# distributional completeness

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##### 2: 29.3 Definitions and Basic Properties
For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. …
###### §29.3(ii) Distribution
In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. …
$\int_{0}^{K}\operatorname{dn}\left(x,k\right)\left(\mathit{Ec}^{2m}_{\nu}\left% (x,k^{2}\right)\right)^{2}\mathrm{d}x=\frac{1}{4}\pi,$
To complete the definitions, $\mathit{Ec}^{m}_{\nu}\left(K,k^{2}\right)$ is positive and $\left.\ifrac{\mathrm{d}\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)}{\mathrm{d}z}% \right|_{z=K}$ is negative. …
##### 3: 27.2 Functions
Tables of primes (§27.21) reveal great irregularity in their distribution. They tend to thin out among the large integers, but this thinning out is not completely regular. …
27.2.3 $\pi\left(x\right)\sim\frac{x}{\ln x}.$
##### 4: 22.4 Periods, Poles, and Zeros
###### §22.4(i) Distribution
Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2iK^{\prime}$, $2iK^{\prime}$. … For the distribution of the $k$-zeros of the Jacobian elliptic functions see Walker (2009). … Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\mbox{s}=0$, $\mbox{c}=K$, $\mbox{d}=K+iK^{\prime}$, $\mbox{n}=iK^{\prime}$. The set of points $z=mK+niK^{\prime}$, $m,n\in\mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+niK^{\prime}$, where again $m,n\in\mathbb{Z}$. …
##### 5: Bibliography J
• E. Jahnke, F. Emde, and F. Lösch (1966) Tafeln höherer Funktionen (Tables of Higher Functions). 7th edition, B. G. Teubner, Stuttgart (Bilingual).
• A. T. James (1964) Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.
• N. L. Johnson, S. Kotz, and N. Balakrishnan (1994) Continuous Univariate Distributions. 2nd edition, Vol. I, John Wiley & Sons Inc., New York.
• N. L. Johnson, S. Kotz, and N. Balakrishnan (1995) Continuous Univariate Distributions. 2nd edition, Vol. II, John Wiley & Sons Inc., New York.
• ##### 6: 2.6 Distributional Methods
###### §2.6 Distributional Methods
Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … The Dirac delta distribution in (2.6.17) is given by … These equations again hold only in the sense of distributions. …
##### 7: Philip J. Davis
He returned to Harvard after the war and completed a Ph. … He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010. …
##### 8: DLMF Project News
error generating summary
##### 9: 28.12 Definitions and Basic Properties
To complete the definition we require … To complete the definitions of the $\mathrm{me}_{\nu}$ functions we set …
##### 10: 1.16 Distributions
1.16.30 $\mathbf{D}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{1}},\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{2}},\ldots,\frac{1}{\mathrm{i}}\frac{% \partial}{\partial x_{n}}\right).$