§28.4 Fourier Series

§28.4(i) Definitions

The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane. For $n=0,1,2,3,\dots$,

 28.4.1 $\displaystyle\operatorname{ce}_{2n}\left(z,q\right)$ $\displaystyle=\sum_{m=0}^{\infty}A^{2n}_{2m}(q)\cos 2mz,$ ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\cos\NVar{z}$: cosine function, $m$: integer, $q=h^{2}$: parameter, $n$: integer, $z$: complex variable and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.2.3 (in slightly different form) 20.2.4 (in slightly different form) Permalink: http://dlmf.nist.gov/28.4.E1 Encodings: TeX, pMML, png See also: Annotations for §28.4(i), §28.4 and Ch.28 28.4.2 $\displaystyle\operatorname{ce}_{2n+1}\left(z,q\right)$ $\displaystyle=\sum_{m=0}^{\infty}A^{2n+1}_{2m+1}(q)\cos(2m+1)z,$ 28.4.3 $\displaystyle\operatorname{se}_{2n+1}\left(z,q\right)$ $\displaystyle=\sum_{m=0}^{\infty}B^{2n+1}_{2m+1}(q)\sin(2m+1)z,$ 28.4.4 $\displaystyle\operatorname{se}_{2n+2}\left(z,q\right)$ $\displaystyle=\sum_{m=0}^{\infty}B^{2n+2}_{2m+2}(q)\sin(2m+2)z.$

§28.4(ii) Recurrence Relations

 28.4.5 $\displaystyle aA_{0}-qA_{2}$ $\displaystyle=0,$ $\displaystyle(a-4)A_{2}-q(2A_{0}+A_{4})$ $\displaystyle=0,$ $\displaystyle(a-4m^{2})A_{2m}-q(A_{2m-2}+A_{2m+2})$ $\displaystyle=0$, $m=2,3,4,\dots$, $a=a_{2n}\left(q\right)$, $A_{2m}=A_{2m}^{2n}(q)$. ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^{2}$: parameter, $n$: integer, $a$: parameter and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.2.5 20.2.6 20.2.7 Referenced by: item (d), item (d) Permalink: http://dlmf.nist.gov/28.4.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §28.4(ii), §28.4 and Ch.28
 28.4.6 $\displaystyle(a-1-q)A_{1}-qA_{3}$ $\displaystyle=0,$ $\displaystyle\left(a-(2m+1)^{2}\right)A_{2m+1}-q(A_{2m-1}+A_{2m+3})$ $\displaystyle=0$, $m=1,2,3,\dots$, $a=a_{2n+1}\left(q\right)$, $A_{2m+1}=A_{2m+1}^{2n+1}(q)$. ⓘ Symbols: $a_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^{2}$: parameter, $n$: integer, $a$: parameter and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.2.8 Permalink: http://dlmf.nist.gov/28.4.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(ii), §28.4 and Ch.28
 28.4.7 $\displaystyle(a-1+q)B_{1}-qB_{3}$ $\displaystyle=0,$ $\displaystyle\left(a-(2m+1)^{2}\right)B_{2m+1}-q(B_{2m-1}+B_{2m+3})$ $\displaystyle=0$, $m=1,2,3,\dots$, $a=b_{2n+1}\left(q\right)$, $B_{2m+1}=B_{2m+1}^{2n+1}(q)$. ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^{2}$: parameter, $n$: integer, $a$: parameter and $B_{m}(q)$: Fourier coefficient A&S Ref: 20.2.11 Permalink: http://dlmf.nist.gov/28.4.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(ii), §28.4 and Ch.28
 28.4.8 $\displaystyle(a-4)B_{2}-qB_{4}$ $\displaystyle=0,$ $\displaystyle(a-4m^{2})B_{2m}-q(B_{2m-2}+B_{2m+2})$ $\displaystyle=0$, $m=2,3,4,\dots$, $a=b_{2n+2}\left(q\right)$, $B_{2m+2}=B_{2m+2}^{2n+2}(q).$ ⓘ Symbols: $b_{\NVar{n}}\left(\NVar{q}\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^{2}$: parameter, $n$: integer, $a$: parameter and $B_{m}(q)$: Fourier coefficient A&S Ref: 20.2.9 20.2.10 Referenced by: item (d), item (d) Permalink: http://dlmf.nist.gov/28.4.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(ii), §28.4 and Ch.28

