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 👁 U(a,b,z)
and can be defined by
| 👁 U(a,b,z) | 👁 = | 👁 picsc(pib)[(_1F^~_1(a;b;z))/(Gamma(a-b+1))-(z^(1-b)_1F^~_1(a-b+1;2-b;z))/(Gamma(a))] |
(1)
|
| 👁 Image | 👁 = | 👁 z^(-a)_2F_0(a,1+a-b;;-z^(-1)), |
(2)
|
![👁 picsc(pib)[(_1F^~_1(a;b;z))/(Gamma(a-b+1))-(z^(1-b)_1F^~_1(a-b+1;2-b;z))/(Gamma(a))]](https://mathworld.wolfram.com/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/Inline4.svg){kind=link}
where 👁 _1F^~_1(a;b;z)
is a regularized confluent
hypergeometric function of the first kind, 👁 Gamma(z)
is a gamma function,
and 👁 _2F_0(a,b;;z)
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
for 👁 R[a],R[z]>0
(Abramowitz and Stegun 1972, p. 505).
![👁 R[a],R[z]>0](https://mathworld.wolfram.com/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/Inline11.svg){kind=link}
The confluent hypergeometric function of the second kind is implemented in the Wolfram Language as [a, b, z].
The Whittaker functions give an alternative form of the solution.
The function has a Maclaurin series
![👁 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)+...].](https://mathworld.wolfram.com/images/equations/ConfluentHypergeometricFunctionoftheSecondKind/NumberedEquation3.svg){kind=link}
👁 U(a,b,z)
has derivative
where 👁 G_(p,q)^(m,n)(x|a_1,...,a_p; b_1,...,b_q)
is a Meijer
G-function and 👁 C
is a constant of integration.
See also
Bateman Function, Confluent Hypergeometric Function of the First Kind, Confluent Hypergeometric Limit Function, Coulomb Wave Function, Cunningham Function, Gordon Function, Hypergeometric Function, Poisson-Charlier Polynomial, Toronto Function, Weber Functions, Whittaker FunctionRelated Wolfram sites
http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/Explore with Wolfram|Alpha
References
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 👁 U(a;c;x)
." Ch. 48 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 471-477, 1987.
Referenced on Wolfram|Alpha
Confluent Hypergeometric Function of the Second KindCite this as:
Weisstein, Eric W. "Confluent Hypergeometric Function of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html