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

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

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

§28.5(i) Definitions
§28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation

§28.5(i) Definitions

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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)

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Keywords:: Mathieu’s equation, antiperiodicity, definitions, normalization, reflection properties in , second solutions, values at
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 $\operatorname{fe}_{n}\left(z,q\right)=C_{n}(q)\left(z\operatorname{ce}_{n}% \left(z,q\right)+f_{n}(z,q)\right),$

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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, : integer, : 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 $\operatorname{ge}_{n}\left(z,q\right)=S_{n}(q)\left(z\operatorname{se}_{n}% \left(z,q\right)+g_{n}(z,q)\right),$

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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, : integer, : 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$-antiperiodic, odd},\end{array}$

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Defines:: $f_{n}(z,q)$ : functions (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $q=h^{2}$ : parameter, : integer and : 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$-periodic, even};\end{array}$

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Defines:: $g_{n}(z,q)$ : functions (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $q=h^{2}$ : parameter, : integer and : 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 $(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\,\mathrm{d}x=(S_{n}(q))^{2}\int_% {0}^{2\pi}(g_{n}(x,q))^{2}\,\mathrm{d}x=\pi.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $q=h^{2}$ : parameter, : integer, : 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

$C_{2m}(-q)=C_{2m}(q),$

$C_{2m+1}(-q)=S_{2m+1}(q),$

$S_{2m+2}(-q)=S_{2m+2}(q).$

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Symbols:: : 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 ,

28.5.7

$\operatorname{fe}_{0}\left(z,0\right)=z,$

$\operatorname{fe}_{n}\left(z,0\right)=\sin nz,$

$\operatorname{ge}_{n}\left(z,0\right)=\cos nz$ , $n=1,2,3,\dots$ ;

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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, : integer and : 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 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

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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 $\mathscr{W}\left\{\operatorname{ce}_{n},\operatorname{fe}_{n}\right\}=% \operatorname{ce}_{n}\left(0,q\right)\operatorname{fe}_{n}'\left(0,q\right),$

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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 : 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 $\mathscr{W}\left\{\operatorname{se}_{n},\operatorname{ge}_{n}\right\}=-% \operatorname{se}_{n}'\left(0,q\right)\operatorname{ge}_{n}\left(0,q\right).$

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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 : 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).

§28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation

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Keywords:: Mathieu’s equation, graphics, second solutions
Notes:: Figures 28.5.1–28.5.6 were produced at NIST.
Permalink:: http://dlmf.nist.gov/28.5.ii
See also:: Annotations for §28.5 and Ch.28

Odd Second Solutions

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See also:: Annotations for §28.5(ii), §28.5 and Ch.28

Figure 28.5.1: $\operatorname{fe}_{0}\left(x,0.5\right)$ for $0\leq x\leq 2\pi$ and (for comparison) $\operatorname{ce}_{0}\left(x,0.5\right)$ . Magnify

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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, $\pi$ : the ratio of the circumference of a circle to its diameter and : real variable
Referenced by:: §28.5(ii)
Permalink:: http://dlmf.nist.gov/28.5.F1
Encodings:: pdf, png
See also:: Annotations for §28.5(ii), §28.5(ii), §28.5 and Ch.28

Figure 28.5.2: $\operatorname{fe}_{0}\left(x,1\right)$ for $0\leq x\leq 2\pi$ and (for comparison) $\operatorname{ce}_{0}\left(x,1\right)$ . Magnify

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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, $\pi$ : the ratio of the circumference of a circle to its diameter and : real variable
Permalink:: http://dlmf.nist.gov/28.5.F2
Encodings:: pdf, png
See also:: Annotations for §28.5(ii), §28.5(ii), §28.5 and Ch.28

Figure 28.5.3: $\operatorname{fe}_{1}\left(x,0.5\right)$ for $0\leq x\leq 2\pi$ and (for comparison) $\operatorname{ce}_{1}\left(x,0.5\right)$ . Magnify

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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, $\pi$ : the ratio of the circumference of a circle to its diameter and : real variable
Permalink:: http://dlmf.nist.gov/28.5.F3
Encodings:: pdf, png
See also:: Annotations for §28.5(ii), §28.5(ii), §28.5 and Ch.28

Figure 28.5.4: $\operatorname{fe}_{1}\left(x,1\right)$ for $0\leq x\leq 2\pi$ and (for comparison) $\operatorname{ce}_{1}\left(x,1\right)$ . Magnify

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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, $\pi$ : the ratio of the circumference of a circle to its diameter and : real variable
Permalink:: http://dlmf.nist.gov/28.5.F4
Encodings:: pdf, png
See also:: Annotations for §28.5(ii), §28.5(ii), §28.5 and Ch.28

Even Second Solutions

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See also:: Annotations for §28.5(ii), §28.5 and Ch.28

Figure 28.5.5: $\operatorname{ge}_{1}\left(x,0.5\right)$ for $0\leq x\leq 2\pi$ and (for comparison) $\operatorname{se}_{1}\left(x,0.5\right)$ . Magnify

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