# triangles

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##### 2: Sidebar 9.SB2: Interference Patterns in Caustics
The bright sharp-edged triangle is a caustic, that is a line of focused light. …
##### 3: 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. …
##### 4: 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 $n\neq\ell$.
##### 5: 34.2 Definition: $\mathit{3j}$ Symbol
They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions
##### 6: 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. …
##### 7: 18.1 Notation
• Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$.

• ##### 8: 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 $\mu_{n},\;n=1,2,3,\dots,\mbox{ with \bigtriangleup(\mu_{n})=-2}.$
##### 9: 23.22 Methods of Computation
• (a)

In the general case, given by $cd\neq 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^{3}-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$, $e_{2}$, $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$, then $e_{1}$, $e_{2}$, $e_{3}$ can be identified via (23.5.1), and $k^{2}$, ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$, or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$. In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$, ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^{2}|$, $|{k^{\prime}}^{2}|\leq 1$.

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$, $2\omega_{3}$ are given by

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 2 possible choices of the square root.

• (c)

If $c=0$, then

23.22.3 $2\omega_{1}=2e^{-\pi i/3}\omega_{3}=\frac{\left(\Gamma\left(\frac{1}{3}\right)% \right)^{3}}{2\pi d^{1/6}}.$

There are 6 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and $\omega_{1}>0$.

• ##### 10: 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. …