Confluent hypergeometric function: Difference between revisions

Revision as of 11:46, 27 December 2016

Revision as of 12:29, 2 January 2017

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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 1−{{math|1 − ''b''}}. If we let ''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 ''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> (See [[#Kummer's transformation]] below.)

For most combinations of real (or complex) ''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}MU(a+1-b,2-b,z)</math> willare be definedindependent, and independent.if If''b'' is <math>a=0</math>, thennon-positive wheninteger (so <math>M(a,b,z)</math> isdoesn't definedexist) (thatthen is,we whenmay ''b''be isable notto a non-positive integer)use <math>U(a,z^{1-b,z)=}M(a+1-b,2-b,z)=1</math>, andas whena <math>second solution. But if {{mvar|a+1=b</math>}} is a non-positive integer and ''{{mvar|b''}} is not ana non-positive integer greater than 1, then <{{math>|''U''(a,b,''z'')=z^}} is a multiple of {1-b}{math|''M''(a+1-b,2-b,''z'')=z^{1-b}}.</math> WhenIn <math>a=0</math>that onecase canas usually usewell, <math>z^{1-b}M(a+1-b,2-b,z)</math> can be used as a second solution, butif thisit exists and is undefineddifferent. ifBut when ''b'' is an integer greater than 1 this solution doesn't exist, and if <{{math>|1=''b'' = 1</math>}} then it exists but is a multiple of <math>M(a,b,z),\ U(a,b,z)</math> and of <math>z^{1-b}M(a+1-b,2-b,z).</math> areIn allthose thecases samea second solution, namelyexists of the form <math>w(z)=1</math>.valid Butfor wheneverany <math>a=0</math>positive itinteger is''b'' easyand toany solvereal theor differentialcomplex equation''a'' not equal to find1, as well as for {{math|1=''a'' non-constant= solution,''b'' namely= 1}}):

:<math>wM(a,b,z)=\int_{-\infty}^ln z+z(-u)^{1-b}e^u\mathrmsum_{dk=0}u.^\infty C_kz^k</math>

When ''a'' = 0 we can alternatively use:

:<math>w(z)=z^{1-b}\int_{-\infty}^z(-u)^{b-2b}e^u\mathrm{d}u.</math>

When <math>b=1</math> this is the [[exponential integral]] E₁(''-z'').

A similar problem occurs when ''a''−''b'' is a negative integer and ''b'' is an integer less than 1. In this case <math>M(a,b,z)</math> doesn't exist, and <math>U(a,b,z)</math> is a multiple of <math>z^{1-b}M(a+1-b,2-b,z).</math> A second solution is then of the form:

Similarly, when <math>a+1-b=0</math> we have, in addition to

:<math>U(bz^{1-b}M(a+1-b,2-b,z)=\ln z^+\sum_{1-bk=0},^\infty C_kz^k</math>

the solution

:<math>w(z)=z^{1-b}\int_{-\infty}^z(-u)^{b-2}e^u\mathrm{d}u.</math>

Thus we have two independent solutions for any combination of ''a'' and ''b''.

===Other equations===

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::<math>U(a,a,z)=e^z\int_z^\infty u^{-a}e^{-u}du</math> (a polynomial if ''a'' is a non-positive integer)

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

::<math>U(c-n,c,z)=\frac{\Gamma(c-1)}{\Gamma(c-n)}z^{1-c}M(1-n,2-c,z)</math> when ''n'' is a positive integer. A closed form with powers of ''z''.

::<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).