Confluent hypergeometric function: Difference between revisions

Revision as of 05:52, 27 August 2020

Revision as of 06:03, 27 August 2020

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===Other equations===

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:

:<math>z\frac{d^2w}{dz^2} +(b-z)\frac{dw}{dz} -\left(\sum_{m=0}^M a_m z^m\right)w = 0</math> <ref>{{cite journal|last1=Campos|first1=LMBC|title=On Some Solutions of the Extended Confluent Hypergeometric Differential Equation|journal=Journal of Computational and Applied Mathematics|date=2001|volumepublisher=Elsevier|doi=10.1016/s0377-0427(00)00706-8|pages=177–200|doi-access=free}}</ref>

Note that for {{math|''M'' {{=}} 0}} or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation.

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In terms of [[Laguerre polynomials]], Kummer's functions have several expansions, for example

:<math>M\left(a,b,\frac{x y}{x-1}\right) = (1-x)^a \cdot \sum_n\frac{a^{(n)}}{b^{(n)}}L_n^{(b-1)}(y)x^n</math> {{harv|ErdelyiErdélyi|Magnus|Oberhettinger|Tricomi|1953|loc=6.12}}

==Special cases==

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* {{eom|title=Confluent hypergeometric function|first=E.A. |last=Chistova}}

* {{dlmf|first=Adri B. Olde|last= Daalhuis|id=13}}

* {{cite book | last1= Erdélyi | first1= Arthur | author1-link= Arthur Erdélyi | last2= Magnus | first2= Wilhelm | author2-link= Wilhelm Magnus | last3= Oberhettinger | first3= Fritz | lastauthoramp= yes | last4= Tricomi | first4= Francesco G. | title= Higher transcendental functions. Vol. I | ___location= New York–Toronto–London | publisher= McGraw–Hill Book Company, Inc. | year= 1953 | mr= 0058756 | ref= harv}}

* {{cite journal | last= Kummer | first= Ernst Eduard | authorlink= Ernst Eduard Kummer | title= De integralibus quibusdam definitis et seriebus infinitis | language= Latin | url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002141329 | format= | journal= [[Journal für die reine und angewandte Mathematik]] | year= 1837 | volume= 17 | pages= 228–242 | issn= 0075-4102 | ref= harv | doi=10.1515/crll.1837.17.228}}

* {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Confluent hypergeometric functions | url= https://archive.org/details/confluenthyperge0000slat | url-access= registration | ___location= Cambridge, UK | publisher= Cambridge University Press | year= 1960 | mr= 0107026 | ref= harv}}

* {{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 |series=Series 4 | 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=https://books.google.co.uk/books?id=UrSnNeJW10YC&pg=PA75 | ref=harv | access-date=2017-08-23}}

==External links==