A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions
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...
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| Format: | Article |
| Language: | English |
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Institute for Condensed Matter Physics
2022-09-01
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| Series: | Condensed Matter Physics |
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| Online Access: | https://doi.org/10.5488/CMP.25.33203 |
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| author | W. N. Mathews Jr. M. A. Esrick Z. Y. Teoh J. K. Freericks |
| author_facet | W. N. Mathews Jr. M. A. Esrick Z. Y. Teoh J. K. Freericks |
| author_sort | W. N. Mathews Jr. |
| collection | DOAJ |
| description | 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 ≡ z^(1-b) M(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. |
| first_indexed | 2024-04-12T03:31:01Z |
| format | Article |
| id | doaj.art-4b68e375f9514551864c51175d77bc59 |
| institution | Directory Open Access Journal |
| issn | 1607-324X 2224-9079 |
| language | English |
| last_indexed | 2024-04-12T03:31:01Z |
| publishDate | 2022-09-01 |
| publisher | Institute for Condensed Matter Physics |
| record_format | Article |
| series | Condensed Matter Physics |
| spelling | doaj.art-4b68e375f9514551864c51175d77bc592022-12-22T03:49:33ZengInstitute for Condensed Matter PhysicsCondensed Matter Physics1607-324X2224-90792022-09-012533320310.5488/CMP.25.33203A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functionsW. N. Mathews Jr.https://orcid.org/0000-0003-3111-4265M. A. Esrickhttps://orcid.org/0000-0001-7686-4276Z. Y. Teohhttps://orcid.org/0000-0003-2857-2417J. K. Freerickshttps://orcid.org/0000-0002-6232-9165The 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 ≡ z^(1-b) M(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.https://doi.org/10.5488/CMP.25.33203kummer's equationconfluent hypergeometric equationkummer's functiontricomi function |
| spellingShingle | W. N. Mathews Jr. M. A. Esrick Z. Y. Teoh J. K. Freericks A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions Condensed Matter Physics kummer's equation confluent hypergeometric equation kummer's function tricomi function |
| title | A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions |
| title_full | A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions |
| title_fullStr | A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions |
| title_full_unstemmed | A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions |
| title_short | A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions |
| title_sort | physicist s guide to the solution of kummer s equation and confluent hypergeometric functions |
| topic | kummer's equation confluent hypergeometric equation kummer's function tricomi function |
| url | https://doi.org/10.5488/CMP.25.33203 |
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