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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . …
2: 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. …
3: 31.1 Special Notation
Sometimes the parameters are suppressed.
4: 18.31 Bernstein–Szegő Polynomials
The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( 1 , 1 ) with respect to three types of weight function: ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x ) 1 2 ( 1 + x ) 1 2 ( ρ ( x ) ) 1 . …
5: 19.38 Approximations
Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …
6: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • 7: 15.11 Riemann’s Differential Equation
    §15.11(i) Equations with Three Singularities
    The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … Cases in which there are fewer than three singularities are included automatically by allowing the choice { 0 , 1 } for exponent pairs. … These constants can be chosen to map any two sets of three distinct points { α , β , γ } and { α ~ , β ~ , γ ~ } onto each other. …The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
    8: 27.19 Methods of Computation: Factorization
    Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. …
    9: 31.17 Physical Applications
    §31.17(i) Addition of Three Quantum Spins
    The problem of adding three quantum spins 𝐬 , 𝐭 , and 𝐮 can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. We use vector notation [ 𝐬 , 𝐭 , 𝐮 ] (respective scalar ( s , t , u ) ) for any one of the three spin operators (respective spin values). …
    10: 12.19 Tables
  • Zhang and Jin (1996, pp. 455–473) includes U ( ± n 1 2 , x ) , V ( ± n 1 2 , x ) , U ( ± ν 1 2 , x ) , V ( ± ν 1 2 , x ) , and derivatives, ν = n + 1 2 , n = 0 ( 1 ) 10 ( 10 ) 30 , x = 0.5 , 1 , 5 , 10 , 30 , 50 , 8S; W ( a , ± x ) , W ( a , ± x ) , and derivatives, a = h ( 1 ) 5 + h , x = 0.5 , 1 and a = h ( 1 ) 5 + h , x = 5 , h = 0 , 0.5 , 8S. Also, first zeros of U ( a , x ) , V ( a , x ) , and of derivatives, a = 6 ( .5 ) 1 , 6D; first three zeros of W ( a , x ) and of derivative, a = 0 ( .5 ) 4 , 6D; first three zeros of W ( a , ± x ) and of derivative, a = 0.5 ( .5 ) 5.5 , 6D; real and imaginary parts of U ( a , z ) , a = 1.5 ( 1 ) 1.5 , z = x + i y , x = 0.5 , 1 , 5 , 10 , y = 0 ( .5 ) 10 , 8S.