Confluent hypergeometric function: Definition from Answers.com

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In mathematics, a confluent hypergeometric function 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.

1 Kummer's equation
2 Whittaker's equation
3 Coulomb wave equation
4 Special cases
5 Relations
6 Connection with Laguerre polynomials and similar representations
7 Connection with Fourier Transform
8 Kummer's second formula
9 Multiplication theorem
10 Application to continued fractions
11 See also
12 Notes
13 References

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) which can be obtained by taking a Laplace transform over a suitable contour in the complex plane.

Kummer's function (of the first kind) is given by

$M(a,b,z)= \frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_0^1 e^{zu}u^{a-1}(1-u)^{b-a-1}\,du\,\quad \operatorname{re}\ b > \operatorname{re}\ a > 0.$

Another common notation for this solution is Φ(a,b,z). Differentiation under the integral shows at once that M(a,b,z) solves Kummer's equation, and the normalizing constant is such that the initial conditions

$M(a,b,0) = 1, \quad \left.\frac{d}{dz}M(a,b,z)\right|_{z=0} = a/b.$

Moreover, the integral defines an entire function of z.

A direct power series solution is also possible, which may again be verified by termwise differentiation. To wit,

$M(a,b,z)=\sum_{n=0}^\infty \frac {(a)_n z^n} {(b)_n n!}={}_1F_1(a;b;z),$

is a hypergeometric series, where

$(a)_n=a(a+1)(a+2)\cdots(a+n-1)\,$

is the rising factorial.

The other solution is Kummer's function of the second kind, also known as the Tricomi confluent hypergeometric function.^[1] This is obtained by the Laplace integral

$\begin{align}U(a,b,z) &= \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt, \quad (\operatorname{re}\ a>0) \\ &= \frac {z^{1-b}} {\Gamma(1+a-b)}\int_0^\infty \frac{e^{-z t} t^{a-b}}{(1+t)^a} \mathrm d t \quad (\operatorname{re}\ a-b>-1).\end{align}$

The integral defines a solution in the right half-plane re z > 0. This function is also frequently denoted by Ψ(a,b,z).

The function U is related to Kummer's function M by various identities such as

$\begin{align} U(a,b,z)&=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z)\\ &=\frac{\pi}{\sin\pi b} \left( \frac{M(a,b,z)} {\Gamma(a-b+1)\Gamma(b)} - z^{1-b} \frac{M(a-b+1, 2-b,z)}{\Gamma(a) \Gamma(2-b)} \right).\end{align}$

The existence of such identities is closely tied to the symmetry group of the Kummer equation; see Bateman (1953, Chapter 6).

The asymptotic behavior of U(a,b,z) as z → ∞ can be understood in terms of the integral representation

$U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt, \quad \operatorname{re}\ a>0.$

If z = x is real, then making a change of variables in the integral followed by expanding the binomial series and integrating it formally term by term gives rise to an asymptotic series expansion, valid as x → ∞:^[2]

$U(a,b,x)\sim x^{-a} \, _2F_0\left(a,a-b+1;\, ;-\frac 1 x\right),$

where $_2F_0(\cdot, \cdot; ;-1/x)$ is a generalized hypergeometric function, which converges nowhere but exists as a formal power series in 1/x.

Whittaker's equation

Whittaker's equation is

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

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

M κ,μ (z) = e - z / 2 z μ + 1 / 2 M (μ - κ + 1 / 2,1 + 2μ, z)

W κ,μ (z) = e - z / 2 z μ + 1 / 2 U (μ - κ + 1 / 2,1 + 2μ, z)

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.

Special cases

The error function can be expressed as

$\mathrm{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt= \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\frac{1}{2},\frac{3}{2},-x^2\right).$

The general p-th raw moment (p not necessarily an integer) can be expressed as

$\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right]= \left(2 \sigma^2\right)^\frac p 2 \frac {\Gamma\left(\frac{1+p}2\right)}{\sqrt \pi}\, _1F_1\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2}\right),$

$\operatorname{E} \left[N\left(\mu, \sigma^2 \right)^p \right]=(-2 \sigma^2)^\frac p 2 \cdot U\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2} \right)$ (the function's second branch cut can be chosen by multiplying with

( - 1) p

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:

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

Moreover,

$U(a,b,z)=z^{1-b} U\left(1+a-b,2-b,z\right)$ .

