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Special_Functions.py
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Special_Functions.py
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"""
Special_Functions.py
The Special functions class is a set of special functions in the complex plane involving infinite products in the
complex plane. These products are related to the distribution of prime numbers and the Riemann Hypothesis. The
methods in this class are called by the file Complex_Plotting.py for complex the creation of complex plots.
4/9/2023
@LeoBorcherding
"""
import cmath
import math
import os
import pprint
import mpmath
import numpy as np
import matplotlib.pyplot as plt
import scipy
from scipy import special
from scipy import constants
import re
plt.style.use('dark_background')
# -----------------------------------------------------------------------------------------------------------------
class Special_Functions :
"""
Special_Functions is a class designed to organize the many complex functions that may want to be graphed with the
Complex_Plotting Class.
"""
# -----------------------------------------------------------------------------------------------------------------
def __init__(self, plot_type):
"""
Initialization the Special_Functions Object, defines self arg m = desired magnification exponent
"""
# Regex for identifying product symbols TODO Implement Summation Factory Function
self.pattern = r'([a-z]+)_\(([a-z]+)=([0-9]+)\)\^\(([a-z]+)\=([0-9]+)\)\s*\[(.*)\]'
return
# -----------------------------------------------------------------------------------------------------------------------
# Product of Sin(x) via Sin(x) product Representation.
def product_of_sin(self, z, Normalize_type):
"""
Computes the infinite product of sin(pi*z/n)
f(x) = ∏_(n=2)^x { abs ( sin(pi*x/n) ) ] ) }
Args:
z (complex): A complex number to evaluate.
Returns:
(float): The product of sin(pi*z/n).
"""
# initialize complex components
z_real = np.real(z)
z_imag = np.imag(z)
# initialize scaling coefficients
#TODO implement coefficent slider range of values which generates a folder of pngs each showcasing the given function with different values of m & beta by iterating over that range,
# we then will render multiple frames and combine to a gif or mp4 depending on files req or user input.
if Normalize_type == 'Y':
# self.m = 0.0125
# self.beta = 0.0125
# self.m = 0.005125
# self.beta = 0.006125
self.m = 0.0465
self.beta = 0.178
else:
# self.m = 0.0125
# self.m = 0.005125
# self.beta = 0.006125
self.m = 0.0465
self.beta = 0.178
# calculate infinite product
result = abs(np.prod(
[self.beta * ((z_real) / k) * np.sin(math.pi * (z_real + 1j * z_imag) / k)
for k in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def product_of_product_representation_for_sin(self, z, Normalize_type):
"""
Computes the product of the product representation for sin(pi*z/n).
f(x) = ∏_(k=2)^x (pi*x) ∏_(n=2)^x (1-(x^2)/(k^2)(n^2))
Args:
z (complex): A complex number to evaluate.
Returns:
(float): The product of the product representation for sin(pi*z/n).
"""
#pprint.pprint(z)
#TODO no magnification for bug testing
z_real = np.real(z)
z_imag = np.imag(z)
# pprint.pprint(z_real)
# pprint.pprint(z_imag)
#27 is a fun exponent to try
power_integer = 2
if Normalize_type == 'Y':
# self.m = 0.0125
# self.m = 0.37
# # m = 0.07
# # m = 0.37
# self.beta = 0.078
# self.m = 0.0125
self.m = 0.36
self.beta = 0.1468
else:
# self.m = 0.001
# # self.m = 0.007
# self.m = 0.0125
# # self.m = 0.37
# self.beta = 0.078
self.m = 0.0125
self.beta = 0.078
# calculate the double infinite product via the double for loop
result = abs(np.prod(
[self.beta * ( z_real / n) * (((z_real + 1j * z_imag) * math.pi) * np.prod(
[1 - ( (z_real + 1j * z_imag) ** 2 ) / ((n ** 2) * (k ** 2))
for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
# # print function values for bug testing
# if (z_real % 1 == 0) and (z_imag % 1 == 0):
# print(z, ": ", result)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def complex_playground_magnification_currated_functions_DEMO(self, z, Normalize_type):
""" A playground for fine-tuning the complex magnification for each product series.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
# # check for imaginary magnification
# if im_mag is True:
# imaginary_magnification = z_imag
# else:
# imaginary_magnification = 1
#TODO === PRODUCT OF PRODUCT REPRESENTATION OF SIN ===
# if Normalize_type == 'Y':
# self.m = 0.36
# self.beta = 0.1468
# else:
# self.m = 0.0125
# self.beta = 0.078
#TODO === RIESZ PRODUCT OF COS ===
if Normalize_type == 'Y':
self.m = 0.0125
self.beta = 0.054
else:
self.m = 0.0125
self.beta = 0.054
#TODO === Product OF SIN ===
# result = abs(np.prod(
# [self.beta * ((z_real) / k) * np.sin(math.pi * (z_real + 1j * z_imag) / k)
# for k in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO === PRODUCT OF PRODUCT REPRESENTATION OF SIN ===
# result = abs(np.prod(
# [self.beta * ( z_real / n) * (((z_real + 1j * z_imag) * math.pi) * np.prod(
# [ 1 - pow((( (z_real + 1j * z_imag) ** 2 ) / ( (n ** 2) * (k ** 2) )), (z_real + 1j * z_imag))
# for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO === PRODUCT OF PRODUCT REPRESENTATION OF SIN 2 ===
# result = abs(np.prod(
# [self.beta * ( z_real / n) * (((z_real + 1j * z_imag) * math.pi) * np.prod(
# [ 1 - ( (z_real + 1j * z_imag) ** 2 ) / ( (n ** 2) * (k ** 2) )
# for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO === RIESZ PRODUCT OF COS ===
# result = abs(np.prod(
# [pow((1j * z_imag + np.cos(math.pi * (z_real + 1j * z_imag) * n)), 1j * z_imag)
# for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO === RIESZ PRODUCT OF COS 2 ===
# result = abs(np.prod(
# [pow(((z_real + 1j * z_imag) + (z_real + 1j * z_imag) * np.cos(math.pi * (z_real + 1j * z_imag) * n)), (z_real + 1j * z_imag))
# for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO === RIESZ PRODUCT OF COS 3 ===
result = abs(np.prod(
