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1: 28.35 Tables

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Ince (1932) includes eigenvalues $a_n$, $b_n$, and Fourier coefficients for $n=0$ or $1(1)6$, $q=0(1)10(2)20(4)40$; 7D. Also ${\mathrm{ce}}_n(x,q)$, ${\mathrm{se}}_n(x,q)$ for $q=0(1)10$, $x=1(1)90$, corresponding to the eigenvalues in the tables; 5D. Notation: $a_n={\mathrm{𝑏𝑒}}_n-2q$, $b_n={\mathrm{𝑏𝑜}}_n-2q$.

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Kirkpatrick (1960) contains tables of the modified functions ${\mathrm{Ce}}_n(x,q)$, ${\mathrm{Se}}_{n+1}(x,q)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 4D or 5D.

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Stratton et al. (1941) includes $b_n$, $b_n^{\prime }$, and the corresponding Fourier coefficients for ${\mathrm{Se}}_n(c,x)$ and ${\mathrm{So}}_n(c,x)$ for $n=0$ or $1(1)4$, $c=0(.1\text{or}.2)4.5$. Precision is mostly 5S. Notation: $c=2\sqrt{q}$, $b_n=a_n+2q$, $b_n^{\prime }=b_n+2q$, and for ${\mathrm{Se}}_n(c,x)$, ${\mathrm{So}}_n(c,x)$ see §28.1.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

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Ince (1932) includes the first zero for ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ for $n=2(1)5$ or $6$, $q=0(1)10(2)40$; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$.

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2: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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3: 28.6 Expansions for Small $q$

… ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: … ►

§28.6(ii) Functions ${\mathrm{ce}}_n$ and ${\mathrm{se}}_n$

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28.6.23 $${\mathrm{se}}_1(z,q)=\sin z-\frac{1}{8}q\sin 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\sin 5z+2\sin 3z-\sin z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\sin 7z+\frac{8}{9}\sin 5z-\frac{1}{3}\sin 3z-2\sin z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

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28.6.25 $${\mathrm{se}}_2(z,q)=\sin 2z-\frac{1}{12}q\sin 4z+\frac{1}{128}{q}^2\left(\frac{1}{3}\sin 6z-\frac{4}{9}\sin 2z\right)+\mathrm{\cdots }.$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(ii), §28.6 and Ch.28

… ►For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ everywhere in (28.6.26). …

4: 28.8 Asymptotic Expansions for Large $q$

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28.8.1 $$\begin{array}{c}{a}_m\left(h^2\right)\\ b_{m+1}\left(h^2\right)\end{array}\}\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $b_n\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $m$: integer and $h$: parameter

A&S Ref:

20.2.30 (in different form)

Referenced by:

§28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)

Permalink:

http://dlmf.nist.gov/28.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(i), §28.8 and Ch.28

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$$\frac{{\mathrm{se}}_{m+1}(x,h^2)}{\sin x}={\widehat{S}}_m\left(U_m(\xi )-V_m(\xi )\right),$$

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$$\frac{{\mathrm{se}}_{m+1}(x,h^2)}{{\mathrm{se}}_{m+1}^{\prime }(0,h^2)}=\frac{2^{m-(1/2)}}{{\tau }_{m+1}}\left(W_m^{+}(x)(P_m(x)-Q_m(x))-W_m^{-}(x)(P_m(x)+Q_m(x))\right),$$

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${\sigma }_m\sim 1+{\displaystyle \frac{s}{2^{3}h}}+{\displaystyle \frac{4s^2+3}{2^7{h}^2}}+{\displaystyle \frac{19s^3+59s}{2^{11}{h}^3}}+\mathrm{\cdots },$

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${\tau }_{m+1}\sim 2h-{\displaystyle \frac{1}{4}}s-{\displaystyle \frac{2s^2+3}{2^{6}h}}-{\displaystyle \frac{7s^3+47s}{2^{10}{h}^2}}-\mathrm{\cdots },$

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5: 28.1 Special Notation

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${\mathrm{ce}}_{\nu }(z,q)$,

