Equation of the Related Riemann-Hilbert-Privalov Problem with Zeros and Poles of the Coefficient in the Half-plane* Tatyana G. Voytik (0000-0002-1126-9860)1, Gennadij S. Poletaev (0000-0002-1989-808X)2, and Angela A. Stehun (0000-0003-3140-2689)2(*) 1 Odessa National Maritime University, Odessa, Ukraine beauty5@i.ua 2 Odessa State Academy of Buildings and Architecture, Odessa, Ukraine poletayev_gs@ukr.net, angela.stehun@gmail.com Abstract. Various mathematics and mechanics questions are connected with the Riemann (Riemann-Hilbert-Privalov) problem from the theory of analytic functions. This justifies the relevance of the equation associated with the specified Riemann problem. Until now, the specificity of the location of the singular points of the rational coefficient of the equation has not been considered. The paper aims to obtain general formulas for solving abstract equations in the corresponding ring of rational functions, considering this specificity. The authors propose a way to solve an equation and a related problem. For the first time, the sought general formulas were established and proved to be simpler than those obtained earlier. The research indicates the signs when the use of these formulas is justified. The authors apply the general provisions of (1) the theory of rings and functional analysis, (2) the theory of linear operators, (3) equations in rings with factorization pairs, (4) the problem of factorization by subrings, and (5) the properties of the used ring of rational functions. The authors’ approach is free from the theory of integrals (Cauchy and Fourier types), the requirements of the Hölder condition, and the index. In a similar situation, it applies to (1) integral equations of the Wiener-Hopf type and other equations of the convolution type, as well as to related functional equations; (2) to matrix equations that can be used in mechanics with projectors onto the corresponding subrings and unknown triangular matrices. Keywords: Riemann problem · Equation · Ring · Projector · Factorization pair 1 Introduction 1.1 Relevance, Problem Statement The equations and problems considered in the paper are related to the corresponding *Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). famous Riemann problem (Riemann-Hilbert, Riemann-Hilbert-Privalov) from the theory of analytic functions, but for a narrower class of functions. This problem is important in the theory of differential and integro-differential equations, integral equations of convolution type, the corresponding differential equations of mathematical physics, in the theory of elasticity, torsion problems, and others (Akopyan & Dashtoyan, 