hypergeo2 http://www.openxm.org/... 2003-08-11 2002-08-11, 2003-11-30 0 1 experimental arith1 relation1 calculus1 alg1 interval1 nums1 hypergeo0 hypergeo1 This CD defines some famous hypergeometric functions such as Bessel functions and Airy functions. 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. kummer Kummer's hypergeometric function. kummer(a,c;z) = hypergeo1.hypergeometric1F1(a,c;z) hypergeo2.kummer(a,c,z) ~relation1.eq~ hypergeo1.hypergeometric1F1(a,c,z); besselJ The Bessel function. This function is one of the famous two solutions of the Bessel differential equation at z=0. besselJ(v,z) = (\frac{z}{2})^v \sum_{n=0}^{+\infty} \frac{(-1)^n}{n! \Gamma(v+n+1)} (\frac{z}{2})^2n hypergeo2.besselJ(v,z) ~relation1.eq~ (z ~arith1.divide~ 2 ~arith1.power~ v ~arith1.times~ arith1.sum(interval1.integer_interval(0,infty), OMBIND(fns1.lambda,OMBVAR(n), arith1.unary_minus(1) ~arith1.power~ n ~arith1.divide~ (integer1.factorial(n) ~arith1.times~ hypergeo0.gamma(v ~arith1.plus~ n ~arith1.plus~ 1)) ~arith1.times~ arith1.power(z ~arith1.divide~ 2,2 ~arith1.times~ n)))); z ~arith1.power~ 2 ~arith1.times~ calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), hypergeo2.besselJ(v,z))))) ~arith1.plus~ (z ~arith1.times~ calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), bypergeo2.besselJ(v,z)))) ~arith1.plus~ ((z ~arith1.power~ 2 ~arith1.minus~ (v ~arith1.power~ 2)) ~arith1.times~ besselJ(v,z)) ~relation1.eq~ 0; besselY The Bessel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=0. besselY(v,z) = (\cos(v \pi) besselJ(v,z) - besselJ(-v,z))/\sin(v \pi) hypergeo2.besselY(v,z) ~relation1.eq~ (transc1.cos(v ~arith1.times~ nums1.pi) ~arith1.times~ hypergeo2.besselJ(v,z) ~arith1.minus~ besselJ(arith1.unary_minus(v),z) ~arith1.divide~ transc1.sin(v ~arith1.times~ nums1.pi)); z ~arith1.power~ 2 ~arith1.times~ calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), hypergeo2.besselY(v,z))))) ~arith1.plus~ (z ~arith1.times~ calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), hypergeo2.besselY(v,z)))) ~arith1.plus~ ((z ~arith1.power~ 2 ~arith1.minus~ (v ~arith1.power~ 2)) ~arith1.times~ hypergeo2.besselY(v,z)) ~relation1.eq~ 0; hankel1 The first Hankel function. This function is one of the famous two solutions of the Bessel differential equation at z=\infty. hankel1(v,z) = besselJ(v,z) + i BesselY(v,z) hypergeo2.hankel1(v,z) ~relation1.eq~ (hypergeo2.besselJ(v,z) ~arith1.plus~ (nums1.i ~arith1.times~ hypergeo2.besselY(v,z))); hankel2 The second Hankel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=\infty. hankel2(v,z) = besselJ(v,z) - i BesselY(v,z) hypergeo2.hankel1(v,z) ~relation1.eq~ (hypergeo2.besselJ(v,z) ~arith1.minus~ (nums1.i ~arith1.times~ hypergeo2.besselY(v,z))); airyAi The first Airy function. This function is one of the famous two solutions of the Airy differential equation, and converges to 0 when z->\infty (\frac{d^2}{dz^2} - z) airyAi(z) = 0 calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), hypergeo2.airyAi(z))))) ~arith1.minus~ (z ~arith1.times~ hypergeo2.airyAi(z)) ~relation1.eq~ 0; airyBi The second Airy function. This function is the another one of the famous two solutions of the Airy differential equation, and diverges when z->\infty (\frac{d^2}{dz^2} - z) airyBi(z) = 0 calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), calculus1.diff(OMBIND(fns1.lambda,OMBVAR(z), hypergeo2.airyBi(z))))) ~arith1.minus~ (z ~arith1.times~ hypergeo2.airyBi(z)) ~relation1.eq~ 0;