for derivatives with respect to order

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1: 10.64 Integral Representations

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2: 10.38 Derivatives with Respect to Order

§10.38 Derivatives with Respect to Order

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3: 10.15 Derivatives with Respect to Order

§10.15 Derivatives with Respect to Order

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4: 14.11 Derivatives with Respect to Degree or Order

§14.11 Derivatives with Respect to Degree or Order

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14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.2 $$\frac{\partial }{\partial \nu }{𝖰}_{\nu }^{\mu }\left(x\right)=-\frac{1}{2}{\pi }^2{𝖯}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{𝖰}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(x)+\frac{1}{2}\csc \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(-x),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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5: 11.4 Basic Properties

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11.4.27 $$\frac{d}{dz}\left(z^{\nu }{𝐇}_{\nu }\left(z\right)\right)=z^{\nu }{𝐇}_{\nu -1}\left(z\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§10.22(i)

Permalink:

http://dlmf.nist.gov/11.4.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(v), §11.4 and Ch.11

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11.4.28 $$\frac{d}{dz}\left(z^{-\nu }{𝐇}_{\nu }\left(z\right)\right)=\frac{2^{-\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}-z^{-\nu }{𝐇}_{\nu +1}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve 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:

§10.22(i)

Permalink:

http://dlmf.nist.gov/11.4.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(v), §11.4 and Ch.11

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11.4.31 $${\mathscr{H}}_{\nu -m}(z)=z^{m-\nu }{\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{\nu }{\mathscr{H}}_{\nu }(z)),$$ $m=1,2,3,\mathrm{\dots }$,

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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:

§11.4(v)

Permalink:

http://dlmf.nist.gov/11.4.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(v), §11.4 and Ch.11

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§11.4(vi) Derivatives with Respect to Order

►For derivatives with respect to the order $\nu$, see Apelblat (1989) and Brychkov and Geddes (2005). …

6: 10.40 Asymptotic Expansions for Large Argument

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$\nu$-Derivative

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7: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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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

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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8: Bibliography

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Derivatives and integrals with respect to the order of the Struve functions ${𝐇}_{\nu }(x)$ and ${𝐋}_{\nu }(x)$ . J. Math. Anal. Appl. 137 (1), pp. 17–36.

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External Links:

ISSN 0022-247X, Document, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§11.4(vi), §11.7(iv).

Encodings:

BibTeX

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Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.

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External Links:

ISSN 0044-2275, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.61(v), §10.64.

Encodings:

BibTeX

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9: 10.21 Zeros

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§10.21(xiv) $\nu$-Zeros

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10: 14.10 Recurrence Relations and Derivatives

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14.10.4 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\mu -\nu -1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +1)x{𝖯}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

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

Referenced by:

§14.5(i)

Permalink:

http://dlmf.nist.gov/14.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.5 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){𝖯}_{\nu -1}^{\mu }\left(x\right)-\nu x{𝖯}_{\nu }^{\mu }\left(x\right).$$

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Symbols:

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

A&S Ref:

8.5.4 (modified)

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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