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11: 1.12 Continued Fractions

§1.12 Continued Fractions

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Equivalence

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Series

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Fractional Transformations

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12: 16.14 Partial Differential Equations

§16.14 Partial Differential Equations

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§16.14(i) Appell Functions

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$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

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$$y(1-y)\frac{{\partial }^2{F}_1}{{\partial y}^2}+x(1-y)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_1}{\partial y}-{\beta }^{\prime }x\frac{\partial F_1}{\partial x}-\alpha {\beta }^{\prime }{F}_1=0,$$

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

13: 10.38 Derivatives with Respect to Order

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10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). ►

10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

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10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

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14: 12.6 Continued Fraction

§12.6 Continued Fraction

►For a continued-fraction expansion of the ratio $U(a,x)/U(a-1,x)$ see Cuyt et al. (2008, pp. 340–341).

15: 14.30 Spherical and Spheroidal Harmonics

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14.30.10 $$\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial W}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial W}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}W}{{\partial \varphi }^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $W(\rho ,\theta ,\varphi )$: solution

Permalink:

http://dlmf.nist.gov/14.30.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.12 $$L^2=-{\mathrm{\hslash }}^2\left(\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{{\partial \varphi }^2}\right),$$

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Defines:

$L$: angular momentum operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\sin z$: sine function

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

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16: Bibliography K

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.

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External Links:

Document

Referenced by:

§32.17.

Encodings:

BibTeX

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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17: Bibliography R

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A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

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Notes:

Double-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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18: 10.15 Derivatives with Respect to Order

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel 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 $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

►For $\partial J_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.2.4) and (10.15.3). … ►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: 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 $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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19: 6.16 Mathematical Applications

… ►The $n$th partial sum is given by ►

6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

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Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

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►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

20: 12.17 Physical Applications

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12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

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12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

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