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21: 29.15 Fourier Series and Chebyshev Series

… ►Since (29.2.5) implies that $\cos \varphi =\mathrm{sn}(z,k)$, (29.15.1) can be rewritten in the form …

22: 28.10 Integral Equations

… ►

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

ⓘ

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

… ►

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

ⓘ

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

►

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

ⓘ

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

… ►

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

…

23: 30.2 Differential Equations

… ►

§30.2(ii) Other Forms

►The Liouville normal form of equation (30.2.1) is ►

30.2.2 $$\frac{d^{2}g}{{dt}^2}+\left(\lambda +\frac{1}{4}+{\gamma }^2{\sin }^{2}t-\frac{{\mu }^2-\frac{1}{4}}{{\sin }^{2}t}\right)g=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.2(iii)

Permalink:

http://dlmf.nist.gov/30.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(ii), §30.2 and Ch.30

►

$z=\cos t,$

…

24: 34.8 Approximations for Large Parameters

… ►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►

34.8.1 $j_1,j_2,j_3\gg l_3\gg 1$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than, $\sin z$: sine function, : $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $l,l_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

… ►

34.8.2 $$\cos \theta =\frac{j_1(j_1+1)+j_2(j_2+1)-j_3(j_3+1)}{2\sqrt{j_1(j_1+1)j_2(j_2+1)}},$$

ⓘ

Symbols:

$\cos z$: cosine function and $j,j_r$: non-negative integers or non-negative integers plus one half.

Permalink:

http://dlmf.nist.gov/34.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.8 and Ch.34

… ►Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). …

25: 18.39 Applications in the Physical Sciences

… ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. …

26: 36.13 Kelvin’s Ship-Wave Pattern

… ►The integral is of the form of the real part of (36.12.1) with $y=\varphi$, $u=\theta$, $g=1$, $k=\rho$, and ►

36.13.3 $$f(\theta ,\varphi )=-\frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }.$$

ⓘ

Symbols:

$\cos z$: cosine function

Permalink:

http://dlmf.nist.gov/36.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

… ►

${\theta }_{+}(\varphi )=\frac{1}{2}(\arcsin \left(3\sin \varphi \right)-\varphi ),$

►

${\theta }_{-}(\varphi )=\frac{1}{2}(\pi -\varphi -\arcsin \left(3\sin \varphi \right)).$

… ►

►►

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

…

27: 28.6 Expansions for Small $q$

… ►

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

ⓘ

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

…

28: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►

$x=c\sqrt{({\xi }^2+1)(1-{\eta }^2)}\cos \varphi ,$

►

$y=c\sqrt{({\xi }^2+1)(1-{\eta }^2)}\sin \varphi ,$

… ►

30.14.3 $$h_{\xi }^2=\frac{c^2({\xi }^2+{\eta }^2)}{1+{\xi }^2},$$

ⓘ

Symbols:

$h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients, $\xi$: oblate spheroidal coordinate, $\eta$: oblate spheroidal coordinate and $c$: positive constant

A&S Ref:

21.4.3 (in different form)

Permalink:

http://dlmf.nist.gov/30.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(ii), §30.14 and Ch.30

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation ►

30.14.7 $$\frac{d}{d\xi }\left((1+{\xi }^2)\frac{d{w}_1}{d\xi }\right)-\left(\lambda +{\gamma }^2(1+{\xi }^2)-\frac{{\mu }^2}{1+{\xi }^2}\right)w_1=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\xi$: oblate spheroidal coordinate, $w_j$: solution to DE, ${\gamma }^2={\kappa }^2{c}^2$: parameter, $\lambda$: real parameter and $\mu$: real parameter

A&S Ref:

21.6.3 (in different form)

Referenced by:

§30.14(iv), §30.14(iv)

Permalink:

http://dlmf.nist.gov/30.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.14(iv), §30.14 and Ch.30

…

29: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►

$x=c\sqrt{({\xi }^2-1)(1-{\eta }^2)}\cos \varphi ,$

►

$y=c\sqrt{({\xi }^2-1)(1-{\eta }^2)}\sin \varphi ,$

… ►

30.13.9 $$\frac{d}{d\xi }\left((1-{\xi }^2)\frac{d{w}_1}{d\xi }\right)+\left(\lambda +{\gamma }^2(1-{\xi }^2)-\frac{{\mu }^2}{1-{\xi }^2}\right)w_1=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\xi$: prolate spheroidal coordinate, $w_j$: solution to DE, ${\gamma }^2={\kappa }^2{c}^2$: parameter, $\lambda$: real parameter and $\mu$: real parameter

A&S Ref:

21.6.1 (in different form)

Referenced by:

§30.13(iv), §30.13(iv)

Permalink:

http://dlmf.nist.gov/30.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

►

30.13.10 $$\frac{d}{d\eta }\left((1-{\eta }^2)\frac{d{w}_2}{d\eta }\right)+\left(\lambda +{\gamma }^2(1-{\eta }^2)-\frac{{\mu }^2}{1-{\eta }^2}\right)w_2=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\eta$: prolate spheroidal coordinate, $w_j$: solution to DE, ${\gamma }^2={\kappa }^2{c}^2$: parameter, $\lambda$: real parameter and $\mu$: real parameter

A&S Ref:

21.6.2 (in different form)

Referenced by:

§30.13(iv), §30.13(iv), §30.14(iv), §30.14(iv)

Permalink:

http://dlmf.nist.gov/30.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

… ►

30.13.12 $$w_3(\varphi )=a_3\cos \left(m\varphi \right)+b_3\sin \left(m\varphi \right).$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $b_j$: coefficients, $\varphi$: prolate spheroidal coordinate, $w_j$: solution to DE and $a_j$: coefficients

Referenced by:

§30.13(v), §30.14(v)

Permalink:

http://dlmf.nist.gov/30.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

…

30: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

►

§33.10(i) Large $\rho$

… ►

$$F_{\mathrm{\ell }}(\eta ,\rho )=\sin \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$$

… ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…