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

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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)	$a$	$8q$

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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)

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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)

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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)

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

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

National Bureau of Standards (1967)

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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)

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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)

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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

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