# §28.1 Special Notation

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

$m,n$ integers. real variables. complex variable. order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.) arbitrary small positive number. real or complex parameters of Mathieu’s equation with $q=h^{2}$. unless indicated otherwise, derivatives with respect to the argument

The main functions treated in this chapter are the Mathieu functions

 $\mathrm{ce}_{\nu}\left(z,q\right)$, $\mathrm{se}_{\nu}\left(z,q\right)$, $\mathrm{fe}_{n}\left(z,q\right)$, $\mathrm{ge}_{n}\left(z,q\right)$, $\mathrm{me}_{\nu}\left(z,q\right)$,

and the modified Mathieu functions

 $\mathrm{Ce}_{\nu}\left(z,q\right)$, $\mathrm{Se}_{\nu}\left(z,q\right)$, $\mathrm{Fe}_{n}\left(z,q\right)$, $\mathrm{Ge}_{n}\left(z,q\right)$, $\mathrm{Me}_{\nu}\left(z,q\right)$, ${\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)$, ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$, ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$, $\mathrm{Ie}_{n}\left(z,h\right)$, $\mathrm{Io}_{n}\left(z,h\right)$, $\mathrm{Ke}_{n}\left(z,h\right)$, $\mathrm{Ko}_{n}\left(z,h\right)$.

The functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are also known as the radial Mathieu functions.

The eigenvalues of Mathieu’s equation are denoted by

 $a_{n}\left(q\right),$ $b_{n}\left(q\right),$ $\lambda_{\nu}\left(q\right).$

The notation for the joining factors is

 $g_{\mathit{e},n}(h),$ $g_{\mathit{o},n}(h),$ $f_{\mathit{e},n}(h),$ $f_{\mathit{o},n}(h).$

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

Alternative notations for the functions are as follows.

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

 $\displaystyle\mathrm{Fey}_{n}(z,q)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{e},n}(h)\mathrm{ce}_{n}\left(0,% q\right){\mathrm{Mc}^{(2)}_{n}}\left(z,h\right),$ $\displaystyle\mathrm{Me}_{n}^{(1,2)}(z,q)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{e},n}(h)\mathrm{ce}_{n}\left(0,% q\right){\mathrm{Mc}^{(3,4)}_{n}}\left(z,h\right),$ $\displaystyle\mathrm{Gey}_{n}(z,q)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{o},n}(h)\mathrm{se}_{n}'\left(0% ,q\right){\mathrm{Ms}^{(2)}_{n}}\left(z,h\right),$ $\displaystyle\mathrm{Ne}_{n}^{(1,2)}(z,q)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi}g_{\mathit{o},n}(h)\mathrm{se}_{n}'\left(0% ,q\right){\mathrm{Ms}^{(3,4)}_{n}}\left(z,h\right).$

Arscott (1964b) also uses $-\mathrm{i}\mu$ for $\nu$.

## Campbell (1955)

 $\displaystyle\mathrm{in}_{n}$ $\displaystyle=\mathrm{fe}_{n},$ $\displaystyle\mathrm{ceh}_{n}$ $\displaystyle=\mathrm{Ce}_{n},$ $\displaystyle\mathrm{inh}_{n}$ $\displaystyle=\mathrm{Fe}_{n},$ $\displaystyle\mathrm{jn}_{n}$ $\displaystyle=\mathrm{ge}_{n},$ $\displaystyle\mathrm{seh}_{n}$ $\displaystyle=\mathrm{Se}_{n},$ $\displaystyle\mathrm{jnh}_{n}$ $\displaystyle=\mathrm{Ge}_{n}.$

## Abramowitz and Stegun (1964, Chapter 20)

 $F_{\nu}(z)=\mathrm{Me}_{\nu}\left(z,q\right).$

## National Bureau of Standards (1967)

With $s=4q$,

 $\displaystyle\mathrm{Se}_{n}(s,z)$ $\displaystyle=\dfrac{\mathrm{ce}_{n}\left(z,q\right)}{\mathrm{ce}_{n}\left(0,q% \right)},$ $\displaystyle\mathrm{So}_{n}(s,z)$ $\displaystyle=\dfrac{\mathrm{se}_{n}\left(z,q\right)}{\mathrm{se}_{n}'\left(0,% q\right)}.$

## Stratton et al. (1941)

With $c=2\sqrt{q}$,

 $\displaystyle\mathrm{Se}_{n}(c,z)$ $\displaystyle=\dfrac{\mathrm{ce}_{n}\left(z,q\right)}{\mathrm{ce}_{n}\left(0,q% \right)},$ $\displaystyle\mathrm{So}_{n}(c,z)$ $\displaystyle=\dfrac{\mathrm{se}_{n}\left(z,q\right)}{\mathrm{se}_{n}'\left(0,% q\right)}.$

## Zhang and Jin (1996)

The radial functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are denoted by ${\mathrm{Mc}^{(j)}_{n}}\left(z,q\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,q\right)$, respectively.