Confluent hypergeometric function: Difference between revisions

Revision as of 17:41, 17 May 2023

Revision as of 14:52, 2 November 2023

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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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==Notes==

{{Reflist}}

<references/>

==References==

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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]]