Confluent hypergeometric function

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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:

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.

Kummer's equation[edit]

Kummer's equation is

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

with a regular singular point at 0 and an irregular singular point at ∞. It has two (usually) linearly independent solutions M(a,b,z) and U(a,b,z).

Kummer's function (of the first kind) M is a generalized hypergeometric series introduced in (Kummer 1837), given by

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

where

a^{(0)}=1,
a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\,

is the rising factorial. Another common notation for this solution is Φ(a,b,z). Considered as a function of a, b, or z with the other two held constant, this defines an entire function of a or z, except when b = 0, −1, − 2, ... As a function of b it is analytic except for poles at the non-positive integers.

Some values of a and b yield solutions that can be expressed in terms of other known functions. See #Special cases. When a is a non-positive integer then Kummer's function (if it is defined) is a (generalized) Laguerre polynomial.

Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function

M(a,c,z) = \lim_{b\rightarrow\infty}{}_2F_1(a,b;c;z/b)

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. For this we can usually use the Tricomi confluent hypergeometric function U(a;b;z) introduced by Francesco Tricomi (1947), and sometimes denoted by Ψ(a;b;.z). The function U is defined in terms of Kummer's function M by

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).

This is undefined for integer b, but can be extended to integer b by continuity. Unlike Kummer's function which is an entire function of z, U(z) usually has a singularity at zero. But see #Special cases for some examples where it is an entire function (polynomial).

Note that if \Gamma(b-1)/\Gamma(a) is zero (which can occur if a is a non-positive integer), then U(a,b,z) and M(a,b,z) are not independent and another solution is needed. Also when b is a non-positive integer we need another solution because then M(a,b,z) is not defined. For instance, if a = 0 and b = 0, Kummer's function is undefined, but two independent solutions are w(z)=U(0,0,z)=1 and w(z)=\exp(z). For a = 0 but at other values of b, we have the two solutions:

U(0,b,z)=1
w(z)=\int_{-\infty}^zu^{-b}e^u\mathrm{d}u

When b = 1 this second solution is the exponential integral Ei(z).

See #Special cases for solutions to some other cases.

Other equations[edit]

Confluent hypergeometric functions can be used to solve "most" second-order differential equations in which the coefficients are all linear functions of z:

(A+Bz)\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0

First of all, a substitution of A+Bz with a new z converts the equation to:

z\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0

with new values of C, D, E, and F. (This step simply moves the regular singular point to 0.) If we then replace this z with 1/\sqrt{D^2-4F} times a new z, and multiply the equation by the same factor, we get:

z\frac{d^2w}{dz^2}+\left(C+\frac{D}{\sqrt{D^2-4F}}z\right)\frac{dw}{dz}+\left(\frac{E}{\sqrt{D^2-4F}}+\frac{F}{D^2-4F}z\right)w=0

whose solution is \exp[-(1+D/(D^2-4F)^{1/2})z/2]w(z), where w(z) is a solution to Kummer's equation with a=(1+D/(D^2-4F)^{1/2})C/2-E/\sqrt{D^2-4F} and b=C. Note that the square root may give an imaginary (or complex) number. If it is zero, another solution must be used, namely \exp(-Dz/2)w(z), where w(z) is a confluent hypergeometric limit function satisfying zw''(z)+Cw'(z)+(E-CD/2)w(z)=0.

As noted lower down, even the Bessel equation can be solved using confluent hypergeometric functions.

Integral representations[edit]

If Re b > Re a > 0, M(a,b,z) can be represented as an integral

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 .

thus M(a,a+b,it) is the characteristic function of the beta distribution. For a with positive real part U can be obtained by the Laplace integral

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)

The integral defines a solution in the right half-plane re z > 0.

They can also be represented as Barnes integrals

M(a,b,z) = \frac{1}{2\pi i}\frac{\Gamma(b)}{\Gamma(a)}\int_{-i\infty}^{i\infty} \frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)}(-z)^sds

where the contour passes to one side of the poles of Γ(−s) and to the other side of the poles of Γ(a+s).

