The Solutions of the Mathieu Equation with a Complex Variable and at Least One Parameter Large
Rudolph E. Langer · Transactions of the American Mathematical Society · 1934
An analogous application of the method to a study of the Bessel functions with both the variable and the parameter complex was made by the author in the papers cited below.f These Transactions, as follows: On the asymptotic solutions of ordinary differential equations, etc., vol.33 (1931), p. 23; On the asymptotic solutions of differential equations, etc., vol.34 (1932), p.447;The asymptotic solutions of certain linear ordinary differential equations of the second order, vol.36, p. 90.These papers will be referred to in the text by the designations Li, Ls and L3.* Papers L2 and L3 cited above.In the formulas of paper L3 the variables X, xi, , and £ must be replaced by ip, ¿xi, 20 and 2i% respectively in order that they may appear as given here.* The hypothesis (iv) of papers L2 and L3 is not repeated here.It is obviously satisfied in every case of the present discussion.