solution of triangles and spherical triangles

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2: 15.17 Mathematical Applications
The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\Im z\geq 0$ conformally onto a curvilinear triangle. …
3: Sidebar 9.SB2: Interference Patterns in Caustics
The bright sharp-edged triangle is a caustic, that is a line of focused light. …
4: 34.10 Zeros
In a $\mathit{3j}$ symbol, if the three angular momenta $j_{1},j_{2},j_{3}$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3j}$ symbol is zero. Similarly the $\mathit{6j}$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3j}$ symbols in the summation. …However, the $\mathit{3j}$ and $\mathit{6j}$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …
5: 28.29 Definitions and Basic Properties
28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$
For a given $\nu$, the characteristic equation $\bigtriangleup(\lambda)-2\cos\left(\pi\nu\right)=0$ has infinitely many roots $\lambda$. Conversely, for a given $\lambda$, the value of $\bigtriangleup(\lambda)$ is needed for the computation of $\nu$. …
28.29.16 $\lambda_{n},\;n=0,1,2,\dots,\mbox{ with \bigtriangleup(\lambda_{n})=2},$
28.29.17 $\mu_{n},\;n=1,2,3,\dots,\mbox{ with \bigtriangleup(\mu_{n})=-2}.$
6: 18.37 Classical OP’s in Two or More Variables
Definition in Terms of Jacobi Polynomials
18.37.7 $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)=P^{(\alpha,\beta+\gamma+2n+1)}_{% m-n}\left(2x-1\right)\*x^{n}P^{(\beta,\gamma)}_{n}\left(2x^{-1}y-1\right),$ $m\geq n\geq 0$, $\alpha,\beta,\gamma>-1$.
18.37.8 $\iint\limits_{0 $m\neq j$ and/or $n\neq\ell$.
7: 18.1 Notation
• Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$.

• 8: 34.3 Basic Properties: $\mathit{3j}$ Symbol
Then assuming the triangle conditions are satisfied … Again it is assumed that in (34.3.7) the triangle conditions are satisfied. … In the following three equations it is assumed that the triangle conditions are satisfied by each $\mathit{3j}$ symbol. …
9: 10.47 Definitions and Basic Properties
§10.47(iii) Numerically Satisfactory Pairs of Solutions
For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $\mathsf{j}$, $\mathsf{y}$, $\mathsf{h}$, and $n$, respectively. For (10.47.2) numerically satisfactory pairs of solutions are ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(z\right)$ in the right half of the $z$-plane, and ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ and $\mathsf{k}_{n}\left(-z\right)$ in the left half of the $z$-plane. …
10: 34.2 Definition: $\mathit{3j}$ Symbol
They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions