Confluent hypergeometric function

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In mathematics, a confluent hypergeometric functions is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. (The term "confluent" refers to the merging of singular points of families of differential equations; "confluere" is latin for "to flow together".) There are several common standard forms of confluent hypergeometric functions:

  • Kummer's (confluent hypergeometric) function (for Ernst Kummer) is the family of solutions to a differential equation known as Kummer's equation. There is a different but unrelated Kummer's function bearing the same name.
  • Whittaker functions (for E. T. Whittaker) are solutions to Whittaker's equation.
  • Coulomb wave functions are solutions to Coulomb wave equation.

The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.

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[edit] Kummer's equation

Kummer's equation is

z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0.\,\!

It has two linearly independent solutions M(a,b,z) and U(a,b,z).

Kummer's function (first kind) is given by

M(a,b,z)= \sum_{n=0}^\infty \frac {(a)_n z^n} {(b)_n n!}\,\!, also denoted by {\,}_1F_1(a;b;z) or Φ(a,b,z)

where {\,}_1F_1(a;b;z) is a hypergeometric series and (a)n = a(a + 1)(a + 2)...(a + n − 1) is the rising factorial.

The other solution is Kummer's function (second kind):

U(a,b,z)=\frac{\pi}{\sin\pi b} \left( \frac{M(a,b,z)} {\Gamma(1+a-b)\Gamma(b)} - z^{1-b} \frac{M(1+a-b, 2-b,z)}{\Gamma(a) \Gamma(2-b)} \right), also denoted by Ψ(a,b,z).

[edit] Whittaker's equation

Whittaker's equation is

\frac{d^2w}{dx^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0.

Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer functions M and U by

Mκ,μ(z) = e − z / 2zμ + 1 / 2M(μ − κ + 1 / 2,1 + 2μ,z)
Wκ,μ(z) = e − z / 2zμ + 1 / 2U(μ − κ + 1 / 2,1 + 2μ,z)

[edit] Coulomb wave equation

The Coulomb wave equation equation is

\frac{d^2w}{d\rho^2}+\left(1-\frac{2\nu}{\rho}+\frac{L(L+1)}{\rho^2}\right)w=0

where L is usually a non-negative integer. The solutions are called Coulomb wave functions. Putting x=2iρ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments.

[edit] Special cases

[edit] Relations

There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.

The derivative of Kummer's function M  is given by:

\frac{d}{dz}\,M(a,b,z) = \frac{a}{b}\,M(a+1,b+1,z)

From which follows, by induction, that:

\frac{d^n}{dz^n}M(a,b,z)=\frac{(a)_n}{(b)_n}M(a+n,b+n,z)

Kummer's functions are also related by Kummer's transformations:

M(a,b,z) = e^z\,M(b-a,b,-z)

The derivative of Kummer's function U  is given by:

\frac{d}{dz}\,U(a,b,z) = -a\,U(a+1,b+1,z)

[edit] Application to continued fractions

The series expansion of Kummer's function of the first kind, given by

M(a,b,z) = \sum_{n=0}^\infty \frac {(a)_n z^n} {(b)_n n!}, \,

shows that M(a, b, z) is an entire function of z (provided that b is not a negative integer). and that when a = b, M(a, b, z) is just the familiar exponential function ez.

By applying a limiting argument to the continued fraction of Gauss it can be shown that

\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}} {1 + \cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}} {1 - \cfrac{{\displaystyle\frac{b-a+1}{(b+2)(b+3)}z}} {1 + \cfrac{{\displaystyle\frac{a+2}{(b+3)(b+4)}z}}{1 - \ddots}}}}}

and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole. Moreover, by setting b = 0 and c = 1 and applying an equivalence transformation, an expansion of the exponential function can be obtained:

e^z = \cfrac{1}{1 - \cfrac{z}{1 + \cfrac{z}{2 - \cfrac{z}{3 + \cfrac{z} {2 - \cfrac{z}{5 + \cfrac{z}{2 - \ddots}}}}}}}.

This representation of the exponential function is valid for all values of z.

[edit] References

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