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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 , and hence a solution of (31.2.1) that is analytic at all three finite singularities . …2: 34.10 Zeros
3: 31.1 Special Notation
4: 18.31 Bernstein–Szegő Polynomials
5: 19.38 Approximations
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 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard , 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 for exponent pairs. … ►These constants can be chosen to map any two sets of three distinct points and 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
9: 31.17 Physical Applications
§31.17(i) Addition of Three Quantum Spins
►The problem of adding three quantum spins , , and 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 (respective scalar ) for any one of the three spin operators (respective spin values). …10: 12.19 Tables
Zhang and Jin (1996, pp. 455–473) includes , , , , and derivatives, , , , 8S; , , and derivatives, , and , , , 8S. Also, first zeros of , , and of derivatives, , 6D; first three zeros of and of derivative, , 6D; first three zeros of and of derivative, , 6D; real and imaginary parts of , , , , , 8S.