${\mathrm{se}}_{\nu }(z,q)$,

${\mathrm{fe}}_n(z,q)$,

${\mathrm{ge}}_n(z,q)$,

${\mathrm{me}}_{\nu }(z,q)$,

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${\mathrm{Se}}_n(s,z)={\displaystyle \frac{{\mathrm{ce}}_n(z,q)}{{\mathrm{ce}}_n(0,q)}},$

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${\mathrm{So}}_n(s,z)={\displaystyle \frac{{\mathrm{se}}_n(z,q)}{{\mathrm{se}}_n^{\prime }(0,q)}}.$

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${\mathrm{Se}}_n(c,z)={\displaystyle \frac{{\mathrm{ce}}_n(z,q)}{{\mathrm{ce}}_n(0,q)}},$

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${\mathrm{So}}_n(c,z)={\displaystyle \frac{{\mathrm{se}}_n(z,q)}{{\mathrm{se}}_n^{\prime }(0,q)}}.$

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6: 28.9 Zeros

… ►For real $q$ each of the functions ${\mathrm{ce}}_{2n}(z,q)$, ${\mathrm{se}}_{2n+1}(z,q)$, ${\mathrm{ce}}_{2n+1}(z,q)$, and ${\mathrm{se}}_{2n+2}(z,q)$ has exactly $n$ zeros in . …For $q\to \mathrm{\infty }$ the zeros of ${\mathrm{ce}}_{2n}(z,q)$ and ${\mathrm{se}}_{2n+1}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. …Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …

7: 28.3 Graphics

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Figure 28.3.5: ${\mathrm{se}}_{2n+1}(x,1)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.6: ${\mathrm{se}}_{2n+1}(x,10)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.7: ${\mathrm{se}}_{2n}(x,1)$ for $0\le x\le \pi /2$, $n=1,2,3,4$. Magnify

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Figure 28.3.8: ${\mathrm{se}}_{2n}(x,10)$ for $0\le x\le \pi /2$, $n=1,2,3,4$. Magnify

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Figure 28.3.10: ${\mathrm{se}}_1(x,q)$ for $0\le x\le 2\pi$, $0\le q\le 10$. Magnify 3D Help

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8: 28.11 Expansions in Series of Mathieu Functions

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28.11.1 $$f(z)={\alpha }_0{\mathrm{ce}}_0(z,q)+\sum _{n=1}^{\mathrm{\infty }}\left({\alpha }_n{\mathrm{ce}}_n(z,q)+{\beta }_n{\mathrm{se}}_n(z,q)\right),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, ${\alpha }_n$ and ${\beta }_n$

Referenced by:

§28.11

Permalink:

http://dlmf.nist.gov/28.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11 and Ch.28

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$${\beta }_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x){\mathrm{se}}_n(x,q)dx.$$

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28.11.6 $$\sin (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+1}^{2n+1}(q){\mathrm{se}}_{2n+1}(z,q),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

9: Bibliography P

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Algorithm 385: Exponential integral $\mathrm{Ei}(x)$ . Comm. ACM 13 (7), pp. 446–447.

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Notes:

Includes Fortran code, maximum accuracy 20D. For certification and remarks see same journal v. 13 (1970), pp. 448–449 and p. 750; v. 15 (1972), p. 1074.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii).

Encodings:

BibTeX

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Exponentially small expansions of the confluent hypergeometric functions. Appl. Math. Sci. (Ruse) 7 (133-136), pp. 6601–6609.

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External Links:

ISSN 1312-885X, Document, MathReview Entry

Referenced by:

§13.7(iii), §13.7(iii).

Encodings:

BibTeX

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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

§36.2(ii).

Encodings:

BibTeX

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La Théorie des Fonctions de Bessel Exposée en vue de ses Applications à la Physique Mathématique. Centre National de la Recherche Scientifique, Paris (French).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.63(ii).

Encodings:

BibTeX

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Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.4.

Encodings:

BibTeX

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10: 23 Weierstrass Elliptic and Modular
Functions

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