THE RESIDUE THEOREM AND AN ANALOG OF P. APPELL’S FORMULA FOR FINITE RIEMANN SURFACES
Journal: SCIENCE EVOLUTION
Volume 1 № 1
, 2016
Rubrics: MATHEMATICAL SCIENCES
Authors:
1.
Kemerovo State University
Russian Federation
Russian Federation
2.
Kemerovo State University
Russian Federation
Russian Federation
Type:
Article
Pages:
from 40
to 45
Status:
Published
Received:
27.06.2016
Accepted:
27.06.2016
Published:
27.06.2016
Subject area:
GRNTI 29.01 Общие вопросы физики
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
GRNTI 27.01 Общие вопросы математики
GRNTI 31.01 Общие вопросы химии
GRNTI 34.01 Общие вопросы биологии
Language:
English
Keywords:
Teichmuller spaces for finite Riemann surfaces, Prym differentials, group of characters, Jacobi manifolds
Abstract (English):
A theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surfaces has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. We have proved analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals.
A theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surfaces has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. We have proved analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals.
Keywords:
Teichmuller spaces for finite Riemann surfaces, Prym differentials, group of characters, Jacobi manifolds
Teichmuller spaces for finite Riemann surfaces, Prym differentials, group of characters, Jacobi manifolds
INTRODUCTION Theory of multiplicative functions and Prym differentials for the case of special characters on compact Riemann surface has found applications in geometrical function theory of complex variable, analytical number theory and in equations of mathematical physics [1-7]. In [2, 4] the construction of the general theory of multiplicative functions and Prym differentials on compact Riemann surface for arbitrary characters was started. Theory of functions on compact Riemann surfaces differs from the theory of functions on finite Riemann surfaces even for the class of single meromorphic functions and Abelian differentials. A series of general spaces of functions and differentials on finite Riemann surface of genus will be infinite-dimensional. In this article we continue the construction of the general function theory on finite Riemann surfaces for multiplicative meromorphic functions and differentials. Analogues of the theorem on the full sum of residues for Prym differentials of every integral order and P. Appell's formula on expansion of the multiplicative function with poles of arbitrary multiplicity in the sum of elementary Prym integrals have been proved. MATERIALS AND METHODS 1. Preliminaries Let be a fixed smooth compact oriented surface of genus with and be a compact Riemann surface with fixed complex analytic structure on Let us fix different points Let be a surface of type Any other structure on is given by some Bertrami differential on i.e. by the expression of the form which is invariant relative to the choice of the local parameter on where is complex-valued function on and This structure on we will denote by It is uniformized by quasi-Fuchsian group In the work of L. Bers [3] Abelian differentials on which form a canonical basis dual to the canonical homotopy basis on have been constructed and also it holomorphically depends on points of the Teichmuller space Moreover, the matrix of periods on consists of complex numbers and holomorphically depends on . For any fixed and let us define a classical Jacobi mapping according to the rule . A quotient space is called marked Jacobi manifold for where is a lattice above generated by columns of the matrix [6, 7]. Any homomorphism is called the character for . Further we will assume that where is a simple loop that avoid puncture on Definition 1.1. Meromorphic function on is called a multiplicative function on for the character if Definition 1.2. Differential is called Prym differential in relation to Fuchsian group for or - differential, if If is a multiplicative function on for without zeros or poles, then characters for such functions we shall call inessential and we shall call a unit. Characters that are not inessential we shall call essential on . A set of inessential characters forms a subgroup in the group of all characters on Theorem (of Abel for characters) [2, 4]. Let be a divisor on a marked variable compact Riemann surface of genus , and let be a character on . Then will be a divisor of the function on for and where is a Jacobi mapping. Class consists of Prym differentials for on which have a finite number of poles on and allow meromorphic continuation on . In [6, 7] it was proved that for any essential character , points of natural number and inessential character , points of natural number exists elementary differential of third kind with a single simple pole on . For any inessential character points when , elementary differential doesn’t exist. There it was also proved that on a variable surface of kind for any natural number exists elementary differential of third kind with simple poles and of second genus with the pole of order These differentials locally holomorphically depend on and 2. An analogue of the residue theorem for Prym differentials on finite Riemann surface Residues for Prym differentials can be defined only for the branches of these multivalued differentials. Let - differential such that with pairwise distinct points on Analytic continuation (hereinafter referred to with this symbol) with on meets conditions Let us introduce the following notations: 1) if is inessential character, then let us choose multiplicative unit on for where 2) if is an essential character, then there exists a single function on for with a single simple pole [6]. Such function has a divisor where in Jacobi manifold This function may be presented in a form of [4]. Let us show uniqueness of such function. If there is a point such that the equality is true, then According to classical Abel’s theorem there is a single-valued function with the divisor that has a single simple pole on compact Riemann surface of a positive genus. A contradiction. Without loss of generality we can find Abelian holomorphic 1-differential such that on because divisors of Abelian holomorphic differentials don’t have base points on [4]. Let us choose any Abelian differential with the divisor on so that there were as few points as possible in its divisor. We have an equality in Jacobi manifold where is a vector of Riemann constant for a marked compact Riemann surface with the base point [4]. This equality is equivalent to another equality in the form where This implies Abelian differential with divisor in the form Let us make Abelian 1-differentials and on where and have character on and differential is analytically continued from on By the theorem on a complete sum of residues for Abelian 1-differentials on we obtain the following analogue of the theorem on a complete sum of residues for differentials. Theorem 2.1. 