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21: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $𝐗$ … ►

2.1.2 $f(x)=o\left(\varphi (x)\right)⟺f(x)/\varphi (x)\to 0.$

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Defines:

$o\left(x\right)$: order less than

Permalink:

http://dlmf.nist.gov/2.1.E2

Encodings:

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See also:

Annotations for §2.1(i), §2.1 and Ch.2

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§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. …

22: 10.64 Integral Representations

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23: Bibliography Q

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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

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External Links:

Document, Link

Referenced by:

§18.38(iii).

Encodings:

BibTeX

24: 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

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Annotations for §14.11 and Ch.14

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14.11.3 $${𝖠}_{\nu }^{\mu }(x)=\sin \left(\nu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\frac{1}{2}x\right)}^k\mathrm{\Gamma }\left(k-\nu \right)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!\mathrm{\Gamma }\left(k-\mu +1\right)}\left(\psi \left(k+\nu +1\right)-\psi \left(k-\nu \right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\sin z$: 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.E3

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:

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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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25: 24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

… ►These identities can be regarded as higher-order recurrences. …

26: 11.4 Basic Properties

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§11.4(i) Half-Integer Orders

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11.4.3 $${𝐇}_{-n-\frac{1}{2}}\left(z\right)={(-1)}^n{J}_{n+\frac{1}{2}}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.1.15

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E3

Encodings:

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

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11.4.4 $${𝐋}_{-n-\frac{1}{2}}\left(z\right)=I_{n+\frac{1}{2}}\left(z\right).$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.2.10

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E4

Encodings:

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See also:

Annotations for §11.4(i), §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). …

27: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

28: Need Help?

… ►We have also tried to use the best technologies available in order to make this information useful and accessible. …

29: 11.7 Integrals and Sums

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11.7.1 $$\int z^{\nu }{𝐇}_{\nu -1}\left(z\right)dz=z^{\nu }{𝐇}_{\nu }\left(z\right),$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.3 $$\int z^{\nu }{𝐋}_{\nu -1}\left(z\right)dz=z^{\nu }{𝐋}_{\nu }\left(z\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.5 $$f_{\nu }(z)={\int }_0^z{t}^{\nu }{𝐇}_{\nu }\left(t\right)dt,$$

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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§11.7(iv) Integrals with Respect to Order

►For integrals of ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ with respect to the order $\nu$, see Apelblat (1989). …

30: 14.14 Continued Fractions

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14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

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See also:

Annotations for §14.14 and Ch.14

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14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

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See also:

Annotations for §14.14 and Ch.14

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