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

ⓘ

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 $q=0$
§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
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 $z$ -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	$\displaystyle\operatorname{ce}_{2n+1}\left(z,q\right)$	$\displaystyle=\sum_{m=0}^{\infty}A^{2n+1}_{2m+1}(q)\cos(2m+1)z,$
ⓘ 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 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	$\displaystyle\operatorname{se}_{2n+1}\left(z,q\right)$	$\displaystyle=\sum_{m=0}^{\infty}B^{2n+1}_{2m+1}(q)\sin(2m+1)z,$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : 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	$\displaystyle\operatorname{se}_{2n+2}\left(z,q\right)$	$\displaystyle=\sum_{m=0}^{\infty}B^{2n+2}_{2m+2}(q)\sin(2m+2)z.$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\sin\NVar{z}$ : 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(ii) Recurrence Relations

ⓘ

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

ⓘ

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

ⓘ

Keywords:: Fourier coefficients, Mathieu functions, values at $q=0$
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		$\displaystyle A^{0}_{0}(0)$	$\displaystyle=1/\sqrt{2},\quad A^{2n}_{2n}(0)=1,$
	$n>0$ ,
		$\displaystyle A^{2n}_{2m}(0)$	$\displaystyle=0,$
	$n\neq m$ ,
ⓘ 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 $q$

ⓘ

Keywords:: Fourier coefficients, Mathieu functions, reflection properties in $q$
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	$\displaystyle A^{2n}_{2m}(-q)$	$\displaystyle=(-1)^{n-m}A^{2n}_{2m}(q),$
ⓘ 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 $q$

ⓘ

Keywords:: Fourier coefficients, Mathieu functions, asymptotic forms for small $q$
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		$A^{0}_{2s}(q)=\left(\dfrac{(-1)^{s}2}{(s!)^{2}}\left(\frac{q}{4}\right)^{s}+O% \left(q^{s+2}\right)\right)A^{0}_{0}(q),$
ⓘ 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		$\rselection{A^{m}_{m+2s}(q)\\ B^{m}_{m+2s}(q)}=\left(\dfrac{(-1)^{s}m!}{s!(m+s)!}\left(\frac{q}{4}\right)^{s% }+O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q),}$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : 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		$\rselection{A^{m}_{m-2s}(q)\\ B^{m}_{m-2s}(q)}=\left(\dfrac{(m-s-1)!}{s!(m-1)!}\left(\frac{q}{4}\right)^{s}+% O\left(q^{s+1}\right)\right)\lselection{A^{m}_{m}(q),\\ B^{m}_{m}(q).}$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : 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).

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

ⓘ

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 $q$ ( $\neq 0$ ) and fixed $n$ ,

28.4.24	$\displaystyle\frac{A^{2n}_{2m}(q)}{A^{2n}_{0}(q)}$	$\displaystyle=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{\pi% \left(1+O\left(m^{-1}\right)\right)}{w_{\mbox{\tiny II}}(\frac{1}{2}\pi;a_{2n}% \left(q\right),q)},$
ⓘ 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 $n!$ ), $m$ : 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 See also: Annotations for §28.4(vii), §28.4 and Ch.28
28.4.25	$\displaystyle\frac{A^{2n+1}_{2m+1}(q)}{A^{2n+1}_{1}(q)}$	$\displaystyle=\frac{(-1)^{m+1}}{\left({\left(\tfrac{1}{2}\right)_{m+1}}\right)% ^{2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}% {w^{\prime}_{\mbox{\tiny II}}(\frac{1}{2}\pi;a_{2n+1}\left(q\right),q)},$
ⓘ 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, $m$ : 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	$\displaystyle\frac{B^{2n+1}_{2m+1}(q)}{B^{2n+1}_{1}(q)}$	$\displaystyle=\frac{(-1)^{m}}{\left({\left(\tfrac{1}{2}\right)_{m+1}}\right)^{% 2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w% _{\mbox{\tiny I}}(\frac{1}{2}\pi;b_{2n+1}\left(q\right),q)},$
ⓘ 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, $m$ : 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 See also: Annotations for §28.4(vii), §28.4 and Ch.28
28.4.27	$\displaystyle\frac{B^{2n+2}_{2m}(q)}{B^{2n+2}_{2}(q)}$	$\displaystyle=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4}\right)^{m}\frac{q\pi% \left(1+O\left(m^{-1}\right)\right)}{w^{\prime}_{\mbox{\tiny I}}(\frac{1}{2}% \pi;b_{2n+2}\left(q\right),q)}.$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : 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 $w_{\mbox{\tiny I}}$ and $w_{\mbox{\tiny II}}$ see §28.2(ii).

28.3 Graphics 28.5 Second Solutions

\operatorname{fe}_{n}

\operatorname{ge}_{n}

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§28.4 Fourier Series

Contents

§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(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 $m$