28 Mathieu Functions and Hill’s Equation Notation 28 Mathieu Functions and Hill’s Equation 28.2 Definitions and Basic Properties

§28.1 Special Notation

ⓘ

Keywords:: Mathieu functions, Mathieu’s equation, definition, eigenvalues (or characteristic values), joining factors, modified Mathieu functions, notation, parameters, radial Mathieu functions, second solutions
Referenced by:: 7th item
Permalink:: http://dlmf.nist.gov/28.1
See also:: Annotations for Ch.28

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

$m, n$	integers.
$x, y$	real variables.
$z=x+\mathrm{i}y$	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

\operatorname{ce}_{\nu}\left(z,q\right)

\operatorname{se}_{\nu}\left(z,q\right)

\operatorname{fe}_{n}\left(z,q\right)

\operatorname{ge}_{n}\left(z,q\right)

\operatorname{me}_{\nu}\left(z,q\right)

and the modified Mathieu functions

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

The functions ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$ 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.

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
See also:: Annotations for §28.1 and Ch.28

Alternative notations for the functions are as follows.

Arscott (1964b) and McLachlan (1947)

ⓘ

See also:: Annotations for §28.1 and Ch.28

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

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

Campbell (1955)

ⓘ

See also:: Annotations for §28.1 and Ch.28

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

Abramowitz and Stegun (1964, Chapter 20)

ⓘ

See also:: Annotations for §28.1 and Ch.28

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

National Bureau of Standards (1967)

ⓘ

See also:: Annotations for §28.1 and Ch.28

With $s=4q$ ,

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

Stratton et al. (1941)

ⓘ

See also:: Annotations for §28.1 and Ch.28

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

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

Zhang and Jin (1996)

ⓘ

See also:: Annotations for §28.1 and Ch.28

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

28 Mathieu Functions and Hill’s Equation 28.2 Definitions and Basic Properties

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov