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1: 4.42 Solution of Triangles
§4.42 Solution of Triangles
§4.42(i) Planar Right Triangles
§4.42(ii) Planar Triangles
See accompanying text
Figure 4.42.2: Planar triangle. Magnify
§4.42(iii) Spherical Triangles
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 3 j 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 3 j symbol is zero. Similarly the 6 j symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four 3 j symbols in the summation. …However, the 3 j and 6 j 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
§18.37(ii) OP’s on the Triangle
Definition in Terms of Jacobi Polynomials
18.37.7 P m , n α , β , γ ( x , y ) = P m - n ( α , β + γ + 2 n + 1 ) ( 2 x - 1 ) x n P n ( β , γ ) ( 2 x - 1 y - 1 ) , m n 0 , α , β , γ > - 1 .
18.37.8 0 < y < x < 1 P m , n α , β , γ ( x , y ) P j , α , β , γ ( x , y ) ( 1 - x ) α ( x - y ) β y γ d x d y = 0 , m j and/or n .
5: 34.2 Definition: 3 j 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 z 0 conformally onto a curvilinear triangle. …
7: 18.1 Notation
  • Triangle: P m , n α , β , γ ( x , y ) .

  • 8: 28.29 Definitions and Basic Properties
    For a given ν , the characteristic equation ( λ ) - 2 cos ( π ν ) = 0 has infinitely many roots λ . Conversely, for a given λ , the value of ( λ ) is needed for the computation of ν . …
    28.29.16 λ n , n = 0 , 1 , 2 , ,  with  ( λ n ) = 2 ,
    28.29.17 μ n , n = 1 , 2 , 3 , ,  with  ( μ n ) = - 2 .
    9: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 - c t - 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 - 27 d 2 > 0 , then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

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

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = - 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 - k 2 ) d ( 1 - k 2 k 2 ) ,

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

  • (c)

    If c = 0 , then

    23.22.3 2 ω 1 = 2 e - π i / 3 ω 3 = ( Γ ( 1 3 ) ) 3 2 π d 1 / 6 .

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

  • 10: 34.3 Basic Properties: 3 j 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 3 j symbol. …