2013; Cherskij, P. Kerekesha & D. Kerekesha, 2010; Gahov, 1963; Gahov & Cherskij, 1978; Krejn, 1958; Mhitaryan, 1968; Mushelishvili, 1968; Popov, Kerekesha & Kruglov, 1976; Voytik, Poletaev & Yatsenko, 2017). Therefore, the authors develop (1) general provisions on the solvability of equations associated with it or related problems; (2) ideas about the possible results of the theory; (3) a way of considering the specifics. The research analyzes the solvability, criteria, and formulas for solutions of an equation in the corresponding ring of rational functions: 𝐴𝐴(𝑧𝑧)𝑋𝑋 + (𝑧𝑧) + 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵(𝑧𝑧); 𝑧𝑧 ∈ ℂ ∪ {∞}. (1) This equation for 𝑧𝑧 = 𝑥𝑥 ∈ {−∞, ∞} expresses the boundary conditions of the following, called related, problem: For given rational coefficient functions 𝐴𝐴(𝑥𝑥), 𝐵𝐵(𝑥𝑥), −∞ < 𝑥𝑥 < ∞ find a pair 𝑟𝑟 , 𝑌𝑌− (𝑧𝑧) ∈ ℜ𝑟𝑟− , 𝑧𝑧 ∈ ℂ. All poles of the first of rational functions 𝑋𝑋 + (𝑧𝑧) ∈ ℜ+ one (if they exist) are located only inside the lower one; the second one is located only inside the upper one of the half-plane bounded by the closed real axis and satisfying the linear equation on this axis: 𝐴𝐴(𝑥𝑥)𝑋𝑋 + (𝑥𝑥) + 𝑌𝑌− (𝑥𝑥) = 𝐵𝐵(𝑥𝑥); 𝑥𝑥 ∈ {−∞; ∞}.” (2) The closed real axis here is the contour [1]. The authors consider the case when all zeros and poles of the coefficient 𝐴𝐴(𝑧𝑧) (if they exist) are located simultaneously in the lower or upper half-plane of the extended complex plane. 1.2 Analysis of Research and Publications The study of the Riemann-Hilbert problem’s solvability by exact methods goes back to the works of I. I. Privalov, F. D. Gahov, Yu. I. Cherskij, M. G. Krejn, and others (Gahov & Cherskij, 1978; Krejn, 1958). The connection between the theory of integral equations of Wiener-Hopf type and this problem was first noted by I. M. Rapoport (1948, 1949). Using the provisions of M. G. Krejn (1958, p. 114) concerning the N. I. Mushelishvili book (1968), the authors conclude that this problem was solved under the assumption that the additional Hölder condition for functions on a contour is fulfilled. The apparatus of the Cauchy integral theorem, an index, was often used. This can lead to the need to overcome serious analytical difficulties, sometimes unjustified. Information of history elements to the Riemann problem, integral equations of convolution type, is in the works of F. D. Gahov, Yu. I. Cherskij, M. G. Krejn, and others (Gahov, 1963; Gahov & Cherskij, 1978; Krejn, 1958). Here arises the question about easing restrictions. In M. G. Krejn work (1958), new ideas and an indication of possible approaches to research based on the theory of Banach algebras, under different