distributional derivative

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1: 1.16 Distributions

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§1.16(ii) Derivatives of a Distribution

… ►If $f$ is a locally integrable function then its distributional derivative is $𝐷f={\mathrm{\Lambda }}_f^{\prime }$. In the situation of (1.16.3_5) we have … ►The derivatives of tempered distributions are defined in the same way as derivatives of distributions. … ►The distributional derivative ${𝐷}^{k}f$ of $f$ is defined by …

2: 2.6 Distributional Methods

… ►Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … ►To define convolutions of distributions, we first introduce the space $K^{+}$ of all distributions of the form ${𝐷}^{n}f$, where $n$ is a nonnegative integer, $f$ is a locally integrable function on $ℝ$ which vanishes on $(-\mathrm{\infty },0]$, and ${𝐷}^{n}f$ denotes the $n$th derivative of the distribution associated with $f$. … ►

2.6.37 $$F\ast G={𝐷}^{n+m}(f\ast g).$$

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Symbols:

$g(t)$: locally integrable function, $\ast$: convolution, $𝐷$: derivative of distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $F$: derivative and $G$: derivative

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E37

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Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.40 $$t^{\mu -1}\ast t^{-s-1}=\frac{{(-1)}^s}{\mu \cdot s!}{𝐷}^{s+1}\left(t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)\right),$$ $t>0$,

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mu$: order, $\ast$: convolution and $𝐷$: derivative of distribution

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E40

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Annotations for §2.6(iii), §2.6 and Ch.2

… ►Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. …

3: 10.21 Zeros

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§10.21(i) Distribution

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§10.21(ix) Complex Zeros

►This subsection describes the distribution in $ℂ$ of the zeros of the principal branches of the Bessel functions of the second and third kinds, and their derivatives, in the case when the order is a positive integer $n$. … ► …

4: 1.1 Special Notation

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$x,y$	real variables.
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$⟨\mathrm{\Lambda },\varphi ⟩$	action of distribution $\mathrm{\Lambda }$ on test function $\varphi$.
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primes	derivatives with respect to the variable, except where indicated otherwise.
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5: Bibliography W

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

BibTeX

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

Encodings:

BibTeX

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

Encodings:

BibTeX

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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6: 9.9 Zeros

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§9.9(i) Distribution and Notation

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►For the distribution in $ℂ$ of the zeros of ${\mathrm{Ai}}^{\prime }\left(z\right)-\sigma \mathrm{Ai}\left(z\right)$, where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►

§9.9(iii) Derivatives With Respect to $k$

… ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

7: Bibliography C

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Ellipsoidal distributions of charge or mass. J. Mathematical Phys. 2, pp. 441–450.

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External Links:

MathReview (H. Bremekamp), ZentralBlatt

Referenced by:

§19.33(iv), §19.37(iv).

Encodings:

BibTeX

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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Calculation of angular distributions in complex angular momentum theories of elastic scattering. Molecular Physics 37 (6), pp. 1703–1712.

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External Links:

Document

Referenced by:

§14.31(iii).

Encodings:

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

BibTeX

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8: 1.17 Integral and Series Representations of the Dirac Delta

… ►In applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\delta \left(x\right)$. … ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

9: 10.73 Physical Applications

… ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … ►

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

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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:

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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

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

… ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …

10: 7.20 Mathematical Applications

… ►Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. … ►The normal distribution function with mean $m$ and standard deviation $\sigma$ is given by …For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).