1303.0876 (Yoon Seok Choun)

Asymptotic behavior of Heun function and its integral formalism    [PDF]

Yoon Seok Choun

We consider an integral formalism and asymptotic behavior of Heun function including all higher terms of $A_n$'s; applying three term recurrence formula by Choun. We show how to transform power series expansion of Heun function to an integral formalism mathematically in an elegant way for cases of infinite series and polynomial. The Heun functions generalize the hypergeometric function and also include the Lame function, Mathieu function and the spheroidal wave functions, etc. This function is the mother of all well-known special functions in $21^{th}$ century. According to Whittaker's hypothesis, `The Heun functions are the simplest class of special functions for which no representations in form of contour integrals of elementary functions exists.' However, by using the three-term recurrence formula, we can have exact analytic representations in form of integrals of Heun function. And we show that integral form of Heun function has $_2F_1$ function in itself surprisingly.

View original: http://arxiv.org/abs/1303.0876