Confluent Hypergeometric Function of the Second Kind
The confluent hypergeometric function of the second kind gives the second linearly independent solution to the confluent
hypergeometric differential equation. It is also known as the Kummer's function
of the second kind, Tricomi function, or Gordon function. It is denoted 
and can
be defined by
 |  | ![picsc(pib)[(_1F^~_1(a;b;z))/(Gamma(a-b+1))-(z^(1-b)_1F^~_1(a-b+1;2-b;z))/(Gamma(a))]](/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/Inline4.gif) |
(1)
|
 |  |  |
(2)
|
where 
is
a regularized confluent
hypergeometric function of the first kind, 
is a gamma function, and 
is
a generalized hypergeometric function
(which converges nowhere but exists as a formal power series; Abramowitz and Stegun
1972, p. 504).
It has an integral representation
 |
(3)
|
for ![R[a],R[z]>0](/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/Inline11.gif)
(Abramowitz
and Stegun 1972, p. 505).
The confluent hypergeometric function of the second kind is implemented in the Wolfram Language as HypergeometricU[a, b, z].
The Whittaker functions give an alternative form of the solution.
The function has a Maclaurin series
 |
(4)
|
![U(a,b,z)∼(1/z)^a[1+a(b-a-1)z^(-1)
+1/2a(a+1)(a+b-1)(2+b-a)z^(-2)+...].](/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/NumberedEquation3.gif) |
(5)
|
has derivative
 |
(6)
|
 |
(7)
|
where 
is a Meijer G-function and 
is a constant
of integration.
RELATED WOLFRAM SITES: http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/
Abramowitz, M. and Stegun, I. A. (Eds.). "Confluent Hypergeometric Functions." Ch. 13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503-515, 1972.
Arfken, G. "Confluent Hypergeometric Functions." §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753-758, 1985.
Buchholz, H. The Confluent Hypergeometric Function with Special Emphasis on its Applications. New York: Springer-Verlag, 1969.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 671-672, 1953.
Slater, L. J. "The Second Form of Solutions of Kummer's Equations." §1.3 in Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, p. 5, 1960.
Spanier, J. and Oldham, K. B. "The Tricomi Function 
."
Ch. 48 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 471-477, 1987.
Weisstein, Eric W. "Confluent Hypergeometric Function of the Second Kind." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html
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