Closed-form expressions for Libby–Fox eigenvalues and norms

Develop exact closed-form formulas for the Libby–Fox eigenvalues Ak and normalization constants Ck associated with the Sturm–Liouville problem (equation (8)) and orthogonality relation (equation (9)) for perturbations to the Blasius boundary layer, beyond existing asymptotic approximations such as Brown’s formula (equation (12)).

Background

The eigenvalues and eigenfunctions governing perturbations to the Blasius boundary layer form an irregular Sturm–Liouville problem whose spectrum is known numerically and asymptotically. The authors derive constraints (e.g., equation (57)) but explicitly note the absence of closed-form expressions for the eigenvalues and norms, indicating a longstanding analytic gap.