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

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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 $z$ -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

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

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

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§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 $m$

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§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

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§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 $m$