Kummer equation
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1: 13.2 Definitions and Basic Properties
Kummer’s Equation
►Standard Solutions
… ►§13.2(v) Numerically Satisfactory Solutions
… ► …2: 13.3 Recurrence Relations and Derivatives
3: 13.14 Definitions and Basic Properties
Whittaker’s Equation
… ►This equation is obtained from Kummer’s equation (13.2.1) via the substitutions , , and . … ►Standard solutions are: … ►The principal branches correspond to the principal branches of the functions and on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). …4: 15.10 Hypergeometric Differential Equation
§15.10(ii) Kummer’s 24 Solutions and Connection Formulas
…5: 13.29 Methods of Computation
§13.29(ii) Differential Equations
►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. … ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of , when and are real and is a positive integer. …6: Errata
There were clarifications made in the conditions on the parameter in of those equations.
A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions and as in and and fixed.
The equality has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation has been changed to .
Reported 2015-02-10 by Adri Olde Daalhuis.
The equality has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.
Reported 2015-02-10 by Adri Olde Daalhuis.
The equality has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation has been introduced.
Reported 2015-02-10 by Adri Olde Daalhuis.