28 Mathieu Functions and Hill’s Equation Mathieu Functions of Integer Order 28.3 Graphics 28.5 Second Solutions $\operatorname{fe}_{n}$ , $\operatorname{ge}_{n}$

§28.4 Fourier Series

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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
See also:: Annotations for Ch.28

§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(i) Definitions

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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
See also:: Annotations for §28.4 and Ch.28

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

28.4.1 $\operatorname{ce}_{2n}\left(z,q\right)=\sum_{m=0}^{\infty}A^{2n}_{2m}(q)\cos 2mz,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, : integer, $q=h^{2}$ : parameter, : integer, : 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 $\operatorname{ce}_{2n+1}\left(z,q\right)=\sum_{m=0}^{\infty}A^{2n+1}_{2m+1}(q)% \cos(2m+1)z,$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cos\NVar{z}$ : cosine function, : integer, $q=h^{2}$ : parameter, : integer, : 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 $\operatorname{se}_{2n+1}\left(z,q\right)=\sum_{m=0}^{\infty}B^{2n+1}_{2m+1}(q)% \sin(2m+1)z,$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, : integer, $q=h^{2}$ : parameter, : integer, : 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 $\operatorname{se}_{2n+2}\left(z,q\right)=\sum_{m=0}^{\infty}B^{2n+2}_{2m+2}(q)% \sin(2m+2)z.$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : sine function, : integer, $q=h^{2}$ : parameter, : integer, : 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(ii) Recurrence Relations

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Keywords:: Fourier coefficients, Mathieu functions, recurrence relations
Notes:: See Meixner and Schäfke (1954, p. 188).
Referenced by:: item (b)
Permalink:: http://dlmf.nist.gov/28.4.ii
See also:: Annotations for §28.4 and Ch.28

28.4.5

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

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

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

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, : integer, : 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

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

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

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, : integer, : 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

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

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

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, : integer, : 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

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

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

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Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, : integer, $q=h^{2}$ : parameter, : integer, : 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

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Keywords:: Fourier coefficients, Mathieu functions, normalization
Notes:: See Meixner and Schäfke (1954, p. 188).
Permalink:: http://dlmf.nist.gov/28.4.iii
See also:: Annotations for §28.4 and Ch.28

28.4.9 $2\left(A^{2n}_{0}(q)\right)^{2}+\sum_{m=1}^{\infty}\left(A^{2n}_{2m}(q)\right)% ^{2}=1,$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $\sum_{m=0}^{\infty}\left(A^{2n+1}_{2m+1}(q)\right)^{2}=1,$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $\sum_{m=0}^{\infty}\left(B^{2n+1}_{2m+1}(q)\right)^{2}=1,$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $\sum_{m=0}^{\infty}\left(B^{2n+2}_{2m+2}(q)\right)^{2}=1.$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 , and by continuity for the other values of .

§28.4(iv) Case

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Keywords:: Fourier coefficients, Mathieu functions, values at
Notes:: See Meixner and Schäfke (1954, p. 188).
Permalink:: http://dlmf.nist.gov/28.4.iv
See also:: Annotations for §28.4 and Ch.28

28.4.13

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

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

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Symbols:: : integer, : 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

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

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

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Symbols:: : integer, : 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

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

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

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Symbols:: : integer, : 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

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

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

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Symbols:: : integer, : 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

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Keywords:: Fourier coefficients, Mathieu functions, reflection properties in
Notes:: See Meixner and Schäfke (1954, p. 189).
Permalink:: http://dlmf.nist.gov/28.4.v
See also:: Annotations for §28.4 and Ch.28

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

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $B^{2n+2}_{2m+2}(-q)=(-1)^{n-m}B^{2n+2}_{2m+2}(q),$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $A^{2n+1}_{2m+1}(-q)=(-1)^{n-m}B^{2n+1}_{2m+1}(q),$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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 $B^{2n+1}_{2m+1}(-q)=(-1)^{n-m}A^{2n+1}_{2m+1}(q).$

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Symbols:: : integer, $q=h^{2}$ : parameter, : 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

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Keywords:: Fourier coefficients, Mathieu functions, asymptotic forms for small
Notes:: See Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33).
Permalink:: http://dlmf.nist.gov/28.4.vi
See also:: Annotations for §28.4 and Ch.28

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : factorial (as in ), $q=h^{2}$ : 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 $\rselection{A^{m}_{m+2s}(q)\\ B^{m}_{m+2s}(q)}=\left(\dfrac{(-1)^{s}m!}{s!(m+s)!}\left(\frac{q}{4}\right)^{s% }+O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q),}$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : factorial (as in ), : 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 $\rselection{A^{m}_{m-2s}(q)\\ B^{m}_{m-2s}(q)}=\left(\dfrac{(m-s-1)!}{s!(m-1)!}\left(\frac{q}{4}\right)^{s}+% O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q).}$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : factorial (as in ), : 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).

§28.4(vii) Asymptotic Forms for Large

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Keywords:: Fourier coefficients, Mathieu functions, asymptotic forms of higher coefficients
Notes:: See Wolf (2008).
Permalink:: http://dlmf.nist.gov/28.4.vii
See also:: Annotations for §28.4 and Ch.28

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer, $q=h^{2}$ : parameter, : 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
See also:: Annotations for §28.4(vii), §28.4 and Ch.28

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $q=h^{2}$ : parameter, : integer, : 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 $\frac{B^{2n+1}_{2m+1}(q)}{B^{2n+1}_{1}(q)}=\frac{(-1)^{m}}{\left({\left(\tfrac% {1}{2}\right)_{m+1}}\right)^{2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O% \left(m^{-1}\right)\right)}{w_{\mbox{\tiny I}}(\frac{1}{2}\pi;b_{2n+1}\left(q% \right),q)},$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, : integer, $q=h^{2}$ : parameter, : 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
See also:: Annotations for §28.4(vii), §28.4 and Ch.28

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

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