# distributional derivative

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## 11—20 of 20 matching pages

##### 11: 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 $\frac{\mathrm{d}z}{\mathrm{d}\alpha}=-\ifrac{\frac{\partial f}{\partial\alpha}% }{\frac{\partial f}{\partial z}}.$
3.8.14 $\frac{\mathrm{d}z}{\mathrm{d}a_{j}}=-\frac{z^{j}}{f^{\prime}(z)}.$
3.8.16 $\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}=(-4.30\dots)\times 1% 0^{7}.$
##### 12: 20.13 Physical Applications
20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$
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). …
##### 13: Bibliography I
• Y. Ikebe (1975) The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.
• A. E. Ingham (1932) The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Cambridge University Press, Cambridge.
• ##### 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.
• 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).
• 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.
• 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).
• 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).
• ##### 15: 14.30 Spherical and Spheroidal Harmonics
14.30.3 $Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^{l}l!}\left(\frac{(l-m% )!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(\sin\theta\right)^{m}\*\left% (\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)^{l+m}\left(\sin\theta\right)% ^{2l}.$
###### Distributional Completeness
14.30.10 ${\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial W% }{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial W}{\partial\theta}\right)}+\frac{1}{\rho% ^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}W}{{\partial\phi}^{2}}=0,$
14.30.12 $\mathrm{L}^{2}=-\hbar^{2}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{{\sin}^% {2}\theta}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right),$
14.30.13 $\mathrm{L}_{z}=-\mathrm{i}\hbar\frac{\partial}{\partial\phi};$
##### 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).
• Harvard University (1945) Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.
• 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.
• G. W. Hill (1970) Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.
• 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.
• ##### 17: 31.15 Stieltjes Polynomials
31.15.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_{% j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\Phi(z)}{\prod_{j=1}^% {N}(z-a_{j})}w=0,$
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: 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.
• 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).
• R. S. Strichartz (1994) A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.
• 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.
• 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.
• ##### 19: 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.
• 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.
• 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.
• 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.
• 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.
• ##### 20: Errata
• Equations (10.15.1), (10.38.1)

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

• Subsection 1.16(vii)

Several changes have been made to

1. (i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$, $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. (ii)

introduce the partial differential operator $\mathbf{D}$ in (1.16.30);

3. (iii)

clarify the definition (1.16.32) of the partial differential operator $P(\mathbf{D})$; and

4. (iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

• Subsection 1.16(viii)

An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

• Equation (31.12.3)
31.12.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0$

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

Reported 2013-10-31 by Henryk Witek.

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

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

Reported 2010-04-02.