Any second-order Fuchsian equation with four singular points is equivalent to Heun’s equation
represented by the Riemann -symbol
is a so-called accessory parameter. Suppose . We consider four classes of boundary value problems on , being an eigenvalue.
I: (1), is smooth at 0, is smooth at 1.
IV: (1), is smooth at 0, is smooth at 1.
Let us consider, for example, class I. Let be the solution of (1) satisfying , and be the solution of (1) satisfying .
Problem. For an eigenvalue of the BVP I, estimate the norm
where , .
where , and is the Wronskian of and .
Note that and are independent of . These quantities are already used during the evaluation-algorithm of the eigenvalue . Thus the theorem tells that the evaluation of the norm can be obtained as a by-product of the search for the eigenvalues, and so that this formula greatly improves the efficiency of numerical procedures involving Heun functions.