An elementary analysis for motivating the definition of the regular singular point of the ordinary linear differential equation

1986 ◽  
Vol 17 (1) ◽  
pp. 39-40 ◽  
Author(s):  
S. Scott
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Linear Differential Equation ◽  
Ordinary Linear Differential Equation ◽  
Regular Singular Point ◽  
Elementary Analysis ◽  
Linear Differential ◽  
Definition Of

scholarly journals Logarithmic A-hypergeometric series

2020 ◽  
Vol 31 (13) ◽  
pp. 2050110
Author(s):  
Mutsumi Saito
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Hypergeometric Series ◽  
Standard Technique ◽  
Classical Case ◽  
Ordinary Linear Differential Equation ◽  
Regular Singular Point ◽  
Series Solutions ◽  
Parameter Vector ◽  
Linear Differential

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular [Formula: see text]-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic [Formula: see text]-hypergeometric series solutions, we adopt a different perturbation in this paper.

scholarly journals Holomorphic solutions about an irregular singular point of an ordinary linear differential equation

1985 ◽  
Vol 39 (2) ◽  
pp. 178-186 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Linear Differential Equation ◽  
Ordinary Linear Differential Equation ◽  
Holomorphic Solution ◽  
Linearly Independent ◽  
Irregular Singular Point ◽  
Zero Radius ◽  
Linear Differential ◽  
Indicial Equation

AbstractIt is shown that that an ordinary linear differential equation may possess a holomorphic solution in a neighbourhood of an irregular singular point even though the usual linearly independent solutions corresponding to the two roots of the indicial equation both have zero radius of convergence.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

Studies in multiplicative solutions to linear differential equations

1987 ◽  
Vol 106 (3-4) ◽  
pp. 277-305 ◽  
Author(s):  
F. M. Arscott
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Floquet Theory ◽  
Special Functions ◽  
Linear Differential Equations ◽  
Ordinary Linear Differential Equation ◽  
Closed Path ◽  
Starting Point ◽  
Linear Differential

SynopsisGiven an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.

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