§28.4(iii) Normalization

 28.4.9 $2\left(A^{2n}_{0}(q)\right)^{2}+\sum_{m=1}^{\infty}\left(A^{2n}_{2m}(q)\right)% ^{2}=1,$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.5.4 Referenced by: item (d) Permalink: http://dlmf.nist.gov/28.4.E9 Encodings: TeX, pMML, png See also: Annotations for §28.4(iii), §28.4 and Ch.28
 28.4.10 $\displaystyle\sum_{m=0}^{\infty}\left(A^{2n+1}_{2m+1}(q)\right)^{2}$ $\displaystyle=1,$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $A_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E10 Encodings: TeX, pMML, png See also: Annotations for §28.4(iii), §28.4 and Ch.28 28.4.11 $\displaystyle\sum_{m=0}^{\infty}\left(B^{2n+1}_{2m+1}(q)\right)^{2}$ $\displaystyle=1,$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $B_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E11 Encodings: TeX, pMML, png See also: Annotations for §28.4(iii), §28.4 and Ch.28 28.4.12 $\displaystyle\sum_{m=0}^{\infty}\left(B^{2n+2}_{2m+2}(q)\right)^{2}$ $\displaystyle=1.$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $B_{m}(q)$: Fourier coefficient Referenced by: item (d) Permalink: http://dlmf.nist.gov/28.4.E12 Encodings: TeX, pMML, png See also: Annotations for §28.4(iii), §28.4 and Ch.28

Ambiguities in sign are resolved by (28.4.13)–(28.4.16) when $q=0$, and by continuity for the other values of $q$.

§28.4(iv) Case $q=0$

 28.4.13 $\displaystyle A^{0}_{0}(0)$ $\displaystyle=1/\sqrt{2},\quad A^{2n}_{2n}(0)=1,$ $n>0$, $\displaystyle A^{2n}_{2m}(0)$ $\displaystyle=0,$ $n\neq m$, ⓘ Symbols: $m$: integer, $n$: integer and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.2.29 Referenced by: §28.4(iii) Permalink: http://dlmf.nist.gov/28.4.E13 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(iv), §28.4 and Ch.28
 28.4.14 $\displaystyle A^{2n+1}_{2n+1}(0)$ $\displaystyle=1,$ $\displaystyle A^{2n+1}_{2m+1}(0)$ $\displaystyle=0,$ $n\neq m$, ⓘ Symbols: $m$: integer, $n$: integer and $A_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(iv), §28.4 and Ch.28
 28.4.15 $\displaystyle B^{2n+1}_{2n+1}(0)$ $\displaystyle=1,$ $\displaystyle B^{2n+1}_{2m+1}(0)$ $\displaystyle=0,$ $n\neq m$, ⓘ Symbols: $m$: integer, $n$: integer and $B_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(iv), §28.4 and Ch.28
 28.4.16 $\displaystyle B^{2n+2}_{2n+2}(0)$ $\displaystyle=1,$ $\displaystyle B^{2n+2}_{2m+2}(0)$ $\displaystyle=0,$ $n\neq m$. ⓘ Symbols: $m$: integer, $n$: integer and $B_{m}(q)$: Fourier coefficient Referenced by: §28.4(iii) Permalink: http://dlmf.nist.gov/28.4.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.4(iv), §28.4 and Ch.28

