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1: 19.8 Quadratic Transformations

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§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►As $n\to \mathrm{\infty }$, $a_n$ and $g_n$ converge to a common limit $M(a_0,g_0)$ called the AGM (Arithmetic-Geometric Mean) of $a_0$ and $g_0$. …showing that the convergence of $c_n$ to 0 and of $a_n$ and $g_n$ to $M(a_0,g_0)$ is quadratic in each case. … ►

19.8.5 $$K\left(k\right)=\frac{\pi }{2M(1,k^{\prime })},$$ .

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.8.E5

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Annotations for §19.8(i), §19.8 and Ch.19

… ►Again, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. …

2: 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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Abramowitz and Stegun (1964, Chapter 20)

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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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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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National Bureau of Standards (1967) includes the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$, and $n=4(1)15$ with $q=0(2)100$; Fourier coefficients for ${\mathrm{ce}}_n(x,q)$ and ${\mathrm{se}}_n(x,q)$ for $n=0(1)15$, $n=1(1)15$, respectively, and various values of $q$ in the interval $[0,100]$; joining factors $g_{e,n}(\sqrt{q})$, $f_{e,n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\text{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$. Precision is generally 8D.

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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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4: 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

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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). …

5: 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

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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

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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

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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

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

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6: 18.42 Software

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

7: 28.2 Definitions and Basic Properties

… ►Period $\pi$ means that the eigenfunction has the property $w(z+\pi )=w(z)$, whereas antiperiod $\pi$ means that $w(z+\pi )=-w(z)$. Even parity means $w(-z)=w(z)$, and odd parity means $w(-z)=-w(z)$. … ►

${\mathrm{ce}}_0(z,0)=1/\sqrt{2},$

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${\mathrm{ce}}_n(z,0)=\cos \left(nz\right),$

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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:

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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

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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:

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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:

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Annotations for §28.11, §28.11 and Ch.28

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9: 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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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:

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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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10: 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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