1) For any differential of the class on Riemann surface of type with any polar divisor of any integer and unit for inessential character on the following equality is true: ++ += 0; 2) For any differential of the class on Riemann surface of type with any polar divisor of any integer and single, accurate to multiplication by a nonzero constant, function for essential character on the following equality is true: ++ += 0. In both cases is an Abelian holomorphic differential with a divisor in Jacobi manifold and on Remark 2.1. Note that in the preceding theorem when there is no second sum in the assertion of the theorem. Remark 2.2. P. Appell considered the residue theorem only when on compact Riemann surface of kind Let us find some corollaries of the residue theorem and reciprocity laws for multiplicative functions on finite Riemann surface. First we find corollary for 1-differential with any character in a special case, when where is a multiplicative function on of the class. Let be zeros of with multiplicity and let be poles for with multiplicity when the function is continued analytically from to Let us also consider single-valued function on with poles of multiplicity accordingly where points are not included in the support of the divisor Let us note that will be an Abelian differential with simple poles and residues in them accordingly. Then, because of uniqueness of the expression under integral and by the residue theorem for Abelian 1-differentials, we obtain that: (1) where is a connected fundamental domain for group in domain [3, 5]. From (1) we will obtain formulas that are connected with a special choice of a function on 1) Let be an analytical function on and be a constant. Then is a classical fact that on [4]; 2) If has multiple poles in points i.e. then we obtain the equality Remark 2.3. These two equalities are some reciprocity laws that connect zeros and poles for the multiplicative function of the class on with poles of single-valued meromorphic functions on 3. An analogue of Appell’s formula for a multiplicative function expansion on a variable finite Riemann surface. Let us denote through an elementary Prym integral of the second kind on for essential character with a single simple pole in and residue +1 in which holomorphically depends on and where has a zero residue in the point Let be a function on of the class for essential character with simple poles and residues in them accordingly for one of its branches. Let us take analytical continuation of this function (and denote it with the same symbol) with on . Let us consider the expression where and are the basis of Prym differentials of the first kind for essential character on that holomorphically depends on and [2]. Then is a meromorphic single-valued branch of the Prym integral with essential character on fundamental polygon with divisor on Among other things, Prym integral for has a branch whose principal parts of Laurent series match with principal parts of Laurent series in points for and zeros are periods, on [2]. Therefore If is a pole of order, then in the preceding formula a summand should be replaced with the sum of the form + where are coefficients in the principal part of Laurent series for some branch of the function in the point Indeed, in the neighborhood of the point we have expansion where is an order of a pole in the point for , k = 1,…,s. It follows that Theorem 3.1. Let be a branch of the function of the class for essential character on a variable Riemann surface of type with pairwise different poles in of the multiplicities with given principal parts in them. Then for the analytical continuation it is true that on and where for some branch in the neighborhood on and all summands holomorphically depend on and Now let be an inessential character. The proof of the preceding expansion formula for essential character is not applicable because in this case Prym integral of the second kind with a single simple pole on doesn’t exist. That is Prym differential of the second kind for inessential character has to have at least two second-order poles in different arbitrary points and on and with zero residues in and . In this case Prym integrals of the second kind with two simple poles and should be used as prime elements of expansion. Let us consider another Prym differential of the third kind on where is a unit for on and is a normalized Abelian differential with simple poles and on and residues +1 and -1 in these points accordingly, which holomorphically depend on and It has been known that and Abelian differential is expressed through the Riemann theta function for the surface . In such case it is equal to the sum of two functions, one of which only depends on and another only depends on [2, p. 117]. Thus a derivative doesn’t depend on . Prym differential has expansion in the neighborhood of the point where [6]. Prym differential also has expansion in the neighborhood of the point where . A differential with two poles of the second order and zero residues in these points can be set in the following way Let us denote that the principal part for in the point takes the form of and in the point takes the form of . It follows that the constructed differential has two poles of the second order in and , and two residues in this points. In the neighborhood of the point its principal part takes the form of that is similar in the point It is clear that a constructed differential holomorphically depends on and From all has been said it follows that derivative doesn’t depend on Theorem 3.2. Let be a branch of the function of the class with inessential character and pairwise different poles of the multiplicity with set principal parts in them on a variable Riemann surface of type Then for the analytical continution on it is true that and where for some branch in the neighborhood on when on and all summands holomorphically depend on and Proof. It suffices to verify a coincidence of principal parts on the left and right in this formula. For the neighborhood of the point on we have expansion in Laurent series For the neighborhood of the point we have expansion because according to the formula on full sum of residues for Abelian differentials of third kind on which in the point has residue The theorem is proved. Remark 3.1. P. Appell [2] has proven the theorem 3.2 for compact Riemann surface and simple poles. Every simple element (summand) depended on additional poles. In our work the theorem has been proved for a variable finite Riemann surface of genus and poles of any multiplicity. Moreover, every summand in our work has either one or two poles. Also when we obtain a classical fact about expansion of single-valued meromorphic function in sum of Abelian integrals on a compact Riemann surface. CONCLUSION Analogues of the residue theorem for Prym differentials of any entire order on variable finite Riemann surfaces are obtained for the first time. Thus three reciprocity laws have been proved. Analogues of P.Appell’s expansion formula for functions with any characters on variable finite Riemann surfaces have been proved. In this case simple elements (summands) only have one or two poles.
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