# finite

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## 1—10 of 99 matching pages

##### 1: 16.9 Zeros
Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many zeros if and only if the $a_{j}$ can be re-indexed for $j=1,\dots,p$ in such a way that $a_{j}-b_{j}$ is a nonnegative integer. … Then ${{}_{p}F_{p}}\left(\mathbf{a};\mathbf{b};z\right)$ has at most finitely many real zeros. …
##### 2: 36.15 Methods of Computation
###### §36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\Phi$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
##### 3: Daniel W. Lozier
Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …
##### 4: 34.6 Definition: $\mathit{9j}$ Symbol
The $\mathit{9j}$ symbol may be defined either in terms of $\mathit{3j}$ symbols or equivalently in terms of $\mathit{6j}$ symbols:
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
The $\mathit{9j}$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
##### 5: 29.20 Methods of Computation
A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $\mathbf{M}$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
##### 6: 5.16 Sums
For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
##### 7: 29.19 Physical Applications
Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …
##### 8: 27.10 Periodic Number-Theoretic Functions
Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form … is a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion … The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 9: 26.1 Special Notation
 $x$ real variable. … number of elements of a finite set $A$. …
##### 10: 15.15 Sums
For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …