scholarly journals On the Regular Integral Solutions of a Generalized Bessel Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
L. M. B. C. Campos ◽  
F. Moleiro ◽  
M. J. S. Silva ◽  
J. Paquim
Keyword(s):  
Differential Equation ◽  
Complex Plane ◽  
Complex Variable ◽  
Finite Complex ◽  
Regular Singularity ◽  
Bessel Differential Equation ◽  
Zero Degree ◽  
Nonzero Degree ◽  
Fuchs Series ◽  
Integral Solutions

The original Bessel differential equation that describes, among many others, cylindrical acoustic or vortical waves, is a particular case of zero degree of the generalized Bessel differential equation that describes coupled acoustic-vortical waves. The solutions of the generalized Bessel differential equation are obtained for all possible combinations of the two complex parameters, order and degree, and finite complex variable, as Frobenius-Fuchs series around the regular singularity at the origin; the series converge in the whole complex plane of the variable, except for the point-at-infinity, that is, the only other singularity of the differential equation. The regular integral solutions of the first and second kinds lead, respectively, to the generalized Bessel and Neumann functions; these reduce to the original Bessel and Neumann functions for zero degree and have alternative expressions for nonzero degree.

The asymptotic solution of differential equations with a turning point and singularities

1957 ◽  
Vol 53 (2) ◽  
pp. 382-398 ◽  
Author(s):  
R. C. Thorne
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Complex Variable ◽  
Turning Point ◽  
Asymptotic Solutions ◽  
Regular Singularity ◽  
Image Position ◽  
Interior Points

ABSTRACTAsymptotic solutions of the differential equationfor large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals The Uniformisation of the Equation $$z^w=w^z$$

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Positive Solutions ◽  
Complex Plane ◽  
Complex Variable ◽  
Holomorphic Functions ◽  
Parametric Form ◽  
Lambert Function ◽  
The Real ◽  
Complex Equation ◽  
Real Equation ◽  
Expository Paper

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Solution to the Bessel differential equation with interactive fuzzy boundary conditions

2021 ◽  
Vol 41 (1) ◽  
Author(s):  
Daniel Eduardo Sánchez ◽  
Vinícius Francisco Wasques ◽  
Estevão Esmi ◽  
Laécio Carvalho de Barros
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Bessel Differential Equation ◽  
Fuzzy Boundary

The approximate solution of nonlinear Fredholm implicit integro-differential equation in the complex plane

2021 ◽  
Author(s):  
Samir Lemita ◽  
Sami Touati ◽  
Kheireddine Derbal
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Numerical Process

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

Asymptotic Values Along Julia Rays

1976 ◽  
Vol 28 (6) ◽  
pp. 1210-1215
Author(s):  
P. M. Gauthier ◽  
J. S. Hwang
Keyword(s):  
Complex Plane ◽  
Finite Complex ◽  
Asymptotic Values ◽  
The Family ◽  
Definition Of

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.

Complex Approximation and Simultaneous Interpolation on Closed Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 701-706 ◽  
Author(s):  
P. M. Gauthier ◽  
W. Hengartner
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Closed Subset ◽  
Finite Complex ◽  
Complex Approximation ◽  
Closed Sets ◽  
Limit Points ◽  
Simultaneous Interpolation ◽  
Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

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