orthpoly
http://www.openxm.org/...
2003-08-11
2002-08-11, 2003-11-30
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experimental
arith1
relation1
calculus1
alg1
interval1
nums1
hypergeo0
hypergeo1
This CD defines orthogonal polynomials which are hypergeometric polynomials.
These functions are described in the following books.
(1) Handbook of Mathematical Functions, Abramowitz, Stegun
(2) Higher transcendental functions. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G.
legendreP
The first Legendre function.
This function is one of the two famous solutions of Legendre
differential equation.
Binary
legendreP(v;z) = hypergeo1.hypergeometric2F1(-v,v+1,1;(1-z)/2)
orthpoly1.legendreP(v,z) ~relation1.eq~
hypergeo1.hypergeometric2F1(arith1.unary_minus(v),
v ~arith1.plus~ 1, 1, 1 ~arith1.minus~ z ~arith1.divide~ 2);
legendreQ
The second Legendre function.
This function is the another one of the famous two solutions of Legendre
differential equation.
Binary
legendreQ(v;z) = \frac{\sqrt{\pi}\Gamma(v+1)}{\Gamma(v+3/2)}
/(2z)^{v+1}
hypergeo1.hypergeometric2F1((v+1)/2,v/2+1,v+3/2;1/z^2)
orthpoly1.legendreQ(v,z) ~relation1.eq~
( arith1.root(nums1.pi,2) ~arith1.times~
hypergeo0.gamma(v ~arith1.plus~ 1) ~arith1.divide~
hypergeo0.gamma(v ~arith1.plus~ (3 ~arith1.divide~ 2))
~arith1.divide~ (2 ~arith1.times~ z ~arith1.power~
(v ~arith1.plus~ 1)) ~arith1.times~
hypergeo1.hypergeometric2F1(v ~arith1.plus~ 1 ~arith1.divide~ 2,
v ~arith1.divide~ 2 ~arith1.plus~ 1,
v ~arith1.plus~ (3 ~arith1.divide~ 2),
1 ~arith1.divide~ z ~arith1.power~ 2));
jacobiG
The Jacobi polynomial.
4ary
jacobiG(n,a,c;z)
= hypergeometric2F1(-n,a+n,c,z) (c \not\in Z_{<=0})
arith1.unary_minus(c) ~set1.notin~ setname1.N ~logic1.implies~
(orthpoly1.jacobiG(n,a,c,z) ~relation1.eq~
hypergeo1.hypergeometric2F1(arith1.unary_minus(n), a ~arith1.plus~ n, c, z));