representation for Xmn, we rewrite (13) as

> („ ) _ \ i(ma+nß) <mn

Kaß (U)_ Z^ e < (u) .

Hence, paß can be rewritten as

p (u) _ C + ^^ ei(ma+nß) (amn(u)) - <mn(u)<Jmn(U)

aß aß „(u)

We note that if we consider the product H amn(u), then we obtain the Weierstrass ^-function.

Let q,p E C*, Iql < 1.

Definition 5. [5] A meromorphic in C* function f is said to be p-loxodromic with the multiplicator q if f(qz) _ pf(z) for each, z E C*.

u

u

We denote by the class of p-loxodromic functions with the multiplieator q.

The case p =1 was studied earlier in the works of O. Rausenberger [7], G. Valiron [8] and

2m 2-Kt

Let a1 = e "i, a2 = e "^d f1 e Caiq, f2 e Ca2P. Then

We define

Then g E Q£. Indeed,

fi(a i z) = q fl(z), f 2( a2 z) = p f 2(z).

g(u) := fi(e2m^)^(e2m).

, . \ p ( 2Trí — \r ( 2ni— 2tTÍ

g(u + Ui)= fi[e ^J f2[e ^2e "2J

p ( 2TTÍ — \r ( 2TTÍ ^

=fi [e ^J f2 (a.2e =P fi(e 2711 ^) Í2(e 2m ^) =pg(u),

and

/ . \ r f 2TTÍ2tTÍ^\ , ( 2TTÍ

g(u + (J2)= fi[e ^ J f2[e "2J

p ( 2TTÍ —\P ( 2TTÍ

=fi\aie ^J f2[e "2J =qfi(e 2711 ^) f 2(e 2m ^) = qg(u). Vice versa, let g E QS, p = 1, q = 1, that is

g(u + ui) = g(u), g(u + ^) = qg(u).

We denote

f(z):=g (^ log z) . (16)

The function f is well-defined since g admits the period ul and therefore, the substitution of logz by logz + 2nik, k E Z does not change the value of g in the right hand side of (16), In other words, here the composition of a multivalent mapping with a univalent one is a univalent

27i

function. Hence, if we let a = e 7"í, Im — > 0, we obtain

' ' Ui '

f(a z) = (2tlog( a z))=9 logz) =Q9 logz) = 1 f(z).

Thus, f E £aq. The case p = 1, q = 1 is similar. We let

f(z) :=K2Ilogz)

2-KÍ ^í

and a = e .Then f E Cap. Indeed,

f(a z) = (21log( a z 0=9 +21logz) =pg log^) =pf(z).

In the case p = 1, g = 1 the functions g logz) are multivalent, k =1, 2,

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Andriv Yaroslavovyeh Khrystiyanvn, Ivan Franko National University of Lviv, Universvtetska str,, 1, 79000, Lviv, Ukraine E-mail: [email protected]

Dzvenvslava Volodymvrivna Lukivska, Ivan Franko National University of Lviv, Universvtetska str., 1, 79000, Lviv, Ukraine E-mail: [email protected]