Connection with Laguerre polynomials and similar representations

In terms of Laguerre polynomials, Kummer's functions of the first kind has the expansion

$M(a,b,z)=\frac{{b-1 \choose a}}{{b-1+\alpha \choose a}} \sum_{k=0} (-1)^k L_k^{(-\alpha-k)}(z) \frac{{-a \choose k}}{{-b-\alpha \choose k}},$

and

$\begin{align}U(a,b,z)&=\frac{(1+\alpha-b)!}{(1+\alpha-b+a)!} \sum_{k=0}L_k^{(\alpha)}(z) \frac{{-a \choose k}}{{b-a-\alpha-2 \choose k}}\\ &=z^{1-b}\frac{(b+\alpha-1)!}{(a+\alpha)!} \sum_{k=0}L_k^{(\alpha)}(z) \frac{{a-b+k \choose k}}{{a+\alpha+k \choose k}} \\ &= \sum_{k=0} {\beta-b \choose k} {-a \choose k} k! U(a+k,\beta, z)\\ &= \frac{\Gamma(\alpha)}{\Gamma(a)}\sum_{k=0} (-1)^k {a-\alpha \choose k} U(\alpha, b-k,z)= \frac{e^z}{\Gamma(a)} \sum_{k=0} {k-a \choose k} z^{k+1-b} \Gamma(b-k-1,z)\\ &= \frac{\Gamma(\alpha-b+1)}{\Gamma(a-b+1)} \sum_{k=0} {\alpha-a-1+k \choose k} z^k U(\alpha+k,b+k,z)= \frac{e^z}{\Gamma(a-b+1)} \sum_{k=0} {b-a-1+k \choose k} z^k \Gamma(1-b-k,z).\end{align}$

which converges for $\alpha > 2 b - \frac 5 2$ .

Connection with Fourier Transform

Kummer's functions of the first and second kind are each other's Fourier transform, as

$\int_{-\infty}^\infty e^{-z^2 \pi} U\left(a,1+\frac b 2, z^2 \pi\right) e^{-2 \pi i k z}\mathrm d z= \frac{\Gamma\left(\frac{1-b}2\right)}{\Gamma\left(a+\frac{1-b}2\right)} e^{-k^2 \pi} M\left(a,a+\frac{1-b}2,k^2\pi\right),$

and conversely

$\int_{-\infty}^\infty e^{-z^2 \pi} M\left(a,1+\frac b 2, z^2 \pi\right) e^{-2 \pi i k z}\mathrm d z= \frac{\Gamma\left(1+\frac b 2\right)}{\Gamma\left(1+\frac b 2-a\right)} e^{-k^2 \pi} U\left(a,a+\frac{1-b}2,k^2\pi\right).$

Kummer's second formula

In the special case $b = 2 a$ the function reduces to a Bessel function:

$\begin{align}\, _1F_1(a,2a,x)&= e^{\frac x 2}\, _0F_1 (; a+\tfrac{1}{2}; \tfrac{1}{16}x^2) \\ &= e^{\frac x 2} \left(\tfrac{1}{4}x\right)^{\tfrac{1}{2}-a} \Gamma\left(a+\tfrac{1}{2}\right) I_{a-\frac 1 2}\left(\tfrac{1}{2}x\right).\end{align}$

This identity is sometimes also referred to as Kummer's second transformation.

Derived thus,

$U(a,2a,x)= \frac{e^\frac x 2}{\sqrt \pi} x^{\frac 1 2 -a} K_{a-\frac 1 2} \left(\frac x 2 \right),$

where $K$ is related to Bessel polynomial for integer $a$ .

Multiplication theorem

The following multiplication theorems hold true:

$\begin{align}U(a,b,z)&= e^{(1-t)z} \sum_{i=0} \frac{(t-1)^i z^i}{i!} U(a,b+i,z t)=\\ &= e^{(1-t)z} t^{b-1} \sum_{i=0} \frac{\left(1-\frac 1 t\right)^i}{i!} U(a-i,b-i,z t).\end{align}$

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 e^z.

By applying a limiting argument to Gauss's continued fraction 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.

Notes

^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 13", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 See also chapter 14.; Bateman 1953, §6.5
^ Andrews, G.E.; Askey, R.; Roy, R. (2000), Special functions, Cambridge University Press .

References

Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 13", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 See also chapter 14.
Bateman, Harry (1953), Higher transcendental functions, 1, McGraw-Hill .
Brychkov, Yu.A.; Prudnikov, A.P. (2001), "Whittaker function", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
Chistova, E.A. (2001), "Confluent hypergeometric function", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
Rozov, N.Kh. (2001), "Whittaker equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026 .
Wall, H. S. (1948), Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc. (chapter XVIII)

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