# triangle conditions

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##### 1: 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. …
##### 2: 34.2 Definition: $\mathit{3j}$ Symbol
They therefore satisfy the triangle conditions
##### 3: 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. …
##### 4: 34.5 Basic Properties: $\mathit{6j}$ Symbol
In the following equation it is assumed that the triangle conditions are satisfied. …
##### 5: 18.37 Classical OP’s in Two or More Variables
The following three conditions, taken together, determine $R^{(\alpha)}_{m,n}\left(z\right)$ uniquely: …
###### 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 $n\neq\ell$.
##### 6: 28.29 Definitions and Basic Properties
Given $\lambda$ together with the condition (28.29.6), the solutions $\pm\nu$ of (28.29.9) are the characteristic exponents of (28.29.1). … 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 $\mu_{n},\;n=1,2,3,\dots,\mbox{ with \bigtriangleup(\mu_{n})=-2}.$
##### 7: 10.22 Integrals
(Thus if $a,b,c$ are the sides of a triangle, then $A^{\frac{1}{2}}$ is the area of the triangle.) … Sufficient conditions for the validity of (10.22.77) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $\nu\geq-\tfrac{1}{2}$, or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\,\mathrm{d}x<\infty$ when $-1<\nu<-\tfrac{1}{2}$; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62). … A sufficient condition for the validity is $\int_{a}^{\infty}|f(y)|\,\mathrm{d}y<\infty$. …Sufficient conditions for the validity of (10.22.79) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $0<\nu\leq\tfrac{1}{2}$, or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\frac{1}{2}-\nu}|f(x)|\,\mathrm{d}x<\infty$ when $\tfrac{1}{2}<\nu<1$; see Titchmarsh (1962a, pp. 88–90). …
##### 8: Bibliography G
• W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
• A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.