assumptions and without the requirement that the functions be Hölder, appeared for the first time. Related to the Riemann- Hilbert problem, integral equations of convolution type, the study of abstract equations with an arbitrary right-hand side in associative rings with a special pair of subrings, and their realizations in concrete rings, the results of individual works by G. S. Poletaev (1973–2020) and in co-authorship (Poletaev, 1988; Voytik et al., 2017). The works by V. N. Akopyan & L. L. Dashtoyan (2013); A. McNabb & A. Schumitzky (1972); and G. Ya. Popov, P. V. Kerekesha & V. E. Kruglov (1976) confirmed the relevance of a range of issues close to the Riemann-Hilbert problem. Along with others, the case is important when in this type of problems and related, the coefficients are rational functions (Gahov, 1963; Mushelishvili, 1968; Popov et al., 1976). However, it does not consider the specifics of the case when the zeros and poles of the coefficient of a related problem and the equation under consideration are simultaneously located in one of the half-planes of the extended complex plane. Therefore, the search for solutions to equations (1) in the considered setting is relevant. 1.3 Research Objective The research aims to establish the form of general formulas for solutions of abstract equations (1), considering the specifics of assumptions in cases where the coefficient 𝐴𝐴(𝑧𝑧) is a function of the considered ring of rational ones. Moreover, a function, all zeros, and poles of which (if they exist) are finite, immaterial, and are concentrated simultaneously either inside the upper half-plane or the lower half- plane of the extended complex plane. The aim can be achieved using the corresponding statements and theorems, decision formulas established in the works of the abovelisted authors (Krejn, 1958; McNabb & Schumitzky, 1972; Poletaev, 1988; Voytik et al., 2017). 2 Materials and Methods 2.1 Designations, Definitions, and General Provisions The current research uses designations and statements from the works of the abovementioned authors. The symbol 𝑅𝑅 denotes an arbitrary commutative and associative ring with unit 𝑒𝑒. Let 𝑝𝑝+ , 𝑝𝑝− - be the commuting projectors 𝑅𝑅 → 𝑅𝑅. Therefore: 𝑝𝑝0 : = 𝑝𝑝+ 𝑝𝑝− (= 𝑝𝑝− 𝑝𝑝 + ), 𝑝𝑝∓ : = 𝑝𝑝∓ − 𝑝𝑝0 , 𝑝𝑝∗ : = 𝑝𝑝+ + 𝑝𝑝− . For any subset 𝐵𝐵 ⊆ 𝑅𝑅 let us denote 𝐵𝐵 : = 𝑝𝑝 𝐵𝐵; 𝐵𝐵∓ : = 𝑝𝑝∓ 𝐵𝐵; 𝐵𝐵 ∗ = 𝐵𝐵 + + 𝐵𝐵 − ; ∓, 0 ∓, 0 𝐵𝐵∗ = 𝐵𝐵+ + 𝐵𝐵− . For any element 𝑥𝑥 ∈ 𝑅𝑅, we assume: 𝑥𝑥 ∓, 0 : = 𝑝𝑝 ∓, 0 𝑥𝑥, 𝑥𝑥∓ : = 𝑝𝑝∓ 𝑥𝑥, 𝑥𝑥∗ : = 𝑝𝑝∗ 𝑥𝑥. Definition 1. A pair of subrings , 𝑅𝑅 of a ring 𝑅𝑅 with unit 𝑒𝑒 will be called a (𝑅𝑅+ −) factorization pair [FP], if it is generated in force in 𝑅𝑅 commuting projectors 