【亚博体育qee9.com】达人捕鱼游戏在线玩7sE

SearchAdvancedHelp

(0.001 seconds)

11—20 of 200 matching pages

11: 28.10 Integral Equations

… ►

28.10.5 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+1}(t,h^2)dt=\frac{h{B}_1^{2n+1}(h^2)}{{\mathrm{se}}_{2n+1}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

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

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E5

Encodings:

pMML, png, TeX

See also:

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

►

28.10.6 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin z\sin t\cos \left(2h\cos z\cos t\right){\mathrm{se}}_{2n+1}(t,h^2)dt=\frac{B_1^{2n+1}(h^2)}{2{\mathrm{se}}_{2n+1}(\frac{1}{2}\pi ,h^2)}{\mathrm{se}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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 $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.22 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E6

Encodings:

pMML, png, TeX

See also:

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

►

28.10.7 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin z\sin t\sin \left(2h\cos z\cos t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=-\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{se}}_{2n+2}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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 $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.22 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E7

Encodings:

pMML, png, TeX

See also:

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

►

28.10.8 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+2}(z,h^2).$$

ⓘ

Symbols:

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

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E8

Encodings:

pMML, png, TeX

See also:

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

…

12: 28.13 Graphics

… ►

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

… ►

►

►

Figure 28.13.4: ${\mathrm{se}}_{\nu }(x,1)$ for , $0\le x\le 2\pi$. Magnify 3D Help

…

13: 28.22 Connection Formulas

… ►

28.22.2 $${\mathrm{Ms}}_m^{(1)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{o,m}(h){\mathrm{se}}_m^{\prime }(0,h^2)}{\mathrm{Se}}_m(z,h^2),$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.22.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

28.22.4 $${\mathrm{Ms}}_m^{(2)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{o,m}(h){\mathrm{se}}_m^{\prime }(0,h^2)}\left(-f_{o,m}(h){\mathrm{Se}}_m(z,h^2)-\frac{2}{\pi S_m(h^2)}{\mathrm{Ge}}_m(z,h^2)\right).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ge}}_n(z,q)$: modified Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Se}}_{\nu }(z,q)$: modified 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 $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.22.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

28.22.7 $$g_{o,2m+1}(h)={(-1)}^m\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)}{h{B}_1^{2m+1}(h^2)},$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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 $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E7

Encodings:

pMML, png, TeX

See also:

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

►

28.22.8 $$g_{o,2m+2}(h)={(-1)}^{m+1}\sqrt{\frac{2}{\pi }}\frac{{\mathrm{se}}_{2m+2}^{\prime }(\frac{1}{2}\pi ,h^2)}{h^2{B}_2^{2m+2}(h^2)},$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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 $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.22.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

$${\mathrm{ge}}_m(0,h^2)=\frac{1}{2}\pi S_m(h^2){\left(g_{o,m}(h)\right)}^2{\mathrm{se}}_m^{\prime }(0,h^2).$$

…

14: Bibliography L

… ►

Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

ⓘ

External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

… ►

Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Gauthier-Villars, Paris (French).

ⓘ

Notes:

Reprinted by Éditions Jacques Gabay, Sceaux, 1989.

External Links:

ISBN 2-87647-060-8, MathReview, ZentralBlatt

Referenced by:

(25.5.10), (25.5.11), (25.5.12).

Encodings:

BibTeX

… ►

Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

ⓘ

Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iv).

Encodings:

BibTeX

… ►

The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

…

15: 28.12 Definitions and Basic Properties

… ►

$${\mathrm{me}}_{-n}(z,q)=-\sqrt{2}\mathrm{i}{\mathrm{se}}_n(z,q),$$ $n=1,2,\mathrm{\dots }$;

… ►

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

… ►

28.12.13 $${\mathrm{se}}_{\nu }(z,q)=-\frac{1}{2}\mathrm{i}\left({\mathrm{me}}_{\nu }(z,q)-{\mathrm{me}}_{\nu }(-z,q)\right).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.12.E13

Encodings:

pMML, png, TeX

See also:

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

… ►

28.12.15 $${\mathrm{se}}_{\nu }(z,q)=-{\mathrm{se}}_{\nu }(-z,q)=-{\mathrm{se}}_{-\nu }(z,q).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.12.E15

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). …

16: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►

28.5.2 $${\mathrm{ge}}_n(z,q)=S_n(q)\left(z{\mathrm{se}}_n(z,q)+g_n(z,q)\right),$$

ⓘ

Defines:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $g_n(z,q)$: functions and $S_n(q)$: factor

A&S Ref:

20.3.7 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

28.5.9 $$𝒲\left\{{\mathrm{se}}_n,{\mathrm{ge}}_n\right\}=-{\mathrm{se}}_n^{\prime }(0,q){\mathrm{ge}}_n(0,q).$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.5.E9

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). … ►

►►

Figure 28.5.5: ${\mathrm{ge}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{se}}_1(x,0.5)$. Magnify

►

►►

Figure 28.5.6: ${\mathrm{ge}}_1(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{se}}_1(x,1)$. Magnify

17: Bibliography H

… ►

Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

ⓘ

Notes:

Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.

External Links:

ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§25.10(i), §27.2(i).

Encodings:

BibTeX

… ►

Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

ⓘ

Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

ISSN 0001-0782, Document, Remark 1, Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

►

Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: $I_1(x)/I_0(x)$, $I_{1.5}(x)/I_{0.5}(x)$ and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

ⓘ

Notes:

Single-precision Fortran, minimum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 571; package TOMS)

Referenced by:

5th item.

Encodings:

BibTeX

…

18: 9.4 Maclaurin Series

… ►

9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

19: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.3 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

28.28.5 $$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n^{\prime }(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E5

Encodings:

pMML, png, TeX

See also:

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

… ►

28.28.36 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{se}}_n(t,h^2){\mathrm{se}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(s)}{\mathrm{Ds}}_0(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E36

Encodings:

pMML, png, TeX

See also:

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

►

28.28.37 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{se}}_n^{\prime }(t,h^2){\mathrm{se}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(s)}{\mathrm{Ds}}_1(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E37

Encodings:

pMML, png, TeX

See also:

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

… ►

28.28.38 $${\widehat{\alpha }}_{n,m}^{(s)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{se}}_n(t,h^2){\mathrm{se}}_m(t,h^2)dt={(-1)}^p\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{se}}_n^{\prime }(0,h^2){\mathrm{se}}_m^{\prime }(0,h^2)}{h{\mathrm{Ds}}_2(n,m,0)}.$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: 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, $n$: integer and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E38

Encodings:

pMML, png, TeX

See also:

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

…

20: 28.2 Definitions and Basic Properties

… ► … ►

${\mathrm{se}}_n(z,0)=\sin \left(nz\right)$ , $n=1,2,3,\mathrm{\dots }$.

… ►

28.2.32 $${\int }_0^{2\pi }{\mathrm{se}}_m(x,q){\mathrm{se}}_n(x,q)dx=0,$$ $n\ne m$,

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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.E32

Encodings:

pMML, png, TeX

See also:

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

… ►

28.2.37 $${\mathrm{se}}_{2n+2}(z,-q)={(-1)}^n{\mathrm{se}}_{2n+2}(\frac{1}{2}\pi -z,q).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: 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.E37

Encodings:

pMML, png, TeX

See also:

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

… ►

28.2.39 $$\frac{{\mathrm{se}}_n(z,q)}{{\mathrm{se}}_n^{\prime }(0,q)}=w_{\text{II}}(z;b_n\left(q\right),q),$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, ${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.2.E39

Encodings:

pMML, png, TeX

See also:

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