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1: 18.42 Software

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

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

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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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3: 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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4: 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 $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …

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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${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
…

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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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6: 28.10 Integral Equations

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28.10.1 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

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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 $x$ , $\int$: integral, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.2 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(0,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.3 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=-\frac{h{A}_1^{2n+1}(h^2)}{{\mathrm{ce}}_{2n+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

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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 $x$ , $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.4 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=\frac{A_1^{2n+1}(h^2)}{2{\mathrm{ce}}_{2n+1}(0,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

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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, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.10 $${\int }_0^{\pi }{J}_0\left(2\sqrt{q}(\cos \tau +\cos \zeta )\right){\mathrm{ce}}_n(\tau ,q)d\tau =w_{\text{II}}(\pi ;a_n\left(q\right),q){\mathrm{ce}}_n(\zeta ,q).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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7: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.8 $$𝒲\left\{{\mathrm{ce}}_n,{\mathrm{fe}}_n\right\}={\mathrm{ce}}_n(0,q){\mathrm{fe}}_n^{\prime }(0,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $𝒲$: Wronskian, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►For further information on $C_n(q)$, $S_n(q)$, and expansions of $f_n(z,q)$, $g_n(z,q)$ in Fourier series or in series of ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). … ►

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Figure 28.5.1: ${\mathrm{fe}}_0(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_0(x,0.5)$. Magnify

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Figure 28.5.2: ${\mathrm{fe}}_0(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_0(x,1)$. Magnify

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Figure 28.5.3: ${\mathrm{fe}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_1(x,0.5)$. Magnify

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8: 28.13 Graphics

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§28.13(ii) Solutions ${\mathrm{ce}}_{\nu }(x,q)$, ${\mathrm{se}}_{\nu }(x,q)$, and ${\mathrm{me}}_{\nu }(x,q)$ for General $\nu$

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Figure 28.13.3: ${\mathrm{ce}}_{\nu }(x,1)$ for , $0\le x\le 2\pi$. Magnify 3D Help

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9: 28.22 Connection Formulas

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28.22.1 $${\mathrm{Mc}}_m^{(1)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{e,m}(h){\mathrm{ce}}_m(0,h^2)}{\mathrm{Ce}}_m(z,h^2),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors

A&S Ref:

20.6.15 (in different form)

Permalink:

http://dlmf.nist.gov/28.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.3 $${\mathrm{Mc}}_m^{(2)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{e,m}(h){\mathrm{ce}}_m(0,h^2)}\left(-f_{e,m}(h){\mathrm{Ce}}_m(z,h^2)+\frac{2}{\pi C_m(h^2)}{\mathrm{Fe}}_m(z,h^2)\right),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Fe}}_n(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors, $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.5 $$g_{e,2m}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)}{A_0^{2m}(h^2)},$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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28.22.6 $$g_{e,2m+1}(h)={(-1)}^{m+1}\sqrt{\frac{2}{\pi }}\frac{{\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{h{A}_1^{2m+1}(h^2)},$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $h$: parameter, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(i), §28.22 and Ch.28

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$${\mathrm{fe}}_m^{\prime }(0,h^2)=\frac{1}{2}\pi C_m(h^2){\left(g_{e,m}(h)\right)}^2{\mathrm{ce}}_m(0,h^2),$$

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10: 28.12 Definitions and Basic Properties

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

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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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28.12.12 $${\mathrm{ce}}_{\nu }(z,q)=\frac{1}{2}\left({\mathrm{me}}_{\nu }(z,q)+{\mathrm{me}}_{\nu }(-z,q)\right),$$

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Symbols:

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

A&S Ref:

20.5.2 (in different form)

Permalink:

http://dlmf.nist.gov/28.12.E12

Encodings:

pMML, png, TeX

See also:

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

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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). …