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solution of triangles and spherical triangles

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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
§4.42(iii) Spherical Triangles
See accompanying text
Figure 4.42.3: Spherical triangle. Magnify
2: 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. …
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 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. …
5: 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 .
6: 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 .
7: 18.1 Notation
  • Triangle: P m , n α , β , γ ( x , y ) .

  • 8: 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. …
    9: 10.47 Definitions and Basic Properties
    §10.47(ii) Standard Solutions
    §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 ν replaced by j , y , h , and n , respectively. For (10.47.2) numerically satisfactory pairs of solutions are i n ( 1 ) ( z ) and k n ( z ) in the right half of the z -plane, and i n ( 1 ) ( z ) and k n ( - z ) in the left half of the z -plane. …
    10: 34.2 Definition: 3 j Symbol
    They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions