Confluent hypergeometric function: Difference between revisions

Revision as of 12:29, 2 January 2017

Revision as of 10:58, 3 January 2017

Line 47:

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'', the functions <math>M(a,b,z)</math> and <math>U(a,b,z)</math> are independent, and if ''b'' is a non-positive integer (so <math>M(a,b,z)</math> doesn't exist) then we may be able to use <math>z^{1-b}M(a+1-b,2-b,z)</math> 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)</math> can be used as a second solution if it exists and is different. But when ''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)</math> and of <math>M(a,b,z).</math> In those cases a second solution exists of the form (valid for any positivereal integeror complex ''ba'' and any realpositive or complexinteger ''ab'' notexcept equalwhen to 1, as well as for {{math|1=''a'' =is a positive integer less than ''b'' = 1}}):

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

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

Line 175:

::<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-cb,z)</math> whenfor non-positive integer ''n'' is a positive[[generalized integer.Laguerre A closed form with powers of ''z''polynomial]].

::<math>MU(n,bc,z)</math> for non-positive integer ''n'' is a [[multiple of a generalized Laguerre polynomial]]., Thisequal implies thatto <math>z^\frac{\Gamma(1-bc)}M{\Gamma(a+1-bc)}M(n,2-bc,z)</math> is also a closed-form solution to Kummer's equation when <math>a+1-b</math>the islatter a non-positive integerexists.

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

::<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(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</math> etc.

::<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.

::Using the contiguous relation <math>aM(a+)=(a+z)M+z(a-b)M(b+)/b</math> we get, for example, <math>M(2,1,z)=(1+z)e^z.</math>

*[[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]]: