First kind !! Second kind |- | Integral | | |- | Integral | | |- | Fractional ( non-integral) | | |} Explicit representation and computation First kind Mathieu functions of the first kind can be represented as Fourier series: The expansion coefficients and are functions of but independent of . By substitution into the Mathieu equation, they can be shown to obey three-term recurrence relations in the lower index. For instance, for each one finds Being a second-order recurrence in the index , one can always find two independent solutions and such that the general solution can be expressed as a linear combination of the two: . Moreover, in this particular case, an asymptotic analysis shows that one possible choice of fundamental solutions has the property In particular, is finite whereas diverges. Writing , we therefore see that in order for the Fourier series representation of to converge, must be chosen such that These choices of correspond to the characteristic numbers. In general, however, the solution of a three-term recurrence with variable coefficients cannot be represented in a simple manner, and hence there is no simple way to determine from the condition . Moreover, even if the approximate value of a characteristic number is known, it cannot be used to obtain the coefficients by numerically iterating the recurrence towards increasing . The reason is that as long as only approximates a characteristic number, is not identically and the divergent solution eventually dominates for large enough . To overcome these issues, more sophisticated semi-analytical/numerical approaches are required, for instance using a continued fraction expansion, casting the recurrence as a matrix eigenvalue problem, or implementing a backwards recurrence algorithm. The complexity of the three-term recurrence relation is one of the reasons there are few simple formulas and identities involving Mathieu functions. In practice, Mathieu functions and the corresponding characteristic numbers can be calculated using pre-packaged software, such as Mathematica, Maple, MATLAB, and SciPy. For small values of and low order , they can also be expressed perturbatively as power series of , which can be useful in physical applications. Second kind There are several ways to represent Mathieu functions of the second kind. One representation is in terms of Bessel functions: where , and and are Bessel functions of the first and second kind. Modified functions A traditional approach for numerical evaluation of the modified Mathieu functions is through Bessel