Connection formulas between the solutions of Mathieu’s equation

The problem of connecting the various types of solutions of Mathieu’s equation is solved by the introduction of a new parameter $\Phi$ which is a function of the the two equation parameters $a$ and $q$. This quantity $\Phi$ is introduced and enclosed between two very close analytic limits in section 2. In sections 3, 4, 5 precise definitions are given and information is collected for the three main types of functions which are to be connected. Section 6 contains the connection formulas. Section 7 reviews the status of knowledge achieved. Section 8 is an appendix on integral equations which are more general than those developed earlier in the text, but which appear to be of no use for the main purpose of this paper.