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Added commonplace terminology "confluent hypergeometric function of the first/second kind". |
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In [[mathematics]], a '''confluent [[hypergeometric function]]''' is a solution of a '''confluent hypergeometric equation''', which is a degenerate form of a [[hypergeometric differential equation]] where two of the three [[regular singular point|regular singularities]] merge into an [[irregular singularity]]. (The term "[[Confluence|confluent]]" refers to the merging of singular points of families of differential equations; "confluere" is Latin for "to flow together".) There are several common standard forms of confluent hypergeometric functions:
* '''Kummer's (confluent hypergeometric) function''' {{math|''M''(''a'', ''b'', ''z'')}}, introduced by {{harvs|txt|authorlink=Ernst Kummer| last=Kummer |year=1837}}, is a solution to '''Kummer's differential equation'''. This is also know as the confluent hypergeometric function of the first kind. There is a different and unrelated [[Kummer's function]] bearing the same name.
* '''Tricomi's (confluent hypergeometric) function''' {{math|''U''(''a'', ''b'', ''z'')}} introduced by {{harvs|txt|authorlink=Francesco Tricomi|first=Francesco|last=Tricomi|year=1947}}, sometimes denoted by {{math|Ψ(''a''; ''b''; ''z'')}}, is another solution to Kummer's equation. This is also know as the confluent hypergeometric function of the second kind.
* '''[[Whittaker function]]s''' (for [[Edmund Taylor Whittaker]]) are solutions to '''Whittaker's equation'''.
* '''[[Coulomb wave function]]s''' are solutions to the '''Coulomb wave equation'''. The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.
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* {{cite journal | last= Tricomi | first= Francesco G. | authorlink= Francesco Giacomo Tricomi | title= Sulle funzioni ipergeometriche confluenti | language= Italian | journal= Annali di Matematica Pura ed Applicata. Serie Quarta | year= 1947 | volume= 26 | pages= 141–175 | issn= 0003-4622 | mr= 0029451 | ref= harv | doi=10.1007/bf02415375}}
* {{cite book | last= Tricomi | first= Francesco G. | title= Funzioni ipergeometriche confluenti | language= Italian | ___location= Rome | publisher= Edizioni cremonese | year= 1954 | series= Consiglio Nazionale Delle Ricerche Monografie Matematiche | volume= 1 | isbn= 978-88-7083-449-9 | mr= 0076936 | ref=harv}}
* {{cite book | last=Oldham | first=K.B. | last2=Myland | first2=J. | last3=Spanier | first3=J. | title=An Atlas of Functions: with Equator, the Atlas Function Calculator | publisher=Springer New York | series=An Atlas of Functions | year=2010 | isbn=978-0-387-48807-3 | url=http://books.google.co.uk/books?id=UrSnNeJW10YC&pg=PA75 | ref=harv | access-date=2017-08-23}}
==External links==
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