Higher order terms of asymptotic expansions for spheroidal eigenvalues

The convergence of the spheroidal eigenvalues ${\lambda _{mn}}$ strongly depends upon their initial values. At moderate values of the focal parameter $c$ (which is a product of the wave number and the semi-focal distance of spheroid) and intermediate numbers $n\left ( {n > m} \right )$, the asymptotic expansions for ${\lambda _{mn}}$ of both prolate and oblate spheroidal differential equations require additional higher order terms so that a specific initial value of ${\lambda _{mn}}$ could be confined within the convergence circle and therefore rapidly reach an accurate value. The previously published results of the asymptotic expansions for both prolate and oblate spheroidal eigenvalues have the higher order terms up to ${c^{ - 5}}$. The author has added two additional higher order terms ${c^{ - 6}}$ and ${c^{ - 7}}$ to the asymptotic expansions, greatly improving the convergence of ${\lambda _{mn}}$ and speeding up the computational process. In addition, the higher order terms for the eigenvalues of the Mathieu differential equation may be readily obtained by using the transformation from the prolate spheroidal differential equation.