Confluent hypergeometric function
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 M(a,b,z), introduced by Kummer (1837), is a solution to Kummer's differential equation. There is a different and unrelated Kummer's function bearing the same name.
- Tricomi's (confluent hypergeometric) function U(a;b;z) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(a;b;.z), is another solution to Kummer's equation.
- Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation.
- Coulomb wave functions are solutions to the 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 Integral representations
- 3 Asymptotic behavior
- 4 Relations
- 5 Multiplication theorem
- 6 Connection with Laguerre polynomials and similar representations
- 7 Special cases
- 8 Application to continued fractions
- 9 Notes
- 10 References
- 11 External links
Kummer's equation is
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).
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
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
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 is zero (which can occur if a is a non-positive integer), then and are not independent and another solution is needed. Also when b is a non-positive integer we need another solution because then is not defined. For instance, if a = 0 and b = 0, Kummer's function is undefined, but two independent solutions are and For a = 0 but at other values of b, we have the two solutions:
When b = 1 this second solution is the exponential integral Ei(z).
See #Special cases for solutions to some other cases.
Confluent hypergeometric functions can be used to solve "most" second-order differential equations in which the coefficients are all linear functions of z:
First of all, a substitution of A+Bz with a new z converts the equation to:
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 times a new z, and multiply the equation by the same factor, we get:
whose solution is , where w(z) is a solution to Kummer's equation with and . Note that the square root may give an imaginary (or complex) number. If it is zero, another solution must be used, namely , where w(z) is a confluent hypergeometric limit function satisfying
As noted lower down, even the Bessel equation can be solved using confluent hypergeometric functions.
If Re b > Re a > 0, M(a,b,z) can be represented as an integral
The integral defines a solution in the right half-plane re z > 0.
They can also be represented as Barnes integrals
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).
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 → ∞:
where 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
The asymptotic behavior of Kummer's solution for large |z| is:
The powers of z are taken using . 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 as z goes to minus infinity. Usually this will be a combination of both and but can also be expressed as .
There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.
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
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 functions are also related by Kummer's transformations:
The following multiplication theorems hold true:
Connection with Laguerre polynomials and similar representations
In terms of Laguerre polynomials, Kummer's functions have several expansions, for example
- (Erdelyi 1953, 6.12)
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):
- (a polynomial if a is a non-positive integer)
- for integer n is a Bessel polynomial (see lower down).
- for non-positive integer n is a generalized Laguerre polynomial.
- Bateman's function
- Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions.
For example, the special case the function reduces to a Bessel function:
This identity is sometimes also referred to as Kummer's second transformation. Similarly
When is a non-positive integer, this equals where θ is a Bessel polynomial.
- The error function can be expressed as
- Coulomb wave function
- Cunningham functions
- Exponential integral and related functions such as the sine integral, logarithmic integral
- Hermite polynomials
- Incomplete gamma function
- Laguerre polynomials
- Parabolic cylinder function (or Weber function)
- Poisson–Charlier function
- Toronto functions
- Whittaker functions Mκ,μ(z), Wκ,μ(z) are solutions of Whittaker's equation that can be expressed in terms of Kummer functions M and U by
- The general p-th raw moment (p not necessarily an integer) can be expressed as
- (the function's second branch cut can be chosen by multiplying with ).
Application to continued fractions
By applying a limiting argument to Gauss's continued fraction it can be shown that
and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole.
- Andrews, G.E.; Askey, R.; Roy, R. (2001). Special functions. Cambridge University Press. ISBN 978-0521789882..
- 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.
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 13", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 504, ISBN 978-0486612720, MR 0167642.
- Chistova, E.A. (2001), "c/c024700", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Daalhuis, Adri B. Olde (2010), "Confluent hypergeometric function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions. Vol. I. New York–Toronto–London: McGraw–Hill Book Company, Inc. MR 0058756.
- Kummer, Ernst Eduard (1837). "De integralibus quibusdam definitis et seriebus infinitis". Journal für die reine und angewandte Mathematik (in Latin) 17: 228–242. doi:10.1515/crll.1837.17.228. ISSN 0075-4102.
- Slater, Lucy Joan (1960). Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press. MR 0107026.
- Tricomi, Francesco G. (1947). "Sulle funzioni ipergeometriche confluenti". Annali di Matematica Pura ed Applicata. Serie Quarta (in Italian) 26: 141–175. doi:10.1007/bf02415375. ISSN 0003-4622. MR 0029451.
- Tricomi, Francesco G. (1954). Funzioni ipergeometriche confluenti. Consiglio Nazionale Delle Ricerche Monografie Matematiche (in Italian) 1. Rome: Edizioni cremonese. ISBN 978-88-7083-449-9. MR 0076936.