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1: 10.28 Wronskians and Cross-Products

§10.28 Wronskians and Cross-Products

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2: 10.63 Recurrence Relations and Derivatives

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§10.63(ii) Cross-Products

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10.63.7 $$p_{\nu }{s}_{\nu }=r_{\nu }^2+q_{\nu }^2.$$

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Symbols:

$\nu$: complex parameter, $p_{\nu }$: cross-product, $q_{\nu }$: cross-product, $r_{\nu }$: cross-product and $s_{\nu }$: cross-product

A&S Ref:

9.9.20

Referenced by:

§10.63(ii)

Permalink:

http://dlmf.nist.gov/10.63.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.63(ii), §10.63 and Ch.10

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3: 10.6 Recurrence Relations and Derivatives

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§10.6(iii) Cross-Products

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10.6.10 $$p_{\nu }{s}_{\nu }-q_{\nu }{r}_{\nu }=4/({\pi }^{2}ab).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\nu$: complex parameter, $p_{\nu }$: cross-product, $q_{\nu }$: cross-product, $r_{\nu }$: cross-product and $s_{\nu }$: cross-product

A&S Ref:

9.1.34

Permalink:

http://dlmf.nist.gov/10.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.6(iii), §10.6 and Ch.10

4: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

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5: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.25 $$\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: 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, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E25

Encodings:

pMML, png, TeX

See also:

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

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28.28.26 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: 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, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E26

Encodings:

pMML, png, TeX

See also:

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

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28.28.29 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: 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, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E29

Encodings:

pMML, png, TeX

See also:

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

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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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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6: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

8: 1.6 Vectors and Vector-Valued Functions

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Cross Product (or Vector Product)

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1.6.9

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Symbols:

$det$: determinant, $\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\theta$: angle between $𝐚$ and $𝐛$ , $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Referenced by:

§1.6(i)

Permalink:

http://dlmf.nist.gov/1.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

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Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

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9: 10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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10: 14.2 Differential Equations

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§14.2(iv) Wronskians and Cross-Products

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14.2.5 $${𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)-{𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)=\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.2(iv)

Permalink:

http://dlmf.nist.gov/14.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14