𝑝𝑝+ , 𝑝𝑝− :𝑅𝑅 ∓ = 𝑝𝑝 ∓ (𝑅𝑅), and the following axioms (McNabb & Schumitzky, 1972): 𝑒𝑒 ∈ 𝑅𝑅0 �= 𝑅𝑅∓ ∩ 𝑅𝑅 ± �; (*) 𝑝𝑝0 (= 𝑝𝑝∓ 𝑝𝑝± ) - is the ring homomorphism 𝑅𝑅 + and 𝑅𝑅 − in 𝑅𝑅0 ; (**) + − − + 𝑅𝑅 𝑅𝑅 , 𝑅𝑅 𝑅𝑅 ⊆ 𝑅𝑅 + 𝑅𝑅 + − (: = 𝑅𝑅 ∗ ). (***) Definition 2. Any ring 𝑅𝑅 with unit 𝑎𝑎1′ , considered together with a fixed FP of the subrings 𝑎𝑎2′ 𝑅𝑅, in other words, subrings with axiomatically specified properties (*), (**), (***), will be called a “ring with a factorization pair” or, in short - a ring with FP. 2.2 Factorization of an Element of a ring by its Factorization Pair Let us say that the element 𝑎𝑎 ∈ 𝑅𝑅 admits factorization in the commutative ring 𝑅𝑅 by the factorization pair 𝑎𝑎2′ = 𝑡𝑡2− 𝑠𝑠20 𝑟𝑟2+ (– by the FP 𝑎𝑎2′ = 𝑡𝑡2− 𝑠𝑠20 𝑟𝑟2+ ), if there are elements 𝑟𝑟 + ∈ 𝑅𝑅+ , 𝑠𝑠 0 ∈ 𝑅𝑅0 , 𝑡𝑡 − ∈ 𝑅𝑅− such that: 𝑎𝑎 = 𝑟𝑟 + 𝑠𝑠 0 𝑡𝑡 − . This factorization is called, namely: 1. Correct factorization факторизацией [CF], if 𝑟𝑟 + ∈ 𝑅𝑅 + , 𝑠𝑠 0 ∈ 𝑅𝑅0 , 𝑡𝑡 − ∈ 𝑅𝑅 − - are invertible in their subrings, respectively; 2. Normalized factorization [NF], if 𝑡𝑡 0 = 𝑟𝑟 0 = 𝑒𝑒; 3. Normalized regular factorization [NRF] if it is CF and 𝑡𝑡 0 = 𝑟𝑟 0 = 𝑒𝑒. The correct factorization of an element from 𝑅𝑅 − by the FP 𝑡𝑡1− ′𝑠𝑠10 ′𝑟𝑟1+ ′𝑥𝑥+ 𝑟𝑟2+ ′𝑠𝑠20 ′𝑡𝑡2− ′ = 𝑏𝑏+ + 𝑞𝑞 − can be normalized (McNabb & Schumitzky, 1972; Poletaev, 1988). There is only one normalized correct factorization. 2.3 Ring 𝕽𝕽𝒓𝒓 with Factorization Pair (𝕽𝕽𝒓𝒓+ , 𝕽𝕽𝒓𝒓− ) Further, we will use definitions, designations, and provisions from work by T. G. Voytik, G. S. Poletaev, & S. A. Yatsenko (2017). By ℜ𝑟𝑟 , we denote the collection of all rational functions of a complex variable 𝑧𝑧 ∈ ℂ, all whose poles (if they exist) are finite and immaterial. The limits of functions from ℜ𝑟𝑟 at infinity exist and are finite. Let ℜ𝑟𝑟+ ( ℜ−𝑟𝑟 ) – be subsets of functions from ℜ𝑟𝑟 , all poles of which (if they exist) are located inside the lower (upper) half-plane 𝐻𝐻− (𝐻𝐻+ ), respectively (Krejn, 1958, pp.14–15). It can be shown that ℜ±,0 𝑟𝑟 = 𝑝𝑝 ±,0 (ℜ𝑟𝑟 ), where 𝑝𝑝0 : = 𝑝𝑝 + 𝑝𝑝 − (= 0 + − 𝑝𝑝 𝑝𝑝 ), ℜ𝑟𝑟 = ℜ𝑟𝑟 ∩ ℜ𝑟𝑟 , − + ℜ𝑟𝑟 - ring with FP (ℜ𝑟𝑟+ , ℜ𝑟𝑟− ). For any function A( z ) ∈ ℜ𝑟𝑟 the decomposition is true: 𝐴𝐴(𝑧𝑧): = 𝐴𝐴0 + 𝐴𝐴+ (𝑧𝑧) + 𝐴𝐴− (𝑧𝑧). Let us establish formulas for solutions for (1) considering corresponding to the research objective. 