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

§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
§28.4(v) Change of Sign of
§28.4(vi) Behavior for Small
§28.4(vii) Asymptotic Forms for Large

§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 -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, : 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

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, : 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

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, : 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

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, : 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

§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, : 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:: §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, : 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

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, : 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

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

§28.4(iv) Case

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

28.4.13

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

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

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

28.4.14

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

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

Symbols:: : integer, : 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:: : integer, : 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:: : 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

§28.4(v) Change of Sign of

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:: : 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

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

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

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

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

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

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

§28.4(vi) Behavior for Small

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, : : 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, : : factorial, : 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, : : factorial, : 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

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

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

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

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

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, : 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

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, : 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

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, : : factorial, : integer, $q=h^{2}$ : parameter, : integer, : 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.3 Graphics 28.5 Second Solutions $\mathop{\mathrm{fe}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{ge}_{{n}}\/}\nolimits$