§28.4 Fourier Series

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

28.4.1		${\mathrm{ce}}_{2n}(z,q)$	$={\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{A}_{2m}^{2n}(q)\cos 2mz,$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: 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		${\mathrm{ce}}_{2n+1}(z,q)$	$={\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{A}_{2m+1}^{2n+1}(q)\cos (2m+1)z,$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E2 Encodings: TeX, pMML, png See also: Annotations for §28.4(i), §28.4 and Ch.28
28.4.3		${\mathrm{se}}_{2n+1}(z,q)$	$={\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{B}_{2m+1}^{2n+1}(q)\sin (2m+1)z,$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E3 Encodings: TeX, pMML, png See also: Annotations for §28.4(i), §28.4 and Ch.28
28.4.4		${\mathrm{se}}_{2n+2}(z,q)$	$={\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{B}_{2m+2}^{2n+2}(q)\sin (2m+2)z.$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E4 Encodings: TeX, pMML, png See also: Annotations for §28.4(i), §28.4 and Ch.28

28.4.5		$a{A}_0-q{A}_2$	$=0,$
		$(a-4)A_2-q(2A_0+A_4)$	$=0,$
		$(a-4m^2)A_{2m}-q(A_{2m-2}+A_{2m+2})$	$=0$,
	$m=2,3,4,\mathrm{\dots }$, $a=a_{2n}\left(q\right)$, $A_{2m}=A_{2m}^{2n}(q)$.
ⓘ Symbols: $a_n\left(q\right)$: 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		$(a-1-q)A_1-q{A}_3$	$=0,$
		$\left(a-{(2m+1)}^2\right)A_{2m+1}-q(A_{2m-1}+A_{2m+3})$	$=0$,
	$m=1,2,3,\mathrm{\dots }$, $a=a_{2n+1}\left(q\right)$, $A_{2m+1}=A_{2m+1}^{2n+1}(q)$.
ⓘ Symbols: $a_n\left(q\right)$: 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		$(a-1+q)B_1-q{B}_3$	$=0,$
		$\left(a-{(2m+1)}^2\right)B_{2m+1}-q(B_{2m-1}+B_{2m+3})$	$=0$,
	$m=1,2,3,\mathrm{\dots }$, $a=b_{2n+1}\left(q\right)$, $B_{2m+1}=B_{2m+1}^{2n+1}(q)$.
ⓘ Symbols: $b_n\left(q\right)$: 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		$(a-4)B_2-q{B}_4$	$=0,$
		$(a-4m^2)B_{2m}-q(B_{2m-2}+B_{2m+2})$	$=0$,
	$m=2,3,4,\mathrm{\dots }$, $a=b_{2n+2}\left(q\right)$, $B_{2m+2}=B_{2m+2}^{2n+2}(q).$
ⓘ Symbols: $b_n\left(q\right)$: 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

28.4.9		$$2{\left(A_0^{2n}(q)\right)}^2+\sum _{m=1}^{\mathrm{\infty }}{\left(A_{2m}^{2n}(q)\right)}^2=1,$$
ⓘ 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		${\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{\left(A_{2m+1}^{2n+1}(q)\right)}^2$	$=1,$
ⓘ 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		${\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{\left(B_{2m+1}^{2n+1}(q)\right)}^2$	$=1,$
ⓘ 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		${\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{\left(B_{2m+2}^{2n+2}(q)\right)}^2$	$=1.$
ⓘ 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 $q$.

28.4.13		$A_0^0(0)$	$=1/\sqrt{2},A_{2n}^{2n}(0)=1,$
	$n>0$,
		$A_{2m}^{2n}(0)$	$=0,$
	$n\ne m$,
ⓘ 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		$A_{2n+1}^{2n+1}(0)$	$=1,$
		$A_{2m+1}^{2n+1}(0)$	$=0,$
	$n\ne m$,
ⓘ 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		$B_{2n+1}^{2n+1}(0)$	$=1,$
		$B_{2m+1}^{2n+1}(0)$	$=0,$
	$n\ne m$,
ⓘ 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		$B_{2n+2}^{2n+2}(0)$	$=1,$
		$B_{2m+2}^{2n+2}(0)$	$=0,$
	$n\ne m$.
ⓘ 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

