# distributional completeness

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## 1—10 of 16 matching pages

##### 1: 14.30 Spherical and Spheroidal Harmonics

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###### Distributional Completeness

…##### 2: 29.3 Definitions and Basic Properties

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►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 $4K$.
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###### §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 $[0,2K)$. … ►
${\int}_{0}^{K}}\mathrm{dn}(x,k){\left({\mathit{Ec}}_{\nu}^{2m}(x,{k}^{2})\right)}^{2}dx={\displaystyle \frac{1}{4}}\pi ,$

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►To complete the definitions, ${\mathit{Ec}}_{\nu}^{m}(K,{k}^{2})$ is positive and ${d{\mathit{Es}}_{\nu}^{m}(z,{k}^{2})/dz|}_{z=K}$ is negative.
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##### 3: 27.2 Functions

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27.2.1
$$n=\prod _{r=1}^{\nu \left(n\right)}{p}_{r}^{{a}_{r}},$$

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►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.
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27.2.3
$$\pi \left(x\right)\sim \frac{x}{\mathrm{ln}x}.$$

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##### 4: 22.4 Periods, Poles, and Zeros

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###### §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 $0$, $2K$, $2K+2\mathrm{i}{K}^{\prime}$, $2\mathrm{i}{K}^{\prime}$. … ►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 $\text{s}=0$, $\text{c}=K$, $\text{d}=K+\mathrm{i}{K}^{\prime}$, $\text{n}=\mathrm{i}{K}^{\prime}$. The set of points $z=mK+n\mathrm{i}{K}^{\prime}$, $m,n\in \mathbb{Z}$, comprise the*lattice*for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime}$, where again $m,n\in \mathbb{Z}$. …##### 5: Bibliography J

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Tafeln höherer Funktionen (Tables of Higher Functions).
7th edition, B. G. Teubner, Stuttgart (Bilingual).
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Distributions of matrix variates and latent roots derived from normal samples.
Ann. Math. Statist. 35 (2), pp. 475–501.
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Continuous Univariate Distributions.
2nd edition, Vol. I, John Wiley & Sons Inc., New York.
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Continuous Univariate Distributions.
2nd edition, Vol. II, John Wiley & Sons Inc., New York.
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##### 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. … ►###### §2.6(iv) Regularization

…##### 7: Philip J. Davis

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►He returned to Harvard after the war and completed a Ph.
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►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.
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►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.
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##### 8: DLMF Project News

*error generating summary*

##### 9: 28.12 Definitions and Basic Properties

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►To complete the definition we require
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►To complete the definitions of the ${\mathrm{me}}_{\nu}$ functions we set
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##### 10: 1.16 Distributions

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1.16.30
$$\mathbf{D}=(\frac{1}{\mathrm{i}}\frac{\partial}{\partial {x}_{1}},\frac{1}{\mathrm{i}}\frac{\partial}{\partial {x}_{2}},\mathrm{\dots},\frac{1}{\mathrm{i}}\frac{\partial}{\partial {x}_{n}}).$$

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