Confluent hypergeometric function: Difference between revisions

Revision as of 19:57, 19 December 2016

Revision as of 11:46, 27 December 2016

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<!--:<math>z\frac{d^2v}{dz^2}+2(1-b)\frac{dv}{dz}-b(1-b)z^{-1}v + (b-z)\left[\frac{dv}{dz}+(1-b)z^{-1}v\right] - av = 0</math>-->

:<math>z\frac{d^2v}{dz^2}+(2-b-z)\frac{dv}{dz} - (a+1-b)v = 0.</math>

This means that <math>z^{1-b}M(a-b+1-b,2-b,z)</math> is a solution so long as ''b'' is not an integer greater than 1, just as <math>M(a,b,z)</math> is a solution so long as ''b'' is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function {{math|''U''(''a'', ''b'', ''z'')}} introduced by {{harvs|txt|authorlink=Francesco Tricomi| first=Francesco |last=Tricomi | year=1947}}, and sometimes denoted by {{math|Ψ(''a''; ''b''; ''z'')}}. It is a combination of the above two solutions, defined by

:<math>U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z).</math>

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 ''z'', ''U''(''z'') usually has a [[singularity (mathematics)|singularity]] at zero. For example, if ''b''=0 and ''a''≠0 then <math>\Gamma(a+1)U(a,b,c)-1</math> is asymptotic to <math>az\ln z</math> as ''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}U(a+1-b,2-b,z)</math> to Kummer's equation is the same as the solution <math>U(a,b,z).</math>

For most combinations of ''a'' and ''b'', at least two of the three functions <math>M(a,b,z),\ U(a,b,z)</math> and <math>z^{1-b}M(a+1-b,2-b,z)</math> will be defined and independent. If <math>a=0</math>, then when <math>M(a,b,z)</math> is defined (that is, when ''b'' is not a non-positive integer) <math>U(a,b,z)=M(a,b,z)=1</math>, and when <math>a+1=b</math> and ''b'' is not an integer greater than 1 then <math>U(a,b,z)=z^{1-b}M(a+1-b,2-b,z)=z^{1-b}.</math> When <math>a=0</math> one can usually use <math>z^{1-b}M(a+1-b,2-b,z)</math> as a second solution, but this is undefined if ''b'' is an integer greater than 1, and if <math>b=1</math> then <math>M(a,b,z),\ U(a,b,z)</math> and <math>z^{1-b}M(a+1-b,2-b,z)</math> are all the same solution, namely <math>w(z)=1</math>. But whenever <math>a=0</math> it is easy to solve the differential equation to find a non-constant solution, namely

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::<math>\frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}e^z</math>

::<math>U(a,a+1,z)= z^{-a}</math>

::<math>U(-n,-2n,z)</math> for non-negative integer ''n'' is a Bessel polynomial (see lower down).

::<math>M(n,b,z)</math> for non-positive integer ''n'' is a [[generalized Laguerre polynomial]]. This implies that <math>z^{1-b}M(a+1-b,2-b,z)</math> is also a closed-form solution to Kummer's equation when <math>a+1-b</math> is a non-positive integer.

*[[Bateman's function]]

*[[Bessel function]]s and many related functions such as [[Airy function]]s, [[Kelvin function]]s, [[Hankel function]]s. For example, in the special case {{math|''b'' {{=}} 2''a''}} the function reduces to a [[Bessel function]]: