§28.4 Fourier Series

Keywords:: Fourier series, Mathieu functions
Referenced by:: §28.10(i), ¶ ‣ §28.11, §28.23, §28.28(i), §28.28(ii), §28.31(ii)
Permalink:: http://dlmf.nist.gov/28.4

§28.4(i) Definitions
§28.4(ii) Recurrence Relations
§28.4(iii) Normalization
§28.4(iv) Case $q=0$
§28.4(v) Change of Sign of $q$
§28.4(vi) Behavior for Small $q$
§28.4(vii) Asymptotic Forms for Large $m$

§28.4(i) Definitions

Notes:: See Arscott (1964b, §3.3) and Meixner and Schäfke (1954, §2.71).
Referenced by:: §28.22(i)
Permalink:: http://dlmf.nist.gov/28.4.i

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 $\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,q\right)=\sum _{{m=0}}^{{\infty}}A^{{2n}}_{{2m}}(q)\mathop{\cos\/}\nolimits 2mz,$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : 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

28.4.2 $\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(z,q\right)=\sum _{{m=0}}^{{\infty}}A^{{2n+1}}_{{2m+1}}(q)\mathop{\cos\/}\nolimits(2m+1)z,$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : 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

28.4.3 $\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(z,q\right)=\sum _{{m=0}}^{{\infty}}B^{{2n+1}}_{{2m+1}}(q)\mathop{\sin\/}\nolimits(2m+1)z,$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : 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

28.4.4 $\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(z,q\right)=\sum _{{m=0}}^{{\infty}}B^{{2n+2}}_{{2m+2}}(q)\mathop{\sin\/}\nolimits(2m+2)z.$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sin\/}\nolimits z$ : 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

§28.4(ii) Recurrence Relations

Notes:: See Meixner and Schäfke (1954, p. 188).
Keywords:: Mathieu functions
Referenced by:: §28.34(i)
Permalink:: http://dlmf.nist.gov/28.4.ii

28.4.5

$aA_{0}-qA_{2}=0,$

$(a-4)A_{2}-q(2A_{0}+A_{4})=0,$

$(a-4m^{2})A_{{2m}}-q(A_{{2m-2}}+A_{{2m+2}})=0$ , $m=2,3,4,\dots$ , $a=\mathop{a_{{2n}}\/}\nolimits\!\left(q\right)$ , $A_{{2m}}=A_{{2m}}^{{2n}}(q)$ .

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : 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:: §28.34(ii), §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.4.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

28.4.6

$(a-1-q)A_{1}-qA_{3}=0,$

$\left(a-(2m+1)^{2}\right)A_{{2m+1}}-q(A_{{2m-1}}+A_{{2m+3}})=0$ , $m=1,2,3,\dots$ , $a=\mathop{a_{{2n+1}}\/}\nolimits\!\left(q\right)$ , $A_{{2m+1}}=A_{{2m+1}}^{{2n+1}}(q)$ .

Symbols:: $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : 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

28.4.7

$(a-1+q)B_{1}-qB_{3}=0,$

$\left(a-(2m+1)^{2}\right)B_{{2m+1}}-q(B_{{2m-1}}+B_{{2m+3}})=0$ , $m=1,2,3,\dots$ , $a=\mathop{b_{{2n+1}}\/}\nolimits\!\left(q\right)$ , $B_{{2m+1}}=B_{{2m+1}}^{{2n+1}}(q)$ .

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : 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

28.4.8

$(a-4)B_{2}-qB_{4}=0,$

$(a-4m^{2})B_{{2m}}-q(B_{{2m-2}}+B_{{2m+2}})=0$ , $m=2,3,4,\dots$ , $a=\mathop{b_{{2n+2}}\/}\nolimits\!\left(q\right)$ , $B_{{2m+2}}=B_{{2m+2}}^{{2n+2}}(q).$

Symbols:: $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : 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:: §28.34(ii), §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.4.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

§28.4(iii) Normalization

Notes:: See Meixner and Schäfke (1954, p. 188).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.4.iii

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:: §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.4.E9
Encodings:: TeX, pMML, png

28.4.10 $\sum _{{m=0}}^{{\infty}}\left(A^{{2n+1}}_{{2m+1}}(q)\right)^{2}=1,$

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

28.4.11 $\sum _{{m=0}}^{{\infty}}\left(B^{{2n+1}}_{{2m+1}}(q)\right)^{2}=1,$

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

28.4.12 $\sum _{{m=0}}^{{\infty}}\left(B^{{2n+2}}_{{2m+2}}(q)\right)^{2}=1.$

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer and $B_{{m}}(q)$ : Fourier coefficient
Referenced by:: §28.34(iii)
Permalink:: http://dlmf.nist.gov/28.4.E12
Encodings:: TeX, pMML, png

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 $q=0$

Notes:: See Meixner and Schäfke (1954, p. 188).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.4.iv

28.4.13

$A^{{0}}_{{0}}(0)=1/\sqrt{2},\quad A^{{2n}}_{{2n}}(0)=1,$ $n>0$ ,

$A^{{2n}}_{{2m}}(0)=0,$ $n\neq m$ ,

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

28.4.14

$A^{{2n+1}}_{{2n+1}}(0)=1,$

$A^{{2n+1}}_{{2m+1}}(0)=0,$ $n\neq m$ ,

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

28.4.15

$B^{{2n+1}}_{{2n+1}}(0)=1,$

$B^{{2n+1}}_{{2m+1}}(0)=0,$ $n\neq m$ ,

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

28.4.16

$B^{{2n+2}}_{{2n+2}}(0)=1,$

$B^{{2n+2}}_{{2m+2}}(0)=0,$ $n\neq m$ .

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

§28.4(v) Change of Sign of $q$

Notes:: See Meixner and Schäfke (1954, p. 189).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.4.v

28.4.17 $A^{{2n}}_{{2m}}(-q)=(-1)^{{n-m}}A^{{2n}}_{{2m}}(q),$

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

28.4.18 $B^{{2n+2}}_{{2m+2}}(-q)=(-1)^{{n-m}}B^{{2n+2}}_{{2m+2}}(q),$

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

28.4.19 $A^{{2n+1}}_{{2m+1}}(-q)=(-1)^{{n-m}}B^{{2n+1}}_{{2m+1}}(q),$

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

28.4.20 $B^{{2n+1}}_{{2m+1}}(-q)=(-1)^{{n-m}}A^{{2n+1}}_{{2m+1}}(q).$

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

§28.4(vi) Behavior for Small $q$

Notes:: See Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.4.vi

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

28.4.21 $A^{0}_{{2s}}(q)=\left(\dfrac{(-1)^{s}2}{(s!)^{2}}\left(\frac{q}{4}\right)^{s}+\mathop{O\/}\nolimits\!\left(q^{{s+2}}\right)\right)A^{0}_{0}(q),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $!$ : $n!$ : factorial, $q=h^{2}$ : 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

28.4.22 $\rselection{A^{m}_{{m+2s}}(q)\\ B^{m}_{{m+2s}}(q)}=\left(\dfrac{(-1)^{s}m!}{s!(m+s)!}\left(\frac{q}{4}\right)^{s}+\mathop{O\/}\nolimits\!\left(q^{{s+1}}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q),}$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $!$ : $n!$ : factorial, $m$ : 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

28.4.23 $\rselection{A^{m}_{{m-2s}}(q)\\ B^{m}_{{m-2s}}(q)}=\left(\dfrac{(m-s-1)!}{s!(m-1)!}\left(\frac{q}{4}\right)^{s}+\mathop{O\/}\nolimits\!\left(q^{{s+1}}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q).}$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $!$ : $n!$ : factorial, $m$ : 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

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

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

Notes:: See Wolf (2008).
Keywords:: Mathieu functions
Permalink:: http://dlmf.nist.gov/28.4.vii

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

28.4.24 $\frac{A^{{2n}}_{{2m}}(q)}{A^{{2n}}_{{0}}(q)}=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{\pi\left(1+\mathop{O\/}\nolimits\!\left(m^{{-1}}\right)\right)}{w_{{\mbox{\tiny II}}}(\frac{1}{2}\pi;\mathop{a_{{2n}}\/}\nolimits\!\left(q\right),q)},$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $!$ : $n!$ : factorial, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $w_{{\mbox{\tiny II}}}(z;a,q)$ : fundamental solution and $A_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.4.E24
Encodings:: TeX, pMML, png

28.4.25 $\frac{A^{{2n+1}}_{{2m+1}}(q)}{A^{{2n+1}}_{{1}}(q)}=\frac{(-1)^{{m+1}}}{\left(\left(\tfrac{1}{2}\right)_{{m+1}}\right)^{2}}\left(\frac{q}{4}\right)^{{m+1}}\frac{2\left(1+\mathop{O\/}\nolimits\!\left(m^{{-1}}\right)\right)}{w^{{\prime}}_{{\mbox{\tiny II}}}(\frac{1}{2}\pi;\mathop{a_{{2n+1}}\/}\nolimits\!\left(q\right),q)},$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $m$ : 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

28.4.26 $\frac{B^{{2n+1}}_{{2m+1}}(q)}{B^{{2n+1}}_{{1}}(q)}=\frac{(-1)^{m}}{\left(\left(\tfrac{1}{2}\right)_{{m+1}}\right)^{2}}\left(\frac{q}{4}\right)^{{m+1}}\frac{2\left(1+\mathop{O\/}\nolimits\!\left(m^{{-1}}\right)\right)}{w_{{\mbox{\tiny I}}}(\frac{1}{2}\pi;\mathop{b_{{2n+1}}\/}\nolimits\!\left(q\right),q)},$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $m$ : integer, $q=h^{2}$ : parameter, $n$ : integer, $w_{{\mbox{\tiny I}}}(z;a,q)$ : fundamental solution and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.4.E26
Encodings:: TeX, pMML, png

28.4.27 $\frac{B^{{2n+2}}_{{2m}}(q)}{B^{{2n+2}}_{{2}}(q)}=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{q\pi\left(1+\mathop{O\/}\nolimits\!\left(m^{{-1}}\right)\right)}{w^{{\prime}}_{{\mbox{\tiny I}}}(\frac{1}{2}\pi;\mathop{b_{{2n+2}}\/}\nolimits\!\left(q\right),q)}.$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{b_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $!$ : $n!$ : factorial, $m$ : 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

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

§28.4 Fourier Series

Contents

§28.4(i) Definitions

§28.4(ii) Recurrence Relations

§28.4(iii) Normalization

§28.4(iv) Case

§28.4(v) Change of Sign of

§28.4(vi) Behavior for Small

§28.4(vii) Asymptotic Forms for Large

§28.4(iv) Case $q=0$

§28.4(v) Change of Sign of $q$

§28.4(vi) Behavior for Small $q$

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