Difference between revisions of "Confluent hypergeometric function"
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− | The logarithmic integral function: | + | The [[logarithmic integral]] function: |
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Revision as of 10:18, 23 December 2014
Kummer function, Pochhammer function
A solution of the confluent hypergeometric equation
![]() | (1) |
The function may be defined using the so-called Kummer series
![]() | (2) |
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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),
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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
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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.
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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:
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Basic integral representations.
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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
:
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Representations of functions by confluent hypergeometric functions.
Bessel functions:
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Laguerre polynomials:
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Probability integrals:
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The exponential integral function:
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The logarithmic integral function:
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Gamma-functions:
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Elementary functions:
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References
[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) |
Confluent hypergeometric function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Confluent_hypergeometric_function&oldid=17182