# boundary conditions and the Weyl alternative

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##### 2: 3.6 Linear Difference Equations
However, $w_{n}$ can be computed successfully in these circumstances by boundary-value methods, as follows. … For a difference equation of order $k$ ($\geq 3$), …or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. Typically $k-\ell$ conditions are prescribed at the beginning of the range, and $\ell$ conditions at the end. …
##### 3: 1.6 Vectors and Vector-Valued Functions
###### §1.6(ii) Vectors: Alternative Notations
Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … Suppose $S$ is an oriented surface with boundary $\,\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. …
##### 4: 32.11 Asymptotic Approximations for Real Variables
Next, for given initial conditions $w(0)=0$ and $w^{\prime}(0)=k$, with $k$ real, $w(x)$ has at least one pole on the real axis. … with boundary conditionand with boundary conditionAlternatively, if $\nu$ is not zero or a positive integer, then …
##### 5: Bibliography L
• D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.
• H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (1923) The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. Methuen and Co., Ltd., London.
• ##### 6: 32.2 Differential Equations
In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. …
##### 7: 34.2 Definition: $\mathit{3j}$ Symbol
They therefore satisfy the triangle conditionsIf either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $\mathit{3j}$ symbol is zero. When both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sum …
34.2.6 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},$
For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
##### 8: 1.10 Functions of a Complex Variable
Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup\partial D$. … (Or more generally, a simple contour that starts at the center and terminates on the boundary.) … Alternatively, take $z_{0}$ to be any point in $D$ and set $F(z_{0})={\mathrm{e}}^{\alpha\ln\left(1-z_{0}\right)}{\mathrm{e}}^{\beta\ln% \left(1+z_{0}\right)}$ where the logarithms assume their principal values. … This result is also true when $b=\infty$, or when $f(z,t)$ has a singularity at $t=b$, with the following conditions. … The last condition means that given $\epsilon$ ($>0$) there exists a number $a_{0}\in[a,b)$ that is independent of $z$ and is such that …
##### 9: 21.9 Integrable Equations
These parameters, including $\boldsymbol{{\Omega}}$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). …
##### 10: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions
The following expansion, with appropriate conditions and together with similar results, is given in Fields and Wimp (1961):
16.10.1 ${{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},\dots,c_{r}\atop b_{1},\dots,b% _{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty}\frac{{\left(\mathbf{% a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}(-z)^{k}}{{% \left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}k!}{{}_{p+2}F_{q+1}}% \left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_% {q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop% \alpha,\beta,d_{1},\dots,d_{s}};\zeta\right).$
16.10.2 ${{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};z\zeta\right% )=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}}{k!}{{}_{p+1% }F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left% (\frac{z}{z-1}\right)^{k}.$