Heun's equation

In mathematics, the Heun's differential equation is a second-order linear ordinary differential equation (ODE) of the form

:$$frac {d^2w}{dz^2} + left [frac{gamma}{z}+ frac{delta}{z-1} + frac{epsilon}{z-d} ight] frac {dw}{dz} + frac {alpha eta z -q} {z(z-1)(z-d)} w = 0.

(Note that $$epsilon=alpha+eta-gamma-delta+1 is needed to ensure regularity of the point at ∞.)Every second-order linear ODE in the complex plane (or on the Riemann sphere, to be more accurate) with four regular singular points can be transformed into this equation. It has four regular singular points: 0,1,"d" and ∞.

The equation is named after Karl L. W. M. Heun. ("Heun" rhymes with "loin.") This equation has 192 local solutions, analogous the the 24 local solutions of the hypergeometric differential equations obtained by Kummer.The solution that possesses a series expansion in the vicinity of the singular point is called Heun's function and is written $$Hl. In his original 1889 work, Heun obtained 48 of the local solutions, but some of the expressions are incorrect.

ee also

* Second-order ODE's with three regular singular points can always be transformed into the hypergeometric differential equation.

References

* Andrew Russell Forsyth [http://www.archive.org/details/theorydiffeq04forsrich Theory of Differential Equations (vol. 4)] (Cambridge University Press, 1906) p. 158
* Karl Heun [http://www.digizeitschriften.de/resolveppn/GDZPPN00225140X Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten] Mathematische Annalen 33 p. 161 (1899).
* A. Erdelyi, F. oberhettinger, W. Magnus and F. Tricomi "Higher Transcendental functions" v. 3 (MacGraw Hill, NY, 1953).
* G. Valent [http://arxiv.org/abs/math-ph/0512006 Heun functions versus elliptic functions]
* Robert S. Maier [http://arxiv.org/abs/math/0408317 The 192 Solutions of the Heun Equation] Math. Comp. 76 , 811-843 (2007).