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11: 3.4 Differentiation

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§3.4(iii) Partial Derivatives

… ► … ►The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. … ►

12: 12.19 Tables

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Miller (1955) includes $W(a,x)$, $W(a,-x)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

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Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

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Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

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13: 28.35 Tables

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

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14: Bibliography K

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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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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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15: 10.73 Physical Applications

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10.73.1 $${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{{\partial \varphi }^2}+\frac{{\partial }^{2}V}{{\partial z}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

… ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►

10.73.3 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

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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 $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

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16: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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On the complex zeros of Airy and Bessel functions and those of their derivatives. Anal. Appl. (Singap.) 12 (5), pp. 537–561.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§9.9(i), §9.9(i).

Encodings:

BibTeX

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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17: Errata

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These equations have been generalized to include the additional cases of $\partial J_{-\nu }\left(z\right)/\partial \nu$, $\partial I_{-\nu }\left(z\right)/\partial \nu$, respectively.

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0$$

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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18: 36.4 Bifurcation Sets

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36.4.1 $$\frac{\partial }{\partial t}{\mathrm{\Phi }}_K(t_j(𝐱);𝐱)=0.$$

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Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Referenced by:

§36.11, §36.12(i), §36.4(i), §36.5(i), §36.5(ii)

Permalink:

http://dlmf.nist.gov/36.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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36.4.3 $$\frac{{\partial }^2}{{\partial t}^2}{\mathrm{\Phi }}_K(t;𝐱)=0.$$

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Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable and $K$: codimension

Permalink:

http://dlmf.nist.gov/36.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.4(i), §36.4(i), §36.4 and Ch.36

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$x=\frac{9}{20}{z}^2.$

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$x=-\frac{3}{20}{z}^2,$

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19: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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20: 14.30 Spherical and Spheroidal Harmonics

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14.30.3 $$Y_{l,m}(\theta ,\varphi )=\frac{{(-1)}^{l+m}}{2^{l}l!}{\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi }{\left(\sin \theta \right)}^m{\left(\frac{d}{d(\cos \theta )}\right)}^{l+m}{\left(\sin \theta \right)}^{2l}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E3

Encodings:

pMML, png, TeX

See also:

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

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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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