3 Results 3.1 Main Result Results of individual works by G. S. Poletaev (1973–2020) and in co-authorship (Poletaev, 1988; Voytik et al., 2017) suggest that the following is true: Theorem 1. Let the function 𝑨𝑨(𝒛𝒛) ∈ 𝕽𝕽𝒓𝒓 have no real zeros and 𝒍𝒍𝒍𝒍𝒍𝒍𝑨𝑨(𝒛𝒛) = 𝒛𝒛→∞ 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 ≠ 𝟎𝟎. If, in this case, the element inverse in 𝕽𝕽𝒓𝒓 is the function 𝑨𝑨−𝟏𝟏 (𝒛𝒛) admits a normalized regular factorization: 𝑨𝑨−𝟏𝟏 (𝒛𝒛) = 𝑮𝑮 + (𝒛𝒛) 𝑺𝑺𝟎𝟎 (𝒛𝒛) 𝑻𝑻 − (𝒛𝒛), 𝒛𝒛 ∈ ℂ ∪ {∞} concerning the factorization pair (𝕽𝕽+ 𝒓𝒓 , 𝕽𝕽𝒓𝒓 ), then the abstract equation (1) and the − problem concerning 𝑿𝑿 (𝒛𝒛) ∈ 𝕽𝕽𝒓𝒓 , 𝒀𝒀− (𝒛𝒛) ∈ 𝕽𝕽𝒓𝒓− , for any right-hand side of B( z ) ∈ + + 𝕽𝕽𝒓𝒓 , в 𝕽𝕽𝒓𝒓 are uniquely solvable. The formulas find their only solution: 𝑋𝑋 + (𝑧𝑧) = 𝐺𝐺 + (𝑧𝑧) 𝑆𝑆 0 [𝑇𝑇 − (𝑧𝑧)𝐵𝐵 + (𝑧𝑧) ]+ , 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵− (𝑧𝑧) + (𝑇𝑇 − (𝑧𝑧))−1 [𝑇𝑇 − (𝑧𝑧)𝐵𝐵 + (𝑧𝑧)]− , 𝑧𝑧 ∈ ℂ ∪ {∞}, (3) where: 𝑆𝑆 0 : = 𝑆𝑆 0 (𝑧𝑧) = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ∈ ℂ. Using theorem 1, let us establish a statement corresponding to the research objective. Theorem 2. Let the function A( z ) ∈ ℜ𝑟𝑟 с 𝐴𝐴0 : = 𝑝𝑝0 𝐴𝐴(𝑧𝑧) = 𝑙𝑙𝑙𝑙𝑙𝑙 𝐴𝐴(𝑧𝑧) = = 𝑧𝑧→∞ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ≠ 0 has no real zeros and all its zeros and poles (for existence) are located: 1) only inside the lower, or 2) only inside the upper, from the half-planes 𝐻𝐻− , 𝐻𝐻+ of the extended complex plane, bounded by a closed real axis. Then for it there exists in 𝕽𝕽𝒓𝒓 the inverse - function 𝐴𝐴−1 (𝑧𝑧) ∈ ℜ𝑟𝑟 and there is a normalized correct factorization of 𝐴𝐴−1 (𝑧𝑧) = 𝐺𝐺 + (𝑧𝑧)𝑆𝑆 0 (𝑧𝑧)𝑇𝑇 − (𝑧𝑧), 𝑧𝑧 ∈ ℂ ∪ {∞} by the factorization pair (𝑎𝑎1 , 𝑎𝑎2 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖), for which: in the case 1): 𝐺𝐺 + (𝑧𝑧) = [𝐴𝐴−1 (𝑧𝑧)𝐴𝐴0 ]+ = 𝐴𝐴−1 (𝑧𝑧)𝐴𝐴0 , 𝐴𝐴0 : = [𝐴𝐴(𝑧𝑧)]0 , 𝑆𝑆 0 (𝑧𝑧) = [𝐴𝐴0 ]−1 : = [𝐴𝐴−1 (𝑧𝑧)]0 , 𝑇𝑇 − (𝑧𝑧) = 1, 𝑧𝑧 ∈ ℂ ∪ {∞}; and in case 2) it will be: 𝐺𝐺 + (𝑧𝑧) = 1 , 𝑆𝑆 0 (𝑧𝑧) = [𝐴𝐴0 ] −1 , 𝑇𝑇 − (𝑧𝑧) = [𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧)] − = 𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧), 𝑧𝑧 ∈ ℂ ∪ {∞}. For arbitrary 𝐵𝐵(𝑧𝑧) ∈ ℜ𝑟𝑟 there is a unique solution to equation (1) and the problem. This solution in case 1) can be found according to following the formulas: 𝑋𝑋 + (𝑧𝑧) = 𝐴𝐴−1 (𝑧𝑧)𝐵𝐵 + (𝑧𝑧), 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵− (𝑧𝑧), 𝑧𝑧 ∈ ℂ ∪ {∞}; (4) and in case 2) - according to the formulas: 𝑋𝑋 + (𝑧𝑧) = [𝐴𝐴−1 (𝑧𝑧)𝐵𝐵 + (𝑧𝑧)] + , 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵− (𝑧𝑧) + 𝐴𝐴(𝑧𝑧)[𝐴𝐴−1 (𝑧𝑧)𝐵𝐵 + (𝑧𝑧)] − , 𝑧𝑧 ∈ ℂ ∪ {∞}. (5) Proof. Under the conditions of theorem 2, for the coefficient 𝐴𝐴(𝑧𝑧) ∈ ℜ𝑟𝑟 there exists an inverse in ℜ𝑟𝑟 , and by virtue of the definitions of the subrings ℜ𝑟𝑟 + , ℜ𝑟𝑟 − and the assumptions, in case 1) 𝐴𝐴(𝑧𝑧), 𝐴𝐴−1 (𝑧𝑧) ∈ ℜ+ 𝑟𝑟 , and in case 2) 𝐴𝐴(𝑧𝑧), 𝐴𝐴−1 (𝑧𝑧) ∈ ℜ− 𝑟𝑟 . Therefore, there is a normalized correct factorization of 𝐴𝐴−1 (𝑧𝑧) = 𝐺𝐺 + (𝑧𝑧)𝑆𝑆 0 (𝑧𝑧)𝑇𝑇 − (𝑧𝑧), 𝑧𝑧 ∈ ℂ ∪ {∞} over the factorization pair (𝑎𝑎1 , 𝑎𝑎2 