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Confluent Hypergeometric Function of the Second Kind

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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)
=z^(-a)_2F_0(a,1+a-b;;-z^(-1)),
(2)

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

U(a,b,z)=1/(Gamma(a))int_0^inftye^(-zt)t^(a-1)(1+t)^(b-a-1)dt
(3)

for R[a],R[z]>0 (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

U(a,b,z)=-((b+az)Gamma(-b))/(Gamma(1+a-b))+(z^(1-b)Gamma(b-1))/(Gamma(a))+...,
(4)

and asymptotic 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)+...].
(5)

U(a,b,z) has derivative

d/(dz)U(a,b,z)=-aU(a+1,b+1,z)
(6)

and indefinite integral

intU(a,b,z)dz=(G_(2,3)^(2,2)(x|1,2-a; 1,2-b,0))/(Gamma(a)Gamma(a-b+1))+C,
(7)

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 Function

Related Wolfram sites

https://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/

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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 Kind

Cite this as:

Weisstein, Eric W. "Confluent Hypergeometric Function of the Second Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheSecondKind.html

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