misc module#
Created on Sat Aug 17 14:12:41 2024
@author: Pavan Pranjivan Mehta
- class misc.constant#
Bases:
objectSome popular / useful constants
- gamma_function()#
Computes Gamma (x), Euler Gamma function
Parameters#
- xfloat
x > 0
Returns#
- G: float
Gamma of (x)
- binomial_coefficient(k)#
Computes Binomial Coeffcient (n k) = n! / k! (n-k)!
Parameters#
n : int
k : int
Returns#
- B: int, float
B = n! / k! (n-k)!
- jacobi_constant(beta, degree)#
Computes Jacobi Constant, asscoaited with Jacobi Polynomials of nth - degree as a result of orthogonality
Parameters#
- alphafloat
Paramter of Jacobi polynomial, alpha > -1
- betafloat
Paramter of Jacobi polynomial, beta > -1
- degreeint
degree of Jacobi polynomial
Returns#
- C: float
Jacobi Orthogonality Constant
- beta_function(z2)#
Computes Beta function B(z1, z1) = Gamma (z1) * Gamma(z2) / Gamma (z1 + Z2), where, Gamma is the Euler-Gamma function
Parameters#
z1 : float
z2 : float
Returns#
- B: float
B(z1, z1) = Gamma (z1) * Gamma(z2) / Gamma (z1 + Z2)
- pochhammer(n)#
Computes the Pochammer symbol defined as,
(beta + 1)_n := 1, if n = 0
(beta + 1)_n := Product_i^n (beta + i), i = {1, 2, …, n}
Parameters#
beta : float
n : int
Returns#
- B: float
value of the pochehammer
- conv_othro_Qn_xk(n, k)#
Computes convolution orthogonality constant of Q-poly and x^k, with weight (1 - x)^lpha
Parameters#
- alphafloat
alpha > -1
n : int
k : int
Returns#
- C: float
value of the convolution orthogonality constant
Reference#
Allaway, W. R. (1989). Convolution Orthogonality and the Jacobi Polynominals. Canadian Mathematical Bulletin, 32(3), 298-308.
- conv_othro_Qn_Qm(n, m)#
Computes convolution orthogonality constant of Q-poly of n and m, degree, with weight (1 - x)^lpha
Parameters#
- alphafloat
alpha > -1
n : int
m : int
Returns#
- C: float
value of the convolution orthogonality constant
Reference#
Allaway, W. R. (1989). Convolution Orthogonality and the Jacobi Polynominals. Canadian Mathematical Bulletin, 32(3), 298-308.