Confluent hypergeometric function: Difference between revisions

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{{Short description|Solution of a confluent hypergeometric equation}}

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:

[[File:Plot of the Kummer confluent hypergeometric function 1F1(a;b;z) with a=1 and b=2 and input z² with 1F1(1,2,z²) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1.svg|alt=Plot of the Kummer confluent hypergeometric function 1F1(a;b;z) with a=1 and b=2 and input z² with 1F1(1,2,z²) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1|thumb|Plot of the Kummer confluent hypergeometric function 1F1(a;b;z) with a=1 and b=2 and input z² with 1F1(1,2,z²) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1]]

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 known 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 known 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.

==Kummer's equation==

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and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function.

Since Kummer's equation is second order there must be another, independent, solution. The [[indicial equation]] of the method of Frobenius tells us that the lowest power of a [[power series]] solution to the Kummer equation is either 0 or {{math|1 − ''b''}}. If we let {{math|''w''(''z'')}} be

:<math>w(z)=z^{1-b}v(z)</math>

then the differential equation gives

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Although this expression is undefined for integer {{mvar|b}}, it has the advantage that it can be extended to any integer {{mvar|b}} by continuity. Unlike Kummer's function which is an [[entire function]] of {{mvar|z}}, {{math|''U''(''z'')}} usually has a [[singularity (mathematics)|singularity]] at zero. For example, if {{math|''b'' {{=}} 0}} and {{math|''a'' ≠ 0}} then {{math|Γ(''a''+1)''U''(''a'', ''b'', ''z'') − 1}} is asymptotic to {{math|''az'' ln ''z''}} as {{mvar|z}} goes to zero. But see [[#Special cases]] for some examples where it is an entire function (polynomial).

Note that the solution {{math|''z''^1−''b''''MU''(''a'' + 1 − ''b'', 2 − ''b'', ''z'')}} to Kummer's equation is the same as the solution {{math|''U''(''a'', ''b'', ''z'')}}, see [[#Kummer's transformation]].

For most combinations of real or complex {{mvar|a}} and {{mvar|b}}, the functions {{math|''M''(''a'', ''b'', ''z'')}} and {{math|''U''(''a'', ''b'', ''z'')}} are independent, and if {{mvar|b}} is a non-positive integer, so {{math|''M''(''a'', ''b'', ''z'')}} doesn't exist, then we may be able to use {{math|''z''^1−''b''''M''(''a''+1−''b'', 2−''b'', ''z'')}} as a second solution. But if {{mvar|a}} is a non-positive integer and {{mvar|b}} is not a non-positive integer, then {{math|''U''(''z'')}} is a multiple of {{math|''M''(''z'')}}. In that case as well, {{math|''z''^1−''b''''M''(''a''+1−''b'', 2−''b'', ''z'')}} can be used as a second solution if it exists and is different. But when {{mvar|b}} is an integer greater than 1, this solution doesn't exist, and if {{math|1=''b'' = 1}} then it exists but is a multiple of {{math|''U''(''a'', ''b'', ''z'')}} and of {{math|''M''(''a'', ''b'', ''z'')}} In those cases a second solution exists of the following form and is valid for any real or complex {{mvar|a}} and any positive integer {{mvar|b}} except when {{mvar|a}} is a positive integer less than {{mvar|b}}:

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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=LMBCL.M.B.C.|title=On Some Solutions of the Extended Confluent Hypergeometric Differential Equation|journal=Journal of Computational and Applied Mathematics|dateyear=2001|volume=137|publishernumber=Elsevier1|doi=10.1016/s0377-0427(00)00706-8|pages=177–200|doi-accessbibcode=2001JCoAM.137..177C |mr=free1865885}}</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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:<math>U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt, \quad (\operatorname{Re}\ a>0) </math>

The integral defines a solution in the right half-plane {{math|0 < Re ''z'' <> ''π''/20}}.

They can also be represented as [[Barnes integral]]s

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:<math>M(a,b,z)\sim\Gamma(b)\left(\frac{e^zz^{a-b}}{\Gamma(a)}+\frac{(-z)^{-a}}{\Gamma(b-a)}\right)</math>

The powers of {{mvar|z}} are taken using {{math|−3''π''/2 < arg ''z'' ≤ ''π''/2}}.<ref> This is derived from Abramowitz and Stegun (see reference below), [http://people.math.sfu.ca/~cbm/aands/page_508.htm page 508], where a full asymptotic series is given. They switch the sign of the exponent in {{math|exp(''iπa'')}} in the right half-plane but this is immaterial, as the term is negligible there or else {{mvar|a}} is an integer and the sign doesn't matter.</ref> The first term is not needed when {{math|Γ(''b'' − ''a'')}} is finite, that is when {{math|''b'' − ''a''}} is not a non-positive integer and the real part of {{mvar|z}} goes to negative infinity, whereas the second term is not needed when {{math|Γ(''a'')}} is finite, that is, when {{mvar|a}} is a not a non-positive integer and the real part of {{mvar|z}} goes to positive infinity.

