distributional derivative

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11: 9.13 Generalized Airy Functions

… ►

9.13.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=z^{n}w,

n=1,2,3,\ldots

ⓘ

Defines:: $n$ : parameter (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $w$ : function
Source:: Swanson and Headley (1967, (1))
Referenced by:: §9.13(i), §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

… ►The distribution in

\mathbb{C}

and asymptotic properties of the zeros of

A_{n}\left(z\right)

A_{n}'\left(z\right)

B_{n}\left(z\right)

, and

B_{n}'\left(z\right)

are investigated in Swanson and Headley (1967) and Headley and Barwell (1975). … ►

9.13.13

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\tfrac{1}{4}m^{2}t^{m-2}w,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $w$ : function and $m$ : index
Source:: Olver (1977a, (2.01))
Referenced by:: §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

… ►

9.13.17

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=-\tfrac{1}{4}m^{2}t^{m-2}w,

m

even,

ⓘ

Defines:: $W_{m}$ : function (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $w$ : function and $m$ : index
Source:: Olver (1977a, (2.11), p. 131)
Referenced by:: §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(i), §9.13 and Ch.9

… ►

9.13.31

\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-z\frac{\mathrm{d}w}{\mathrm{d}z}+(% p-1)w=0,

ⓘ

Defines:: $w$ : function (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $p$ : parameter
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

…

12: 3.8 Nonlinear Equations

… ►For other efficient derivative-free methods, see Le (1985). … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►

3.8.13

\frac{\mathrm{d}z}{\mathrm{d}\alpha}=-\ifrac{\frac{\partial f}{\partial\alpha}% }{\frac{\partial f}{\partial z}}.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/3.8.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8 and Ch.3

… ►

3.8.14

\frac{\mathrm{d}z}{\mathrm{d}a_{j}}=-\frac{z^{j}}{f^{\prime}(z)}.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$
Referenced by:: §3.8(vi), §3.8(vi)
Permalink:: http://dlmf.nist.gov/3.8.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8 and Ch.3

… ►

3.8.16

\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}=(-4.30\dots)\times 1% 0^{7}.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

…

13: 20.13 Physical Applications

… ►

20.13.1

\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\tau$ : lattice parameter and $z$ : real variable
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §20.13 and Ch.20

… ►

20.13.2

\ifrac{\partial\theta}{\partial t}=\alpha\ifrac{{\partial}^{2}\theta}{{% \partial z}^{2}},

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : time, $\alpha$ : real parameter and $z$ : real variable
Referenced by:: §20.13
Permalink:: http://dlmf.nist.gov/20.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §20.13 and Ch.20

… ►In the singular limit

\Im\tau\rightarrow 0+

, the functions

\theta_{j}\left(z\middle|\tau\right)

j=1,2,3,4

, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

14: Bibliography K

… ►

M. K. Kerimov and S. L. Skorokhodov (1985b) Calculation of the complex zeros of Hankel functions and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (11), pp. 1628–1643, 1741.

ⓘ

Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 25(6), pp. 26–36
External Links:: ISSN 0044-4669, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi), 4th item, 5th item.
Encodings:: BibTeX

… ►

M. K. Kerimov and S. L. Skorokhodov (1986) On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

ⓘ

Notes:: In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).
External Links:: ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(i), §10.74(vi).
Encodings:: BibTeX

►

M. K. Kerimov and S. L. Skorokhodov (1987) On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

ⓘ

Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25
External Links:: ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi), 5th item.
Encodings:: BibTeX

►

M. K. Kerimov and S. L. Skorokhodov (1988) Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

ⓘ

Notes:: In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).
External Links:: ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(ix), §10.74(vi).
Encodings:: BibTeX

… ►

S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

ⓘ

Notes:: In cooperation with W. Rudert
External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

15: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.3

Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : 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

… ►

Distributional Completeness

… ►

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\phi}^{2}}=0,

ⓘ

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

… ►

14.30.12

\mathrm{L}^{2}=-\hbar^{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\phi}^{2}}\right),

ⓘ

Defines:: $\mathrm{L}$ : angular momentum operator (locally)
Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ and $\sin\NVar{z}$ : 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

… ►

14.30.13

\mathrm{L}_{z}=-\mathrm{i}\hbar\frac{\partial}{\partial\phi};

ⓘ

Defines:: $\mathrm{L}_{z}$ : $z$ component of the angular momentum operator (locally)
Symbols:: $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : 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

