# distributional derivative

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##### 1: 1.16 Distributions
###### §1.16(ii) Derivatives of a Distribution
For any locally integrable function $f$, its distributional derivative is $Df=\Lambda^{\prime}_{f}$. … 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: 1.1 Special Notation
 $x,y$ real variables. … distribution. … derivatives with respect to the variable, except where indicated otherwise.
##### 3: 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 $\mathbb{R}$ which vanishes on $(-\infty,0]$, and $D^{n}f$ denotes the $n$th derivative of the distribution associated with $f$. …
2.6.37 $F\ast G=D^{n+m}(f\ast g).$
2.6.40 $t^{\mu-1}\ast t^{-s-1}=\frac{(-1)^{s}}{\mu\cdot s!}D^{s+1}\left(t^{\mu}\left(% \ln t-\gamma-\psi\left(\mu+1\right)\right)\right),$ $t>0$,
Since the function $t^{\mu}\left(\ln t-\gamma-\psi\left(\mu+1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\infty)$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. …
##### 4: 10.21 Zeros
###### §10.21(ix) Complex Zeros
This subsection describes the distribution in $\mathbb{C}$ 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 $n$. …
##### 5: Bibliography W
• P. L. Walker (2007) The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.
• P. L. Walker (2009) The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.
• R. J. Wells (1999) Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.
• J. Wishart (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.
• ##### 6: 9.9 Zeros
###### §9.9(i) Distribution and Notation
They are denoted by $a_{k}$, $a^{\prime}_{k}$, $b_{k}$, $b^{\prime}_{k}$, respectively, arranged in ascending order of absolute value for $k=1,2,\ldots.$For the distribution in $\mathbb{C}$ of the zeros of $\mathrm{Ai}'\left(z\right)-\sigma\mathrm{Ai}\left(z\right)$, 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 $a_{k}$, $b_{k}$, $a^{\prime}_{k}$, and $b^{\prime}_{k}$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …
##### 7: Bibliography C
• B. C. Carlson (1961a) Ellipsoidal distributions of charge or mass. J. Mathematical Phys. 2, pp. 441–450.
• R. Chattamvelli and R. Shanmugam (1997) Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.
• B. K. Choudhury (1995) The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.
• J. N. L. Connor and D. C. Mackay (1979) Calculation of angular distributions in complex angular momentum theories of elastic scattering. Molecular Physics 37 (6), pp. 1703–1712.
• A. G. Constantine (1963) Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.
• ##### 8: 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 $\nabla^{2}V=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial V}{% \partial r}\right)+\frac{1}{r^{2}}\frac{{\partial}^{2}V}{{\partial\phi}^{2}}+% \frac{{\partial}^{2}V}{{\partial z}^{2}}=0,$
10.73.2 $\nabla^{2}\psi=\frac{1}{c^{2}}\frac{{\partial}^{2}\psi}{{\partial t}^{2}},$
10.73.3 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
The analysis of the current distribution in circular conductors leads to the Kelvin functions $\operatorname{ber}x$, $\operatorname{bei}x$, $\operatorname{ker}x$, and $\operatorname{kei}x$. …
##### 9: 7.20 Mathematical Applications
Furthermore, because $\ifrac{\mathrm{d}y}{\mathrm{d}x}=\tan\left(\frac{1}{2}\pi t^{2}\right)$, 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).
##### 10: 9.13 Generalized Airy Functions
9.13.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=z^{n}w,$ $n=1,2,3,\ldots$,
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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\tfrac{1}{4}m^{2}t^{m-2}w,$
9.13.17 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=-\tfrac{1}{4}m^{2}t^{m-2}w,$ $m$ even,
9.13.31 $\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-z\frac{\mathrm{d}w}{\mathrm{d}z}+(% p-1)w=0,$