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, the Tricomi Function, or the Gordon Function. It is denoted $U(a,b,z)$ and has an integral representation

$$U(a,b,z) ={1\over \Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt$$

(Abramowitz and Stegun 1972, p. 505). The Whittaker Functions give an alternative form of the solution. For small $z$, the function behaves as $z^{1-b}$.

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

References

Abramowitz, M. and Stegun, C. 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.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 671-672, 1953.

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.