Characteristic exponents of Mathieu functions

where P(z) is a periodic function of z, and v is the so-called characteristic exponent. This exponent is a function of the parameters p and q only. When v is real, the solutions (2) are called since they are uniformly bounded for all real z; then v = f(p, q) may be chosen to be single-valued. (Note: In the tables and curves shown here, v is not reduced to 0 _ v < 1 as is usually done, but its actual value is taken, thus defining the particular stable region it represents.) It is shown in the references that all values of v are solutions of the continued fraction equation