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖), which will be: in the case 1) 𝐺𝐺 + (𝑧𝑧) = [𝐴𝐴−1 (𝑧𝑧)𝐴𝐴0 )]+ == 𝐴𝐴−1 (𝑧𝑧)𝐴𝐴0 , 𝐴𝐴0 : = [𝐴𝐴(𝑧𝑧)] 0 , 𝑆𝑆 0 (𝑧𝑧) = [𝐴𝐴0 ]−1 = [𝐴𝐴−1 (𝑧𝑧)] 0 , 𝑇𝑇 − (𝑧𝑧) = 1, 𝑧𝑧 ∈ ℂ ∪ {∞}; and in case 2) 𝐺𝐺 + (𝑧𝑧) = 1, 𝑆𝑆 0 (𝑧𝑧) = [𝐴𝐴0 ]−1 , 𝑇𝑇 − (𝑧𝑧) = [𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧)]− = 𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧), 𝑧𝑧 ∈ ℂ ∪ {∞}. Therefore, by theorem 1, for an arbitrary right-hand side 𝐵𝐵(𝑧𝑧) ∈ ℜ𝑟𝑟 there is a unique solution in ℜ𝑟𝑟 to equation (1) and the problem. It can be found by formulas (3), which, in case 1), give: 𝑋𝑋 + (𝑧𝑧) = 𝐴𝐴−1 (𝑧𝑧)𝐴𝐴0 [𝐴𝐴0 ]−1 [1 ⋅ 𝐵𝐵 + (𝑧𝑧)]+ = 𝐴𝐴−1 (𝑧𝑧) 𝐵𝐵 + (𝑧𝑧), 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵− (𝑧𝑧) + (1)−1 [1 ⋅ 𝐵𝐵 + (𝑧𝑧)]− = 𝐵𝐵− (𝑧𝑧). The formulas (4) are valid. In case 2), from the same formulas (3), it follows that: 𝑋𝑋 + (𝑧𝑧) = 1 ⋅ [𝐴𝐴0 ]−1 [𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧) 𝐵𝐵+ (𝑧𝑧)]+ = [𝐴𝐴−1 (𝑧𝑧) 𝐵𝐵+ (𝑧𝑧)]+ , 𝑌𝑌− (𝑧𝑧) = 𝐵𝐵− (𝑧𝑧) + [𝐴𝐴0 ]−1 𝐴𝐴(𝑧𝑧)[𝐴𝐴0 𝐴𝐴−1 (𝑧𝑧)𝐵𝐵 + (𝑧𝑧)]− = = 𝐵𝐵− (𝑧𝑧) + 𝐴𝐴(𝑧𝑧)[𝐴𝐴−1 (𝑧𝑧)𝐵𝐵 + (𝑧𝑧)]− , that is, formulas (5) are valid. The theorem is proved. Consequence. Under the conditions of theorem 2, equation (1) and the problem with the right-hand side 𝐵𝐵(𝑧𝑧) = 1, 𝐵𝐵(𝑧𝑧) ∈ ℜ𝑟𝑟 , 𝑧𝑧 ∈ ℂ ∪ {∞} have in ℜ𝑟𝑟 the only solution 𝑋𝑋𝑒𝑒+ (𝑧𝑧), 𝑌𝑌−𝑒𝑒 (𝑧𝑧). In case 1) from theorem 2, it can be found by the formulas: 𝑋𝑋𝑒𝑒+ (𝑧𝑧) = 𝐴𝐴−1 (𝑧𝑧), 𝑌𝑌−𝑒𝑒 (𝑧𝑧) = 0, 𝑧𝑧 ∈ ℂ ∪ {∞}, (6) and in case 2) - according to the following formulas: 𝑋𝑋𝑒𝑒+ (𝑧𝑧) = [𝐴𝐴−1 (𝑧𝑧)]0 = [𝐴𝐴0 ]−1 , 𝑌𝑌−𝑒𝑒 (𝑧𝑧) = 𝐴𝐴(𝑧𝑧)[𝐴𝐴−1 (𝑧𝑧)]− , 𝑧𝑧 ∈ ℂ ∪ {∞}. (7) The transformation of the second of equalities (7) can be represent in the as: 𝑋𝑋𝑒𝑒+ (𝑧𝑧) = [𝐴𝐴0 (𝑧𝑧)]−1 , 𝑌𝑌−𝑒𝑒 (𝑧𝑧) = 1 − 𝐴𝐴(𝑧𝑧)[𝐴𝐴0 ]−1 , 𝑧𝑧 ∈ ℂ ∪ {∞}. (8) 3.2 Example Let us find a solution to an equation of the form (1) and problem in ℜ𝑟𝑟 for: 11𝑧𝑧 2 −33𝑖𝑖𝑖𝑖−22 5𝑧𝑧+5𝑖𝑖 𝐴𝐴(𝑧𝑧) = ; 𝐵𝐵(𝑧𝑧) = 2 . 3𝑧𝑧 2 −24𝑖𝑖𝑖𝑖−45 𝑧𝑧 +22𝑖𝑖𝑖𝑖−120 In this case, 11 11(𝑧𝑧−𝑖𝑖)(𝑧𝑧−2𝑖𝑖) 3(𝑧𝑧−3𝑖𝑖)(𝑧𝑧−5𝑖𝑖) 𝐴𝐴0 = ; 𝐴𝐴(𝑧𝑧) = ; 𝐴𝐴−1 (𝑧𝑧) = ; 3 3(𝑧𝑧−3𝑖𝑖)(𝑧𝑧−5𝑖𝑖) 11(𝑧𝑧−𝑖𝑖)(𝑧𝑧−2𝑖𝑖) 5(𝑧𝑧 + 𝑖𝑖) 𝐵𝐵(𝑧𝑧) = = 𝐵𝐵 + (𝑧𝑧) = 𝐵𝐵+ (𝑧𝑧), (𝑧𝑧 + 10𝑖𝑖)(𝑧𝑧 + 12𝑖𝑖) 𝐵𝐵 − (𝑧𝑧) = 0 = 𝐵𝐵− (𝑧𝑧). In this example, all zeros and poles of the function 𝐴𝐴(𝑧𝑧) are located in the upper half-plane. Thus, 𝐴𝐴(𝑧𝑧), 𝐴𝐴−1 (𝑧𝑧) ∈ ℜ− 𝑟𝑟 and, by virtue of theorem 2, formulas (5) are applicable. Using these formulas, let us find a unique solution to equation (1) and the problem in the form: + + 3(𝑧𝑧 − 3𝑖𝑖)(𝑧𝑧 − 5𝑖𝑖) 5(𝑧𝑧 + 𝑖𝑖) 𝑋𝑋 + (𝑧𝑧) = � ⋅� � � = 11(𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖) (𝑧𝑧 + 10𝑖𝑖)(𝑧𝑧 + 12𝑖𝑖) +  15( z − 3i )( z − 5i )( z + i )  = =  11( z − i )( z − 2i )( z + 10i )( z + 12i )  15  2805 1755  225 = ⋅ −  =× 11  364( z + 12i ) 264( z + 10i )  242 ⋅ 364 565 z − 1448i 225 565 z − 1448i × = ⋅ 2 , ( z + 10i )( z + 12i ) 88088 z + 22iz − 120 5(𝑧𝑧 + 𝑖𝑖) 11 (𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖) 𝑌𝑌− (𝑧𝑧) = � � + × (𝑧𝑧 + 10𝑖𝑖)(𝑧𝑧 + 12𝑖𝑖) − 3 (𝑧𝑧 − 3𝑖𝑖)(𝑧𝑧 − 5𝑖𝑖) + 3(𝑧𝑧 − 3𝑖𝑖)(𝑧𝑧 − 5𝑖𝑖) 5(𝑧𝑧 + 𝑖𝑖) ×� ⋅� � � = 11(𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖) (𝑧𝑧 + 10𝑖𝑖)(𝑧𝑧 + 12𝑖𝑖) − (𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖) −16 3 =5⋅ ⋅� + �= (𝑧𝑧 − 3𝑖𝑖)(𝑧𝑧 − 5𝑖𝑖) 143(𝑧𝑧 − 𝑖𝑖) 56(𝑧𝑧 − 2𝑖𝑖) 5 (𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖)(−467𝑧𝑧 + 1363𝑖𝑖) = ⋅ = 8008 (𝑧𝑧 − 3𝑖𝑖)(𝑧𝑧 − 5𝑖𝑖)(𝑧𝑧 − 𝑖𝑖)(𝑧𝑧 − 2𝑖𝑖) 5 (−467𝑧𝑧 + 1363𝑖𝑖) = ⋅ . 8008 𝑧𝑧 2 − 8𝑖𝑖𝑖𝑖 − 15 By substitution in (1), one can make sure that the solution is actually found. The desired solution is as follows: 225 565z − 1448i 5 467z − 1363i X + (z) = ⋅ 2 , Y− (z) = − ⋅ 2 . 88088 z + 22iz − 120 8008 z − 8iz − 15 4 Discussion In the range of questions related to integral equations of the convolution type and the Riemann problem, research methods based on the integral theory (Cauchy type) are distinguished by a significant analytical barrier. It is quite challenging to bring research to the level of observable theorems. In different classes of assumptions, it is often necessary to rebuild the theory from scratch. Examples of new approaches go back, particularly to the research of M. G. Krejn (1958). Simultaneously, it is sometimes possible to simplify and obtain the final results and exact formulas for solutions to equation (1). This is how the current research is carried out. Using a new approach, the authors obtained a general solvability theorem that considers the specifics of the assumptions and contains convenient solution formulas. The resulting theorem is consistent with the stated objective. The authors note that the special nature of the used FP ring can be both a strength and a weakness of the research. The proposed approach is simpler and more efficient to use. Other authors did not have similar results. The authors have reflected the research results in many publications. With the received materials (abstract and specific), they participated in scientific conferences of the following structures: 1. Kherson National Technical University (KNTU) in Kherson (Ukraine); 2. Taras Shevchenko National University of Kyiv (KNU); 3. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” (NTUU KPI) in Kyiv (Ukraine); 4. Bauman Moscow State Technical University (BMSTU) in Moscow (Russia); 5. Odessa State Academy of Civil Engineering and Architecture in Odessa (Ukraine) (2015–2020), and others. Experts showed a significant interest to the obtained results. 