Confluent hypergeometric function: Difference between revisions

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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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===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>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'')}}.

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

==Special cases==

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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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and that this continued fraction converges uniformly to a [[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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* {{cite journal | last= Tricomi | first= Francesco G. | author-link= Francesco Giacomo Tricomi | title= Sulle funzioni ipergeometriche confluenti | language= it | 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= it | ___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]]