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1: 28.12 Definitions and Basic Properties

… ►Without loss of generality, from now on we replace $\nu +2n$ by $\nu$. … ►

$${\mathrm{me}}_n(z,q)=\sqrt{2}{\mathrm{ce}}_n(z,q),$$ $n=0,1,2,\mathrm{\dots }$,

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§28.12(iii) Functions ${\mathrm{ce}}_{\nu }(z,q)$, ${\mathrm{se}}_{\nu }(z,q)$, when $\nu \notin \mathbb{Z}$

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28.12.14 $${\mathrm{ce}}_{\nu }(z,q)={\mathrm{ce}}_{\nu }(-z,q)={\mathrm{ce}}_{-\nu }(z,q),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(iii), §28.12 and Ch.28

… ►Again, the limiting values of ${\mathrm{ce}}_{\nu }(z,q)$ and ${\mathrm{se}}_{\nu }(z,q)$ as $\nu \to n$ $(\ne 0)$ are not the functions ${\mathrm{ce}}_n(z,q)$ and ${\mathrm{se}}_n(z,q)$ defined in §28.2(vi). …

2: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

A&S Ref:

20.7.34 20.7.35 20.7.36 20.7.37

Permalink:

http://dlmf.nist.gov/28.28.E2

Encodings:

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See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.11 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}{\mathrm{Ce}}_{\nu }(t,h^2)dt=\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi }{\mathrm{ce}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(3)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E11

Encodings:

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See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.15 $${\int }_0^{\mathrm{\infty }}\cos \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt={(-1)}^{n+1}\frac{1}{2}\pi {\mathrm{Mc}}_{2n}^{(2)}(0,h){\mathrm{ce}}_{2n}(y,h^2),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $y$: real variable

Permalink:

http://dlmf.nist.gov/28.28.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.16 $${\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt=-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.28.E16

Encodings:

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See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

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Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

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See also:

Annotations for §28.28(iv), §28.28 and Ch.28

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3: 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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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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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_n(x,10)$ for $n=1(1)10$, and the first 5 zeros of ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_n^{(j)}(x,\sqrt{10})$ for $n=0$ or $1(1)8$, $j=1,2$. Precision is mostly 9S.

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4: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

5: 28.6 Expansions for Small $q$

… ►(Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).) … ►

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

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28.6.21 $$2^{1/2}{\mathrm{ce}}_0(z,q)=1-\frac{1}{2}q\cos 2z+\frac{1}{32}{q}^2\left(\cos 4z-2\right)-\frac{1}{128}{q}^3\left(\frac{1}{9}\cos 6z-11\cos 2z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable

A&S Ref:

20.2.27

Referenced by:

§28.6(ii)

Permalink:

http://dlmf.nist.gov/28.6.E21

Encodings:

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See also:

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

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28.6.22 $${\mathrm{ce}}_1(z,q)=\cos z-\frac{1}{8}q\cos 3z+\frac{1}{128}{q}^2\left(\frac{2}{3}\cos 5z-2\cos 3z-\cos z\right)-\frac{1}{1024}{q}^3\left(\frac{1}{9}\cos 7z-\frac{8}{9}\cos 5z-\frac{1}{3}\cos 3z+2\cos z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E22

Encodings:

pMML, png, TeX

See also:

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

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28.6.24 $${\mathrm{ce}}_2(z,q)=\cos 2z-\frac{1}{4}q\left(\frac{1}{3}\cos 4z-1\right)+\frac{1}{128}{q}^2\left(\frac{1}{3}\cos 6z-\frac{76}{9}\cos 2z\right)+\mathrm{\cdots },$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $q=h^2$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.6.E24

Encodings:

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See also:

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

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6: 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:

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See also:

Annotations for §28.11 and Ch.28

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

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28.11.3 $$1=2\sum _{n=0}^{\mathrm{\infty }}{A}_0^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.4 $$\cos 2mz=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q){\mathrm{ce}}_{2n}(z,q),$$ $m\ne 0$,

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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28.11.5 $$\cos (2m+1)z=\sum _{n=0}^{\mathrm{\infty }}{A}_{2m+1}^{2n+1}(q){\mathrm{ce}}_{2n+1}(z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

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7: 28.3 Graphics

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Figure 28.3.1: ${\mathrm{ce}}_{2n}(x,1)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.2: ${\mathrm{ce}}_{2n}(x,10)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.3: ${\mathrm{ce}}_{2n+1}(x,1)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.4: ${\mathrm{ce}}_{2n+1}(x,10)$ for $0\le x\le \pi /2$, $n=0,1,2,3$. Magnify

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Figure 28.3.9: ${\mathrm{ce}}_0(x,q)$ for $0\le x\le 2\pi$, $0\le q\le 10$. Magnify 3D Help

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8: 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 $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …

9: 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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${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
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${\mathrm{in}}_n={\mathrm{fe}}_n,$ ${\mathrm{ceh}}_n={\mathrm{Ce}}_n,$ ${\mathrm{inh}}_n={\mathrm{Fe}}_n,$

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

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10: Guide to Searching the DLMF

… ►From there you can also access an advanced search page where you can control certain settings, narrowing the search to certain chapters, or restricting the results to equations, graphs, tables, or bibliographic items. … ►

The following standard special functions: si, Si, ci, Ci, shi, Shi, ce, Ce, se, Se, ln, Ln, Lommels, LommelS, Jacobiphi, and the list is still growing.

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