5 Conclusion The paper considers the specifics of the assumptions about the location of the singular points of the rational coefficient of equation (1). The authors obtained final results that are easy to use, and the main results of the research are consistent with the objective. For the first time, under the conditions of theorem 2, signs are established that consider the specifics of assumptions about the coefficients of equation (1) and general formulas (4)-(8) of solutions in ℜ𝑟𝑟 . They are more convenient than the ones obtained by the authors earlier. The results are applicable when studying the existence, uniqueness, or non-uniqueness of solutions, their construction for specific equations of the form (1), and corresponding related problems. Their practical significance lies in the possibility of facilitating their use for finding solutions of specific equations of the considered form and demonstrating a new general approach. The new research methodology for this problem is also of practical interest. Other researchers did not use the proposed method in the considered ring. The authors want to study the cases further when factorization concerning the factorization pair for 𝐴𝐴−1 (𝑧𝑧) is not correct. The expansion of classes of equations, examples of rings with factorization pairs, the expansion of the scope of the approach based on the approximation of functions, and applied aspects are also promising. The obtained results indicate the fruitfulness of the approach from the theory of equations in rings with factorization pairs for the considered issues and problems. According to assumptions, they deepen the fundamental understanding of issues related to the Riemann problem and simplify the process of solving specific examples. The formulation of the problem of solvability of the equation, considering the specifics of the location of the singular points of the rational coefficient, is given for the first time and the “circular” approach to the research. It demonstrates the possibility of a unified approach based on the provisions on equations in rings with FP and provides an opportunity to see the interpretation and results of research. References Akopyan, V. N., & Dashtoyan, L. L. (2013). Closed solutions of some mixed problems for an orthotropic plane with a cut. Current problems of deformable solid body mechanisms, differential, and integral equations. Abstracts of reports of the international scientific conference. Odessa, Ukraine: Odessa I. I. Mechnikov National University. Cherskij, Yu. I., Kerekesha, P. V., & Kerekesha, D. P. (2010). 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