Analytic continuation and Riemann surface determination of algebraic functions by computer

1996 ◽  
Vol 13 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Kosuke Shihara ◽  
Tateaki Sasaki
Keyword(s):  
Riemann Surface ◽  
Analytic Continuation ◽  
Algebraic Functions

On the connexion of algebraic functions with automorphic functions

1898 ◽  
Vol 63 (389-400) ◽  
pp. 267-268 ◽  
Keyword(s):  
Riemann Surface ◽  
Existence Theorem ◽  
Algebraic Equation ◽  
Rational Functions ◽  
Algebraic Relation ◽  
Periodic Functions ◽  
Algebraic Functions ◽  
Automorphic Functions

If u and z are variables connected by an algebraic equation, they are, in general, multiform functions of each other; the multiformity can be represented by a Riemann surface, to each point of which corresponds a pair of values of u and z . Poincaré and Klein have proved that a variable t exists, of which u and z are uniform automorphic functions; the existence-theorem, however, does not connect t analytically with u and z . When the genus ( genre, Geschlecht ) of the algebraic relation is zero oi unity, t can be found by known methods; the automorphic functions required are rational functions, and doubly periodic functions, in the two case respectively.

The structure and explicit determination of convex-polygonally generated shape-densities

1987 ◽  
Vol 19 (4) ◽  
pp. 896-916 ◽  
Author(s):  
David G. Kendall ◽  
Hui-Lin Le
Keyword(s):  
Analytic Continuation ◽  
Convex Polygon ◽  
Compact Convex ◽  
Finite Sequence ◽  
Analytic Structure ◽  
Explicit Formulae ◽  
Real Analytic ◽  
Explicit Determination ◽  
Tile Function

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a ‘stepping-stone' tile-to-tile route from T0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.

The structure and explicit determination of convex-polygonally generated shape-densities

1987 ◽  
Vol 19 (04) ◽  
pp. 896-916 ◽  
Author(s):  
David G. Kendall ◽  
Hui-Lin Le
Keyword(s):  
Analytic Continuation ◽  
Convex Polygon ◽  
Compact Convex ◽  
Finite Sequence ◽  
Analytic Structure ◽  
Explicit Formulae ◽  
Real Analytic ◽  
Explicit Determination ◽  
Tile Function

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T 0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a ‘stepping-stone' tile-to-tile route from T 0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.

scholarly journals Dispersive analysis of the κ/K0*(700) meson and other light strange resonances

2019 ◽  
Vol 212 ◽  
pp. 03003
Author(s):  
José R. Peláez ◽  
Arkaitz Rodas ◽  
Jacobo Ruiz de Elvira
Keyword(s):  
Analytic Continuation ◽  
Partial Wave ◽  
Dispersion Relations ◽  
Resonance Parameters ◽  
Preliminary Results ◽  
Independent Determination ◽  
Dispersive Analysis ◽  
Hyperbolic Dispersion ◽  
Model Independent

We briefly review our recent works where we use dispersion relations to constrain fits to data on πK → πK and $ \pi \pi \to K\bar K $ providing a simple but consistent description of these processes. Then, simple analytic methods allow to extract parameters of poles associated to light strange resonances without assuming a particular model. We also present preliminary results on a model-independent determination of the controversial κ or $ K_0^*\left( {700} \right) $ resonance parameters, by using those constrained parameterizations as input for partial-wave hyperbolic dispersion relations that allow to perform a rigorous analytic continuation to determine its associated pole.

Calculation of Resonance S-Matrix Poles by Means of Analytic Continuation in the Coupling Constant

2017 ◽  
Vol 21 (4) ◽  
pp. 1154-1172 ◽  
Author(s):  
Jiří Horáček ◽  
Lukáš Pichl
Keyword(s):  
Analytic Continuation ◽  
Resonance Energy ◽  
Original Method ◽  
Separable Potential ◽  
Coupling Constant ◽  
New Approach ◽  
New Variant ◽  
S Matrix ◽  
Limited Accuracy

AbstractThe method of analytic continuation in the coupling constant in combination with the use of statistical Padé approximation is applied to the determination of complex S-matrix poles, i.e. to the determination of resonance energy and widths. These parameters are of vital importance in many physical, chemical and biological processes. It is shown that an alternative to the method of analytic continuation in the coupling constant exists which in principle makes it possible to locate several resonances at once, in contrast to the original method which yields parameters of only one resonance. In addition the new approach appears to be less sensitive to the choice of the perturbation interaction used for the analytical continuation than the original method. In this paper both approaches are compared and tested for model analytic separable potential. It is shown that the new variant of the method of analytic continuation in the coupling constant is more robust and efficient than the original method and yields reasonable results even for data of limited accuracy.

scholarly journals On a functional equation arising from two processors with coupled inputs

2015 ◽  
Vol 4 (3) ◽  
pp. 264
Author(s):  
EL-Sayed El-Hady ◽  
Wolfgang Forg-Rob
Keyword(s):  
Functional Equation ◽  
Boundary Value Problem ◽  
Analytic Continuation ◽  
Generating Functions ◽  
Functional Equations ◽  
Elliptic Functions ◽  
Symmetric Case ◽  
The Other ◽  
Interesting Class

<p>During the last few decades, a certain interesting class of functional equations arises when obtaining the generating functions of many system distributions. Such a class of equations has numerous applications in many modern disciplines like wireless networks and communications. This paper has been motivated by an issue considered by Paul E. Wright in [Advances in applied probability, (1992), 986 􀀀 1007]. The functional equation obtained there has been solved using elliptic functions and analytic continuation, which in turn lead to the determination of the main unknown. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper on one hand, we introduce a solution in the symmetric case using boundary value problem approach. On the other hand, we investigate the potential singularities of the unknowns of the functional equation giving one possible application, and we compute some expectation of interest using the corresponding generating function.</p>

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