…

16: Bibliography H

… ►

J. Hadamard (1896) Sur la distribution des zéros de la fonction

\zeta(s)

et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

ⓘ

Notes:: Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.
External Links:: ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §25.10(i), §27.2(i).
Encodings:: BibTeX

… ►

Harvard University (1945) Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

V. B. Headley and V. K. Barwell (1975) On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

ⓘ

External Links:: FullText, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §9.13(i).
Encodings:: BibTeX

… ►

G. W. Hill (1970) Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

ⓘ

Notes:: Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)
External Links:: ISSN 0001-0782, Document, Remark 1, Remark 2
Referenced by:: §8.28(iv).
Encodings:: BibTeX

►

G. W. Hill (1981) Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions:

I_{1}(x)/I_{0}(x)

I_{1.5}(x)/I_{0.5}(x)

and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

ⓘ

Notes:: Single-precision Fortran, minimum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 571; package TOMS)
Referenced by:: 5th item.
Encodings:: BibTeX

…

17: 31.15 Stieltjes Polynomials

… ►

31.15.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_{% j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\Phi(z)}{\prod_{j=1}^% {N}(z-a_{j})}w=0,

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $\Phi(z)$ : polynomial
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(i), §31.15 and Ch.31

… ►If

t_{k}

is a zero of the Van Vleck polynomial

V(z)

, corresponding to an

n

th degree Stieltjes polynomial

S(z)

, and

z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}

are the zeros of

S^{\prime}(z)

(the derivative of

S(z)

), then

t_{k}

is either a zero of

S^{\prime}(z)

or a solution of the equation … ►then there are exactly

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

polynomials

S(z)

, each of which corresponds to each of the

\genfrac{(}{)}{0.0pt}{}{n+N-2}{N-2}

ways of distributing its

n

zeros among

N-1

intervals

(a_{j},a_{j+1})

j=1,2,\dots,N-1

. …

18: 1.4 Calculus of One Variable

… ►

§1.4(iii) Derivatives

… ►

Higher Derivatives

… ►

Chain Rule

… ►

Leibniz’s Formula

… ►Delta distributions and Dirac

\delta

-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. …

19: Bibliography S

… ►

M. E. Sherry (1959) The zeros and maxima of the Airy function and its first derivative to 25 significant figures. Report AFCRC-TR-59-135, ASTIA Document No. AD214568 Air Research and Development Command, U.S. Air Force, Bedford, MA.

ⓘ

Notes:: Available from U.S. Department of Commerce, Office of Technical Services, Washington, D.C.
Referenced by:: 2nd item, §9.8(iv).
Encodings:: BibTeX

… ►

R. Spigler (1980) Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).

ⓘ

External Links:: MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

… ►

R. S. Strichartz (1994) A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

ⓘ

External Links:: ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt
Referenced by:: §18.17(v).
Encodings:: BibTeX

… ►

R. Szmytkowski (2006) On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39 (49), pp. 15147–15172.

ⓘ

External Links:: ISSN 0305-4470, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

►

R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

ⓘ

External Links:: ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §14.11, §14.11.
Encodings:: BibTeX

…

20: Bibliography B

… ►

J. Baik, P. Deift, and K. Johansson (1999) On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (4), pp. 1119–1178.

ⓘ

External Links:: ISSN 0894-0347, FullText, MathReview (David J. Aldous), ZentralBlatt
Referenced by:: §32.14.
Encodings:: BibTeX

… ►

L. E. Ballentine and S. M. McRae (1998) Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

ⓘ

External Links:: Document
Referenced by:: §24.18.
Encodings:: BibTeX

… ►

A. R. Barnett (1982) COULFG: Coulomb and Bessel functions and their derivatives, for real arguments, by Steed’s method. Comput. Phys. Comm. 27, pp. 147–166.

ⓘ

Notes:: Double-precision Fortran code for positive energies. Maximum accuracy: 31S.
External Links:: Document, Code
Referenced by:: 2nd item, 2nd item.
Encodings:: BibTeX

… ►

C. Bingham, T. Chang, and D. Richards (1992) Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

ⓘ

External Links:: ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt
Referenced by:: §35.10, §35.9.
Encodings:: BibTeX

… ►

Yu. A. Brychkov and K. O. Geddes (2005) On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec. Funct. 16 (3), pp. 187–198.

ⓘ

External Links:: ISSN 1065-2469, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.15, §10.38, §11.4(vi).
Encodings:: BibTeX

…