§28.1 Special Notation

Keywords:: Mathieu functions, Mathieu’s equation, modified Mathieu functions, radial Mathieu functions
Referenced by:: §28.35(i)
Permalink:: http://dlmf.nist.gov/28.1

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

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

The main functions treated in this chapter are the Mathieu functions

$\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ ,

$\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ ,

$\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ ,

$\mathop{\mathrm{ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ ,

$\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ ,

and the modified Mathieu functions

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

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

The eigenvalues of Mathieu’s equation are denoted by

$\mathop{a_{{n}}\/}\nolimits\!\left(q\right),$

$\mathop{b_{{n}}\/}\nolimits\!\left(q\right),$

$\mathop{\lambda _{{\nu}}\/}\nolimits\!\left(q\right).$

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.

Table 28.1.1: Notations for parameters in Mathieu’s equation.

Reference	$a$	$q$
Erdélyi et al. (1955)	$h$	$\theta$
Meixner and Schäfke (1954)	$\lambda$	$h^{2}$
Moon and Spencer (1971)	$\lambda$	$q$
Strutt (1932)	$\lambda$	$h^{2}$
Whittaker and Watson (1927)	$a$	$8q$

Symbols:: $q=h^{2}$ : parameter, $h$ : parameter and $a$ : parameter
Referenced by:: §28.1
Permalink:: http://dlmf.nist.gov/28.1.T1

Alternative notations for the functions are as follows.

¶ Arscott (1964b) and McLachlan (1947)

$\mathrm{Fey}_{n}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{{\mathit{e},n}}(h)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)\mathop{{\mathrm{Mc}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z,h\right),$

$\mathrm{Me}_{n}^{{(1,2)}}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{{\mathit{e},n}}(h)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)\mathop{{\mathrm{Mc}^{{(3,4)}}_{{n}}}\/}\nolimits\!\left(z,h\right),$

$\mathrm{Gey}_{n}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{{\mathit{o},n}}(h){\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)\mathop{{\mathrm{Ms}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z,h\right),$

$\mathrm{Ne}_{n}^{{(1,2)}}(z,q)=\sqrt{\tfrac{1}{2}\pi}g_{{\mathit{o},n}}(h){\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)\mathop{{\mathrm{Ms}^{{(3,4)}}_{{n}}}\/}\nolimits\!\left(z,h\right).$

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

¶ Campbell (1955)

$\displaystyle\mathrm{in}_{n}=\mathop{\mathrm{fe}_{{n}}\/}\nolimits,$ $\displaystyle\mathrm{ceh}_{n}=\mathop{\mathrm{Ce}_{{n}}\/}\nolimits,$ $\displaystyle\mathrm{inh}_{n}=\mathop{\mathrm{Fe}_{{n}}\/}\nolimits,$

$\displaystyle\mathrm{jn}_{n}=\mathop{\mathrm{ge}_{{n}}\/}\nolimits,$ $\displaystyle\mathrm{seh}_{n}=\mathop{\mathrm{Se}_{{n}}\/}\nolimits,$ $\displaystyle\mathrm{jnh}_{n}=\mathop{\mathrm{Ge}_{{n}}\/}\nolimits.$

¶ Abramowitz and Stegun (1964, Chapter 20)

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

¶ National Bureau of Standards (1967)

With $s=4q$ ,

$\mathrm{Se}_{{n}}(s,z)=\dfrac{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)}{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)},$

$\mathrm{So}_{{n}}(s,z)=\dfrac{\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)}{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}.$

¶ Stratton et al. (1941)

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

$\mathrm{Se}_{{n}}(c,z)=\dfrac{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)}{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)},$

$\mathrm{So}_{{n}}(c,z)=\dfrac{\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)}{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}.$

¶ Zhang and Jin (1996)

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