There is always some solution to Kummer's equation asymptotic to {{math|''e^zz''^^{''a''−''b''}}} as {{math|''z'' → −∞}}. Usually this will be a combination of both {{math|''M''(''a'', ''b'', ''z'')}} and {{math|''U''(''a'', ''b'', ''z'')}} but can also be expressed as {{math|''e^z'' (−1)^''a''-''b'' ''U''(''b'' − ''a'', ''b'', −''z'')}}.

==Relations==

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:<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|Erdélyi|Magnus|Oberhettinger|Tricomi|1953|loc=6.12}}

or

:<math>M\left( a,\, b,\, z \right) = \frac{\Gamma\left( 1 - a \right) \cdot \Gamma\left( b \right)}{\Gamma\left( b - a \right)} \cdot L_{-a}^{(b - 1)}\left( z \right)</math>[https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/27/01/0001/]

==Special cases==

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::<math>M_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

::<math>W_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

* The general {{mvar|p}}-th raw moment ({{mvar|p}} not necessarily an integer) can be expressed as<ref>{{CitationCite neededweb|datetitle=MarchAspects 2017of Multivariate Statistical Theory {{!}} Wiley|url=https://www.wiley.com/en-us/Aspects+of+Multivariate+Statistical+Theory-p-9780471769859|access-date=2021-01-23|website=Wiley.com|language=en-us}}</ref>

:: <math>\begin{align}

\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right] &= \frac{\left(2 \sigma^2\right)^{p/2} \Gamma\left(\tfrac{1+p}{2}\right)}{\sqrt \pi} \ {}_1F_1\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2}\right)\\

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==Application to continued fractions==

By applying a limiting argument to [[Gauss's continued fraction]] it can be shown that<ref>{{cite journal|first1=Evelyn|last1=Frank | year=1956|title=A new class of continued fraction expansions for the ratios of hypergeometric functions| journal=Trans. Am. Math. Soc.|volume=81|number=2|pages=453–476|mr= 0076937|jstor=1992927|doi=10.1090/S0002-9947-1956-0076937-0}}</ref>

:<math>\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}}

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</math>

and that this continued fraction converges uniformly to a [[Meromorphic function|meromorphic function]] of {{mvar|z}} in every bounded ___domain that does not include a pole.

==See also==

* [[Composite Bézier curve]]

==Notes==

{{Reflist}}

<references/>

==References==

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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 | lastauthorampname-list-style= yesamp | 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}}

* {{cite journal | last= Kummer | first= Ernst Eduard | authorlinkauthor-link= Ernst Eduard Kummer | title= De integralibus quibusdam definitis et seriebus infinitis | language= Latinla | url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002141329 | format= | journal= [[Journal für die reine und angewandte Mathematik]] | year= 1837 | volume= 1837 | issue= 17 | pages= 228–242 | issn= 0075-4102 | doi=10.1515/crll.1837.17.228| s2cid= 121351583 }}

* {{cite book | last= Slater | first= Lucy Joan | authorlinkauthor-link= 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}}

* {{cite journal | last= Tricomi | first= Francesco G. | authorlinkauthor-link= Francesco Giacomo Tricomi | title= Sulle funzioni ipergeometriche confluenti | language= Italianit | journal= Annali di Matematica Pura ed Applicata |series=Series 4 | year= 1947 | volume= 26 | pages= 141–175 | issn= 0003-4622 | mr= 0029451 | doi=10.1007/bf02415375| s2cid= 119860549 | doi-access= free }}

* {{cite book | last= Tricomi | first= Francesco G. | title= Funzioni ipergeometriche confluenti | language= Italianit | ___location= Rome | publisher= Edizioni cremonese | year= 1954 | series= Consiglio Nazionale Delle Ricerche Monografie Matematiche | volume= 1 | isbn= 978-88-7083-449-9 | mr= 0076936}}

* {{cite book | last1=Oldham | first1=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.com/books?id=UrSnNeJW10YC&pg=PA75 | access-date=2017-08-23}}

==External links==

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* [http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/ Kummer hypergeometric function] on the Wolfram Functions site

* [http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/ Tricomi hypergeometric function] on the Wolfram Functions site

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[[Category:Hypergeometric functions]]

[[Category:Special hypergeometric functions]]

[[Category:Special functions]]