triangle conditions
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8 matching pages ♦
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8 matching pages
1: 34.10 Zeros
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►In a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero.
Similarly the symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four symbols in the summation.
…However, the and symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled.
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2: 34.2 Definition: Symbol
3: 34.3 Basic Properties: Symbol
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►Then assuming the triangle conditions are satisfied
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►Again it is assumed that in (34.3.7) the triangle conditions are satisfied.
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►In the following three equations it is assumed that the triangle conditions are satisfied by each symbol.
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4: 34.5 Basic Properties: Symbol
5: 18.37 Classical OP’s in Two or More Variables
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►The following three conditions, taken together, determine uniquely:
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§18.37(ii) OP’s on the Triangle
►Definition in Terms of Jacobi Polynomials
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18.37.7
, .
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18.37.8
and/or .
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6: 28.29 Definitions and Basic Properties
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►Given together with the condition (28.29.6), the solutions of (28.29.9) are the characteristic
exponents of (28.29.1).
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►For a given , the characteristic equation has infinitely many roots .
Conversely, for a given , the value of is needed for the computation of .
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28.29.16
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28.29.17
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7: 10.22 Integrals
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►(Thus if are the sides of a triangle, then is the area of the triangle.)
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►Sufficient conditions for the validity of (10.22.77) are that when , or that and when ; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62).
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►A sufficient condition for the validity is .
…Sufficient conditions for the validity of (10.22.79) are that when , or that and when ; see Titchmarsh (1962a, pp. 88–90).
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