Confluent hypergeometric function

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Kummer function, Pochhammer function

A solution of the confluent hypergeometric equation


The function may be defined using the so-called Kummer series


where and are parameters which assume any real or complex values except for and is a complex variable. The function is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),

is called the confluent hypergeometric function of the second kind.

The confluent hypergeometric function is an entire analytic function in the entire complex -plane; if is fixed, it is an entire function of and a meromorphic function of with simple poles at the points . The confluent hypergeometric function is an analytic function in the complex -plane with the slit and an entire function of and .

The confluent hypergeometric function is connected with the hypergeometric function by the relation

Elementary relationships. The four functions , are called adjacent (or contiguous) to the function . There is a linear relationship between and any two functions adjacent to it, e.g.

Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function with the associated functions , where and are integers.

Differentiation formulas:

Basic integral representations.

The asymptotic behaviour of confluent hypergeometric functions as can be studied using the integral representations [1], [2], [3]. If , while and are bounded, the behaviour of the function is described by formula (2). In particular, for large and bounded and :

Representations of functions by confluent hypergeometric functions.

Bessel functions:

Laguerre polynomials:

Probability integrals:

The exponential integral function:

The logarithmic integral function:


Elementary functions:

See also [1], [2], [3], [8].


[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1980) (Translated from Russian)
[3] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1964)
[4] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[5] A.L. Lebedev, R.M. Fedorova, "Handbook of mathematical tables" , Moscow (1956) (In Russian)
[6] N.M. Burunova, "Handbook of mathematical tables" , Moscow (1959) (In Russian)
[7] A.A. Fletcher, J.C.P. Miller, L. Rosenhead, L.J. Comrie, "An index of mathematical tables" , 1–2 , Oxford Univ. Press (1962)
[8] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
How to Cite This Entry:
Confluent hypergeometric function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.A. Chistova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article