# §28.1 Special Notation

(For other notation see Notation for the Special Functions.)

 integers. real variables. complex variable. order of the Mathieu function or modified Mathieu function. (When is an integer it is often replaced by .) arbitrary small positive number. real or complex parameters of Mathieu’s equation with . unless indicated otherwise, derivatives with respect to the argument

The main functions treated in this chapter are the Mathieu functions

 , , , , ,

and the modified Mathieu functions

 , , , , , , , , , , , .

The functions and are also known as the radial Mathieu functions.

The eigenvalues of Mathieu’s equation are denoted by

The notation for the joining factors is

Alternative notations for the parameters and are shown in Table 28.1.1.

Table 28.1.1: Notations for parameters in Mathieu’s equation.
Reference
Erdélyi et al. (1955)
Meixner and Schäfke (1954)
Moon and Spencer (1971)
Strutt (1932)
Whittaker and Watson (1927)

Alternative notations for the functions are as follows.

## ¶ Arscott (1964b) and McLachlan (1947)

Arscott (1964b) also uses for .

With ,

With ,

## ¶ Zhang and Jin (1996)

The radial functions and are denoted by and , respectively.