function product series. For large and , the form of the series must be chosen carefully to avoid subtraction errors. Properties There are relatively few analytic expressions and identities involving Mathieu functions. Moreover, unlike many other special functions, the solutions of Mathieu's equation cannot in general be expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable : Since this equation has an irregular singular point at infinity, it cannot be transformed into an equation of the hypergeometric type. Qualitative behavior For small , and behave similarly to and . For arbitrary , they may deviate significantly from their trigonometric counterparts; however, they remain periodic in general. Moreover, for any real , and have exactly simple zeros in , and as the zeros cluster about . For and as the modified Mathieu functions tend to behave as damped periodic functions. In the following, the and factors from the Fourier expansions for and may be referenced (see Explicit representation and computation). They depend on and but are independent of . Reflections and translations Due to their parity and periodicity, and have simple properties under reflections and translations by multiples of : One can also write functions with negative in terms of those with positive : Moreover, Orthogonality and completeness Like their trigonometric counterparts and , the periodic Mathieu functions and satisfy orthogonality relations Moreover, with fixed and treated as the eigenvalue, the Mathieu equation is of Sturm–Liouville form. This implies that the eigenfunctions and form a complete set, i.e. any - or -periodic function of can be expanded as a series in and . Integral identities Solutions of Mathieu's equation satisfy a class of integral identities with respect to kernels that are solutions of More precisely, if solves Mathieu's equation with given and , then the integral where is a path in the complex plane, also solves Mathieu's equation with the same and , provided the following conditions are met: solves In the regions under consideration, exists and is analytic has the same value at the endpoints of Using an appropriate change of variables, the equation for can be transformed into the wave equation and solved. For instance, one solution is . Examples of identities obtained in this way are Identities of the latter type are useful for studying asymptotic properties of the modified Mathieu functions. There also