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

## §28.5(i) Definitions

### Theorem of Ince (1922)

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 $(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\,\mathrm{d}x=(S_{n}(q))^{2}\int_% {0}^{2\pi}(g_{n}(x,q))^{2}\,\mathrm{d}x=\pi.$

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

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

 28.5.8 $\displaystyle\mathscr{W}\left\{\operatorname{ce}_{n},\operatorname{fe}_{n}\right\}$ $\displaystyle=\operatorname{ce}_{n}\left(0,q\right)\operatorname{fe}_{n}'\left% (0,q\right),$ 28.5.9 $\displaystyle\mathscr{W}\left\{\operatorname{se}_{n},\operatorname{ge}_{n}\right\}$ $\displaystyle=-\operatorname{se}_{n}'\left(0,q\right)\operatorname{ge}_{n}% \left(0,q\right).$

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).