the Mathieu differential equation can be deduced from the general theory of ordinary differential equations with periodic coefficients, called Floquet theory. The central result is Floquet's theorem: It is natural to associate the characteristic numbers with those values of which result in . Note, however, that the theorem only guarantees the existence of at least one solution satisfying , when Mathieu's equation in fact has two independent solutions for any given , . Indeed, it turns out that with equal to one of the characteristic numbers, Mathieu's equation has only one periodic solution (that is, with period or ), and this solution is one of the , . The other solution is nonperiodic, denoted and , respectively, and referred to as a Mathieu function of the second kind. This result can be formally stated as Ince's theorem: An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form where is a complex number, the Floquet exponent (or sometimes Mathieu exponent), and is a complex valued function periodic in with period . An example is plotted to the right. Other types of Mathieu functions Second kind Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic. The periodic solution is one of the and , called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either and , respectively, and is called a Mathieu function of the second kind (of integral order). The nonperiodic solutions are unstable, that is, they diverge as . The second solutions corresponding to the modified Mathieu functions and are naturally defined as and . Fractional order Mathieu functions of fractional order can be defined as those solutions and , a non-integer, which turn into and as . If is irrational, they are non-periodic; however, they remain bounded as . An important property of the solutions and , for non-integer, is that they exist for the same value of . In contrast, when is an integer, and never occur for the same value of . (See Ince's Theorem above.) These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly. {| class="wikitable" |+Classification of Mathieu functions |- ! Order !!