Asymptotic behavior[edit]

If a solution to Kummer's equation is asymptotic to a power of z as z goes to infinity, then the power must be −a. This is in fact the case for Tricomi's solution U(a,b,z). Its asymptotic behavior as z → ∞ can be deduced from the integral representations. 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 → ∞:[1]

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 series (with 1 as leading term), which generally converges nowhere but exists as a formal power series in 1/x. This asymptotic expansion is also valid for complex z instead of real x, with |\arg z|<\tfrac 3 2 \pi.

The asymptotic behavior of Kummer's solution for large |z| is:

M(a,b,z)\sim\Gamma(b)\left(\frac{e^zz^{a-b}}{\Gamma(a)}+\frac{(-z)^{-a}}{\Gamma(b-a)}\right)

The powers of z are taken using -\tfrac 3 2\pi<\arg z\le\tfrac 1 2\pi.[2] The first term is only needed when Γ(b-a) is infinite (that is, when b-a is a non-positive integer) or when the real part of z is non-negative, whereas the second term is only needed when Γ(a) is infinite (that is, when a is a non-positive integer) or when the real part of z is non-positive.

There is always some solution to Kummer's equation asymptotic to e^zz^{a-b} as z goes to minus infinity. Usually this will be a combination of both M(a,b,z)\text{ and }U(a,b,z) but can also be expressed as e^z(-1)^{a-b}U(b-a,b,-z).

Relations[edit]

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

Contiguous relations[edit]

Given M(a, b; z), the four functions M(a ± 1, b, z), M(a, b ± 1; z) are called contiguous to M(a, b; z). The function M(a, b; z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, and z. This gives (4
2
)=6 relations, given by identifying any two lines on the right hand side of

\begin{align}
z\frac{dM}{dz} = z\frac{a}{b}M(a+,b+)
&=a(M(a+)-M)\\
&=(b-1)(M(b-)-M)\\
&=(b-a)M(a-)+(a-b+z)M\\
&=z(a-b)M(b+)/b +zM\\
\end{align}

In the notation above, M = M(a, b; z), M(a+) = M(a + 1, b; z), and so on.

Repeatedly applying these relations gives a linear relation between any three functions of the form M(a + m, b + n; z) (and their higher derivatives), where m, n are integers.

There are similar relations for U.

Kummer's transformation[edit]

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

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

Multiplication theorem[edit]

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}

Connection with Laguerre polynomials and similar representations[edit]

In terms of Laguerre polynomials, Kummer's functions have several expansions, for example

M\left(a,b,\frac{x y}{x-1}\right) = (1-x)^a \cdot \sum_n\frac{a^{(n)}}{b^{(n)}}L_n^{(b-1)}(y)x^n (Erdelyi 1953, 6.12)

Special cases[edit]

Functions that can be expressed as special cases of the confluent hypergeometric function include:

  • Some elementary functions (the left-hand side is not defined when b is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation):
M(0,b,z)=1
U(0,c,z)=1
M(b,b,z)=\exp(z)
U(a,a,z)=\exp(z)\int_z^\infty u^{-a}\exp(-u)du (a polynomial if a is a non-positive integer)
\frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}\exp(z)
U(a,a+1,z)= z^{-a}\,
U(-n,-2n,z) for integer n is a Bessel polynomial (see lower down).
M(n,b,z) for non-positive integer n is a generalized Laguerre polynomial.

For example, 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. Similarly

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),

When a is a non-positive integer, this equals 2^{-a}\theta_{-a}(x/2) where θ is a Bessel polynomial.

\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).
M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)
W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\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).

Application to continued fractions[edit]

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.

Notes[edit]

  1. ^ Andrews, G.E.; Askey, R.; Roy, R. (2001). Special functions. Cambridge University Press. ISBN 978-0521789882. .
  2. ^ This is derived from Abramowitz and Stegun (see reference below), page 508. They give a full asymptotic series. They switch the sign of the exponent in exp(iπa) in the right half-plane but this is unimportant because the term is negligible there or else a is an integer and the sign doesn't matter.

References[edit]

External links[edit]