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Heun's Differential Equation

A natural extension of the Riemann p-differential equation given by

(d^2w)/(dx^2)+(gamma/x+delta/(x-1)+epsilon/(x-a))(dw)/(dx)+(alphabetax-q)/(x(x-1)(x-a))w=0

where

alpha+beta-gamma-delta-epsilon+1=0.

See also

Heun Functions, Riemann P-Differential Equation

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References

Decarreau, A.; Dumont-Lepage, M.-C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes canoniques des équations confluentes de l'équation de Heun." Ann. Soc. Sci. de Bruxelles 92, 53-78, 1978.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 3. New York: Krieger, pp. 57-62, 1981.Heun, K. "Zur Theorie der Riemann'schen Functionen Zweiter Ordnung mit Verzweigungspunkten." Math. Ann. 33, 161-179, 1889.Ronveaux, A. (Ed.). Heun's Differential Equations. Oxford, England: Oxford University Press, 1995.Slavyanov, S. Yu. and Lay, W. "The Heun Class of Equations." Ch. 3 in Special Functions: A Unified Theory Based on Singularities. Oxford, England: Oxford University Press, pp. 97-162, 2000.Valent, G. "An Integral Transform Involving Heun Functions and a Related Eigenvalue Problem." SIAM J. Math. Anal. 17, 688-703, 1986.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 576, 1990.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 123, 1997.

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Heun's Differential Equation

Cite this as:

Weisstein, Eric W. "Heun's Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeunsDifferentialEquation.html

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