An equivalent statement of Floquet's theorem is that Mathieu's equation admits a complex-valued solution of form
These classifications are summarized in the table below. The modified Mathieu function counterparts are defined similarly.
The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.
Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below.