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11: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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12: 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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13: 28.2 Definitions and Basic Properties

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

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

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28.2.31 $${\int }_0^{2\pi }{\mathrm{ce}}_m(x,q){\mathrm{ce}}_n(x,q)dx=0,$$ $n\ne m$,

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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 $x$ , $\int$: integral, $m$: integer, $q=h^2$: parameter, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/28.2.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(vi), §28.2 and Ch.28

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28.2.34 $${\mathrm{ce}}_{2n}(z,-q)={(-1)}^n{\mathrm{ce}}_{2n}(\frac{1}{2}\pi -z,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.2.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(vi), §28.2 and Ch.28

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14: 28.23 Expansions in Series of Bessel Functions

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.9 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+1)A_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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15: 28.6 Expansions for Small $q$

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§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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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28.6.26 $${\mathrm{ce}}_m(z,q)=\cos mz-\frac{q}{4}\left(\frac{1}{m+1}\cos (m+2)z-\frac{1}{m-1}\cos (m-2)z\right)+\frac{q^2}{32}\left(\frac{1}{(m+1)(m+2)}\cos (m+4)z+\frac{1}{(m-1)(m-2)}\cos (m-4)z-\frac{2(m^2+1)}{{(m^2-1)}^2}\cos mz\right)+\mathrm{\cdots }.$$

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Symbols:

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

A&S Ref:

20.2.28 (in slightly different form)

Referenced by:

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

Permalink:

http://dlmf.nist.gov/28.6.E26

Encodings:

pMML, png, TeX

See also:

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

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16: 29.14 Orthogonality

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29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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17: 28.14 Fourier Series

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28.14.2 $${\mathrm{ce}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q)\cos (\nu +2m)z,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/28.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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18: 29.5 Special Cases and Limiting Forms

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$lim{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)={\mathrm{ce}}_m(\frac{1}{2}\pi -z,\theta ),$

… ►where ${\mathrm{ce}}_m(z,\theta )$ and ${\mathrm{se}}_m(z,\theta )$ are Mathieu functions; see §28.2(vi).

19: Guide to Searching the DLMF

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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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20: 28.8 Asymptotic Expansions for Large $q$

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$${\mathrm{ce}}_m(x,h^2)={\widehat{C}}_m\left(U_m(\xi )+V_m(\xi )\right),$$

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$$\frac{{\mathrm{ce}}_m(x,h^2)}{{\mathrm{ce}}_m(0,h^2)}=\frac{2^{m-(1/2)}}{{\sigma }_m}\left(W_m^{+}(x)(P_m(x)-Q_m(x))+W_m^{-}(x)(P_m(x)+Q_m(x))\right),$$

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