28.4.17		$A_{2m}^{2n}(-q)$	$={(-1)}^{n-m}{A}_{2m}^{2n}(q),$
ⓘ 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		$B_{2m+2}^{2n+2}(-q)$	$={(-1)}^{n-m}{B}_{2m+2}^{2n+2}(q),$
ⓘ 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		$A_{2m+1}^{2n+1}(-q)$	$={(-1)}^{n-m}{B}_{2m+1}^{2n+1}(q),$
ⓘ 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		$B_{2m+1}^{2n+1}(-q)$	$={(-1)}^{n-m}{A}_{2m+1}^{2n+1}(q).$
ⓘ 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

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

28.4.21		$$A_{2s}^0(q)=\left(\frac{{(-1)}^{s}2}{{(s!)}^2}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+2}\right)\right)A_0^0(q),$$
ⓘ Symbols: $O\left(x\right)$: 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		$$\begin{array}{c}\hfill A_{m+2s}^m(q)\hfill \\ \hfill B_{m+2s}^m(q)\hfill \end{array}\}=\left(\frac{{(-1)}^{s}m!}{s!(m+s)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}\hfill A_m^m(q),\hfill \\ \hfill B_m^m(q),\hfill \end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E22 Encodings: TeX, pMML, png See also: Annotations for §28.4(vi), §28.4 and Ch.28

28.4.23		$$\begin{array}{c}\hfill A_{m-2s}^m(q)\hfill \\ \hfill B_{m-2s}^m(q)\hfill \end{array}\}=\left(\frac{(m-s-1)!}{s!(m-1)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}\hfill A_m^m(q),\hfill \\ \hfill B_m^m(q).\hfill \end{array}$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E23 Encodings: TeX, pMML, png See also: Annotations for §28.4(vi), §28.4 and Ch.28

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

As $m\to \mathrm{\infty }$, with fixed $q$ ($\ne 0$) and fixed $n$,

28.4.24		${\displaystyle \frac{A_{2m}^{2n}(q)}{A_0^{2n}(q)}}$	$={\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{q}{4}}\right)}^m{\displaystyle \frac{\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q)}},$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{II}}(z;a,q)$: fundamental solution and $A_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E24 Encodings: TeX, pMML, png See also: Annotations for §28.4(vii), §28.4 and Ch.28
28.4.25		${\displaystyle \frac{A_{2m+1}^{2n+1}(q)}{A_1^{2n+1}(q)}}$	$={\displaystyle \frac{{(-1)}^{m+1}}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}}{\left({\displaystyle \frac{q}{4}}\right)}^{m+1}{\displaystyle \frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}^{\prime }(\frac{1}{2}\pi ;a_{2n+1}\left(q\right),q)}},$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $A_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E25 Encodings: TeX, pMML, png See also: Annotations for §28.4(vii), §28.4 and Ch.28
28.4.26		${\displaystyle \frac{B_{2m+1}^{2n+1}(q)}{B_1^{2n+1}(q)}}$	$={\displaystyle \frac{{(-1)}^m}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}}{\left({\displaystyle \frac{q}{4}}\right)}^{m+1}{\displaystyle \frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{I}}(\frac{1}{2}\pi ;b_{2n+1}\left(q\right),q)}},$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $b_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{I}}(z;a,q)$: fundamental solution and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E26 Encodings: TeX, pMML, png See also: Annotations for §28.4(vii), §28.4 and Ch.28
28.4.27		${\displaystyle \frac{B_{2m}^{2n+2}(q)}{B_2^{2n+2}(q)}}$	$={\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{q}{4}}\right)}^m{\displaystyle \frac{q\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{I}}^{\prime }(\frac{1}{2}\pi ;b_{2n+2}\left(q\right),q)}}.$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $B_m(q)$: Fourier coefficient Permalink: http://dlmf.nist.gov/28.4.E27 Encodings: TeX, pMML, png See also: Annotations for §28.4(vii), §28.4 and Ch.28

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