# §31.4 Solutions Analytic at Two Singularities: Heun Functions

For an infinite set of discrete values , , of the accessory parameter , the function is analytic at , and hence also throughout the disk . To emphasize this property this set of functions is denoted by

The eigenvalues satisfy the continued-fraction equation

in which are as in §31.3(i).

More generally,

with , denotes a set of solutions of (31.2.1), each of which is analytic at and . The set depends on the choice of and .

The solutions (31.4.3) are called the Heun functions. See Ronveaux (1995, pp. 39–41).