for derivatives with respect to order

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11: 14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

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12: Bibliography C

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Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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13: 14.5 Special Values

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14.5.2 $${\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=-\frac{2^{\mu +1}{\pi }^{1/2}}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(-\frac{1}{2}\nu -\frac{1}{2}\mu \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.6.3 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(i), §14.5 and Ch.14

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14.5.4 $${\frac{d{𝖰}_{\nu }^{\mu }\left(x\right)}{dx}|}_{x=0}=\frac{2^{\mu }{\pi }^{1/2}\cos \left(\frac{1}{2}(\nu +\mu )\pi \right)\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)},$$ $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.6.4

Referenced by:

Erratum (V1.1.3) for Equations (14.5.3), (14.5.4), Erratum (V1.1.3) for Equations (14.5.3), (14.5.4)

Permalink:

http://dlmf.nist.gov/14.5.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The constraint has been corrected to exclude all negative integers.

See also:

Annotations for §14.5(i), §14.5 and Ch.14

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14: 11.2 Definitions

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11.2.7 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=\frac{{(\frac{1}{2}z)}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.13(iv), §11.2(iii), §11.2(iii), §11.9(i)

Permalink:

http://dlmf.nist.gov/11.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

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11.2.9 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}-\left(1+\frac{{\nu }^2}{z^2}\right)w=\frac{{(\frac{1}{2}z)}^{\nu -1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.13(iv), §11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

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15: 11.1 Special Notation

§11.1 Special Notation

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$x$	real variable.
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$\nu$	real or complex order.
$n$	integer order.
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►Unless indicated otherwise, primes denote derivatives with respect to the argument. …

16: 14.2 Differential Equations

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14.2.2 $$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-x^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.19(i), §14.2(ii), §14.2(iii), §14.2(iii), §14.2(iii), §14.2(iii), §14.20(i), §14.3(ii), §14.3(i), §14.3(ii), 2nd item, §30.2(iii), Erratum (V1.0.22) for Subsection 14.2(iii)

Permalink:

http://dlmf.nist.gov/14.2.E2

Encodings:

pMML, png, TeX

See also:

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

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17: 3.4 Differentiation

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3.4.20 $$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{2h}(u_{1,0}-u_{-1,0})+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.21

Permalink:

http://dlmf.nist.gov/3.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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3.4.21 $$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{4h}(u_{1,1}-u_{-1,1}+u_{1,-1}-u_{-1,-1})+O\left(h^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.22

Permalink:

http://dlmf.nist.gov/3.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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3.4.22 $$\frac{{\partial }^2{u}_{0,0}}{{\partial x}^2}=\frac{1}{h^2}(u_{1,0}-2u_{0,0}+u_{-1,0})+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.23

Permalink:

http://dlmf.nist.gov/3.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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3.4.23 $$\frac{{\partial }^2{u}_{0,0}}{{\partial x}^2}=\frac{1}{12h^2}(-u_{2,0}+16u_{1,0}-30u_{0,0}+16u_{-1,0}-u_{-2,0})+O\left(h^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.24

Permalink:

http://dlmf.nist.gov/3.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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3.4.24 $$\frac{{\partial }^2{u}_{0,0}}{{\partial x}^2}=\frac{1}{3h^2}(u_{1,1}-2u_{0,1}+u_{-1,1}+u_{1,0}-2u_{0,0}+u_{-1,0}+u_{1,-1}-2u_{0,-1}+u_{-1,-1})+O\left(h^2\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.25

Permalink:

http://dlmf.nist.gov/3.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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18: 11.9 Lommel Functions

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11.9.1 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=z^{\mu -1}$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.9(i), §11.9(i), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

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19: Bibliography B

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On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec. Funct. 16 (3), pp. 187–198.

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External Links:

ISSN 1065-2469, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.15, §10.38, §11.4(vi).

Encodings:

BibTeX

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20: 11.10 Anger–Weber Functions

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11.10.5 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=f(\nu ,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(ii), §11.10 and Ch.11

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11.10.27 $${\frac{\partial }{\partial \nu }{𝐉}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi {𝐇}_0\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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11.10.28 $${\frac{\partial }{\partial \nu }{𝐄}_{\nu }\left(z\right)|}_{\nu =0}=\frac{1}{2}\pi J_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vii)

Permalink:

http://dlmf.nist.gov/11.10.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vii), §11.10 and Ch.11

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