28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.4 Fourier Series 28.6 Expansions for Small

q

§28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

ⓘ

Permalink:: http://dlmf.nist.gov/28.5
See also:: Annotations for Ch.28

§28.5(i) Definitions

ⓘ

Keywords:: Mathieu’s equation, Theorem of Ince, periodicity, second solutions
Notes:: See McLachlan (1947, pp. 141–155) and Meixner and Schäfke (1954, §2.72). For a proof of the Theorem of Ince see Arscott (1964b, §2.4).
Referenced by:: §28.2(iv), §28.22(i), §28.28(i)
Permalink:: http://dlmf.nist.gov/28.5.i
See also:: Annotations for §28.5 and Ch.28

Theorem of Ince (1922)

ⓘ

Keywords:: Mathieu’s equation, antiperiodicity, definitions, normalization, reflection properties in $q$ , second solutions, values at $q=0$
See also:: Annotations for §28.5(i), §28.5 and Ch.28

If a nontrivial solution of Mathieu’s equation with $q\neq 0$ has period $\pi$ or $2\pi$ , then any linearly independent solution cannot have either period.

Second solutions of (28.2.1) are given by

28.5.1		$\operatorname{fe}_{n}\left(z,q\right)=C_{n}(q)\left(z\operatorname{ce}_{n}% \left(z,q\right)+f_{n}(z,q)\right),$
ⓘ Defines: $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $f_{n}(z,q)$ : functions and $C_{n}(q)$ : factor A&S Ref: 20.3.6 (in different form) Referenced by: §28.5(i), §28.5(i) Permalink: http://dlmf.nist.gov/28.5.E1 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

when $a=a_{n}\left(q\right)$ , $n=0,1,2,\dots$ , and by

28.5.2		$\operatorname{ge}_{n}\left(z,q\right)=S_{n}(q)\left(z\operatorname{se}_{n}% \left(z,q\right)+g_{n}(z,q)\right),$
ⓘ Defines: $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $q=h^{2}$ : parameter, $n$ : integer, $z$ : complex variable, $g_{n}(z,q)$ : functions and $S_{n}(q)$ : factor A&S Ref: 20.3.7 (in different form) Referenced by: §28.5(i), §28.5(i) Permalink: http://dlmf.nist.gov/28.5.E2 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

when $a=b_{n}\left(q\right)$ , $n=1,2,3,\dots$ . For $m=0,1,2,\dots$ , we have

28.5.3		$\begin{array}[]{ll}f_{2m}(z,q)&\mbox{$\pi$-periodic, odd},\\ f_{2m+1}(z,q)&\mbox{$\pi$-antiperiodic, odd},\end{array}$
ⓘ Defines: $f_{n}(z,q)$ : functions (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.5.E3 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

and

28.5.4		$\begin{array}[]{ll}g_{2m+1}(z,q)&\mbox{$\pi$-antiperiodic, even},\\ g_{2m+2}(z,q)&\mbox{$\pi$-periodic, even};\end{array}$
ⓘ Defines: $g_{n}(z,q)$ : functions (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.5.E4 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

compare §28.2(vi). The functions $f_{n}(z,q)$ , $g_{n}(z,q)$ are unique.

The factors $C_{n}(q)$ and $S_{n}(q)$ in (28.5.1) and (28.5.2) are normalized so that

28.5.5		$(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\,\mathrm{d}x=(S_{n}(q))^{2}\int_% {0}^{2\pi}(g_{n}(x,q))^{2}\,\mathrm{d}x=\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer, $x$ : real variable, $f_{n}(z,q)$ : functions, $g_{n}(z,q)$ : functions, $C_{n}(q)$ : factor and $S_{n}(q)$ : factor Permalink: http://dlmf.nist.gov/28.5.E5 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

As $q\to 0$ with $n\neq 0$ , $C_{n}(q)\to 0$ , $S_{n}(q)\to 0$ , $C_{n}(q)f_{n}(z,q)\to\sin nz$ , and $S_{n}(q)g_{n}(z,q)\to\cos nz$ . This determines the signs of $C_{n}(q)$ and $S_{n}(q)$ . (Other normalizations for $C_{n}(q)$ and $S_{n}(q)$ can be found in the literature, but most formulas—including connection formulas—are unaffected since $\operatorname{fe}_{n}\left(z,q\right)/C_{n}(q)$ and $\operatorname{ge}_{n}\left(z,q\right)/S_{n}(q)$ are invariant.)

28.5.6	$\displaystyle C_{2m}(-q)$	$\displaystyle=C_{2m}(q),$
	$\displaystyle C_{2m+1}(-q)$	$\displaystyle=S_{2m+1}(q),$
	$\displaystyle S_{2m+2}(-q)$	$\displaystyle=S_{2m+2}(q).$
ⓘ Symbols: $m$ : integer, $q=h^{2}$ : parameter, $C_{n}(q)$ : factor and $S_{n}(q)$ : factor Permalink: http://dlmf.nist.gov/28.5.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

For $q=0$ ,

28.5.7		$\displaystyle\operatorname{fe}_{0}\left(z,0\right)$	$\displaystyle=z,$
		$\displaystyle\operatorname{fe}_{n}\left(z,0\right)$	$\displaystyle=\sin nz,$
		$\displaystyle\operatorname{ge}_{n}\left(z,0\right)$	$\displaystyle=\cos nz$ ,
	$n=1,2,3,\dots$ ;
ⓘ Symbols: $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/28.5.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

compare (28.2.29).

As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to\pm\infty$ on $\mathbb{R}$ .

Wronskians

ⓘ

Keywords:: Fourier series, Mathieu functions, Mathieu’s equation, Wronskians, expansions in Mathieu functions, second solutions
See also:: Annotations for §28.5(i), §28.5 and Ch.28

28.5.8	$\displaystyle\mathscr{W}\left\{\operatorname{ce}_{n},\operatorname{fe}_{n}\right\}$	$\displaystyle=\operatorname{ce}_{n}\left(0,q\right)\operatorname{fe}_{n}'\left% (0,q\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\mathscr{W}$ : Wronskian, $q=h^{2}$ : parameter and $n$ : integer Permalink: http://dlmf.nist.gov/28.5.E8 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28
28.5.9	$\displaystyle\mathscr{W}\left\{\operatorname{se}_{n},\operatorname{ge}_{n}\right\}$	$\displaystyle=-\operatorname{se}_{n}'\left(0,q\right)\operatorname{ge}_{n}% \left(0,q\right).$
ⓘ Symbols: $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathscr{W}$ : Wronskian, $q=h^{2}$ : parameter and $n$ : integer Permalink: http://dlmf.nist.gov/28.5.E9 Encodings: TeX, pMML, png See also: Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

See (28.22.12) for $\operatorname{fe}_{n}'\left(0,q\right)$ and $\operatorname{ge}_{n}\left(0,q\right)$ .

For further information on $C_{n}(q)$ , $S_{n}(q)$ , and expansions of $f_{n}(z,q)$ , $g_{n}(z,q)$ in Fourier series or in series of $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72).