A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

Abstract

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

“…In most cases, these can be represented in terms of the functions M(a, b, z) and U(a, b, z), but in special cases, the two linearly independent equations are more complicated (see Ref. [11] for more details). Typically, the second solution U is not physically admissible due to its behavior near z = 0.…”

Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

“…Solution that stays finite at the origin is represented by the Kummer confluent hypergeometric function M(b, c, η) [46,47]:…”

On umbral properties of a family of hyperbolic-like functions appearing in magnetic transport problem

“…All the aforementioned problems can be solved in terms of confluent hypergeometric function (more specifically in terms of Kummer's function) [2]. Indeed, the radial function for Landau states can be written as [13]:…”