§28.4(v) Change of Sign of $q$

 28.4.17 $\displaystyle A^{2n}_{2m}(-q)$ $\displaystyle=(-1)^{n-m}A^{2n}_{2m}(q),$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.8.5 Permalink: http://dlmf.nist.gov/28.4.E17 Encodings: TeX, pMML, png See also: Annotations for §28.4(v), §28.4 and Ch.28 28.4.18 $\displaystyle B^{2n+2}_{2m+2}(-q)$ $\displaystyle=(-1)^{n-m}B^{2n+2}_{2m+2}(q),$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer and $B_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E18 Encodings: TeX, pMML, png See also: Annotations for §28.4(v), §28.4 and Ch.28 28.4.19 $\displaystyle A^{2n+1}_{2m+1}(-q)$ $\displaystyle=(-1)^{n-m}B^{2n+1}_{2m+1}(q),$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer, $A_{m}(q)$: Fourier coefficient and $B_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E19 Encodings: TeX, pMML, png See also: Annotations for §28.4(v), §28.4 and Ch.28 28.4.20 $\displaystyle B^{2n+1}_{2m+1}(-q)$ $\displaystyle=(-1)^{n-m}A^{2n+1}_{2m+1}(q).$ ⓘ Symbols: $m$: integer, $q=h^{2}$: parameter, $n$: integer, $A_{m}(q)$: Fourier coefficient and $B_{m}(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E20 Encodings: TeX, pMML, png See also: Annotations for §28.4(v), §28.4 and Ch.28

§28.4(vi) Behavior for Small $q$

For fixed $s=1,2,3,\dots$ and fixed $m=1,2,3,\dots$,

 28.4.21 $A^{0}_{2s}(q)=\left(\dfrac{(-1)^{s}2}{(s!)^{2}}\left(\frac{q}{4}\right)^{s}+O% \left(q^{s+2}\right)\right)A^{0}_{0}(q),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $!$: factorial (as in $n!$), $q=h^{2}$: parameter and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.2.29 (in slightly different form) Permalink: http://dlmf.nist.gov/28.4.E21 Encodings: TeX, pMML, png See also: Annotations for §28.4(vi), §28.4 and Ch.28
 28.4.22 $\rselection{A^{m}_{m+2s}(q)\\ B^{m}_{m+2s}(q)}=\left(\dfrac{(-1)^{s}m!}{s!(m+s)!}\left(\frac{q}{4}\right)^{s% }+O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q),}$
 28.4.23 $\rselection{A^{m}_{m-2s}(q)\\ B^{m}_{m-2s}(q)}=\left(\dfrac{(m-s-1)!}{s!(m-1)!}\left(\frac{q}{4}\right)^{s}+% O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q).}$

For further terms and expansions see Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33).

§28.4(vii) Asymptotic Forms for Large $m$

As $m\to\infty$, with fixed $q$ ($\neq 0$) and fixed $n$,

 28.4.24 $\displaystyle\frac{A^{2n}_{2m}(q)}{A^{2n}_{0}(q)}$ $\displaystyle=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{\pi% \left(1+O\left(m^{-1}\right)\right)}{w_{\mbox{\tiny II}}(\frac{1}{2}\pi;a_{2n}% \left(q\right),q)},$ 28.4.25 $\displaystyle\frac{A^{2n+1}_{2m+1}(q)}{A^{2n+1}_{1}(q)}$ $\displaystyle=\frac{(-1)^{m+1}}{\left({\left(\tfrac{1}{2}\right)_{m+1}}\right)% ^{2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}% {w^{\prime}_{\mbox{\tiny II}}(\frac{1}{2}\pi;a_{2n+1}\left(q\right),q)},$ 28.4.26 $\displaystyle\frac{B^{2n+1}_{2m+1}(q)}{B^{2n+1}_{1}(q)}$ $\displaystyle=\frac{(-1)^{m}}{\left({\left(\tfrac{1}{2}\right)_{m+1}}\right)^{% 2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w% _{\mbox{\tiny I}}(\frac{1}{2}\pi;b_{2n+1}\left(q\right),q)},$ 28.4.27 $\displaystyle\frac{B^{2n+2}_{2m}(q)}{B^{2n+2}_{2}(q)}$ $\displaystyle=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{q\pi% \left(1+O\left(m^{-1}\right)\right)}{w^{\prime}_{\mbox{\tiny I}}(\frac{1}{2}% \pi;b_{2n+2}\left(q\right),q)}.$

For the basic solutions $w_{\mbox{\tiny I}}$ and $w_{\mbox{\tiny II}}$ see §28.2(ii).