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Confluent Hypergeometric Differential Equation

The second-order ordinary differential equation

xy^('')+(c-x)y^'-ay=0,

sometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It has a regular singular point at 0 and an irregular singularity at infty. The solutions

y=b_1_1F_1(a;c;x)+b_2U(a,c,x)

are called confluent hypergeometric function of the first and second kinds, respectively. Note that the confluent hypergeometric function of the first kind is also denoted M(a,c,x) or Phi(a;c;z).

See also

Bateman Function, Confluent Hypergeometric Function of the First Kind, Confluent Hypergeometric Function of the Second Kind, General Confluent Hypergeometric Differential Equation, Hypergeometric Differential Equation, Whittaker Differential Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 504, 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. 551-555, 1953.Slater, L. J. Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1960.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 123-124, 1997.

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Confluent Hypergeometric Differential Equation

Cite this as:

Weisstein, Eric W. "Confluent Hypergeometric Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConfluentHypergeometricDifferentialEquation.html

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