[pow((1j * z_imag + (1j * z_imag) * np.sin(math.pi * (z_real + 1j * z_imag) * n)), 1j * z_imag)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def cos_of_product_of_sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# normalized double infinite product for loop
result = np.cos(abs(np.prod([self.beta * (z_real / n) * np.sin(math.pi * (z_real + 1j * z_imag) / n)
for n in range(2, int(z_real) + 1)])) ** (-self.m))
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def sin_of_product_of_sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# normalized double infinite product for loop
result = np.sin(abs(np.prod([self.beta * (z_real / n) * np.sin(math.pi * (z_real + 1j * z_imag) / n)
for n in range(2, int(z_real) + 1)])) ** (-self.m))
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def cos_of_product_of_product_representation_of_sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# normalized double infinite product for loop
result = np.cos(abs(np.prod([self.beta * (z_real / n) * (
(z_real * math.pi + 1j * z_imag * math.pi) * np.prod(
[1 - ((z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2) for k in range(2, int(z_real) + 1)])) for n in
range(2, int(z_real) + 1)]))) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def sin_of_product_of_product_representation_of_sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# normalized double infinite product for loop
result = np.sin(abs(np.prod([self.beta * (z_real / n) * (
(z_real * math.pi + 1j * z_imag * math.pi) * np.prod(
[1 - ((z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2) for k in range(2, int(z_real) + 1)])) for n in
range(2, int(z_real) + 1)]))) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Riesz_Product_for_Cos(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.0125
self.beta = 0.054
else:
self.m = 0.0125
self.beta = 0.054
# # check for imaginary magnification
# if im_mag is True:
# imaginary_magnification = z_imag
# else:
# imaginary_magnification = 1
# calculate infinite product
result = abs(np.prod(
[pow((1j * z_imag + np.cos(math.pi * (z_real + 1j * z_imag) * n)), 1j * z_imag)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Riesz_Product_for_Sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# calculate infinite product
result = abs(np.prod(
[1 + np.sin(math.pi * (z_real + 1j * z_imag) * n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Riesz_Product_for_Tan(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
# m = 0.26
# m = 0.08
if Normalize_type == 'Y':
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
else:
# self.m = 1
# self.beta = 1
self.m = 0.0125
self.beta = 0.054
# calculate infinite product
result = abs(np.prod(
[1 + np.tan(math.pi * (z_real + 1j * z_imag) * n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Viete_Product_for_Cos(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.27
self.beta = 1
else:
self.m = 0.07
self.beta = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
result = abs(np.prod(
[np.cos(math.pi * (z_real + 1j * z_imag) / (2 ** (n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO add conditional statement for normalization of the function,
# if user norm yes then return norm else return num
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Viete_Product_for_Sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.87
self.beta = 0.4
else:
self.m = 0.87
self.beta = 1
# check for imaginary magnification
# if self.im_mag is True:
# imaginary_magnification = z_imag
# else:
# imaginary_magnification = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
result = abs(np.prod(
[np.sin(math.pi * (z_real + 1j * z_imag) / (2 ** (n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Viete_Product_for_Tan(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.004
self.beta = 0.004
else:
self.m = 0.007
self.beta = 0.004
# # check for imaginary magnification
# if self.im_mag is True:
# imaginary_magnification = z_imag
# else:
# imaginary_magnification = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
result = abs(np.prod(
[np.tan(math.pi * (z_real + 1j * z_imag) / (2 ** (n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
#TODO add conditional statement for normalization of the function,
# if user norm yes then return norm else return num
return result
# -----------------------------------------------------------------------------------------------------------------
def Half_Base_Viete_Product_for_Sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.37
self.beta = 0.07
else:
self.m = 0.07
self.beta = 1
# # check for imaginary magnification
# if self.im_mag is True:
# imaginary_magnification = z_imag
# else:
# imaginary_magnification = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
result = abs(np.prod(
[np.sin(math.pi * (z_real + 1j * z_imag) / (2 ** (-n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
#TODO add conditional statement for normalization of the function,
# if user norm yes then return norm else return num
return result
# -----------------------------------------------------------------------------------------------------------------
def Riesz_Product_for_Tan_and_Prime_indicator_combination(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
m = 0.26
q = 0.08
if Normalize_type == 'Y':
self.m = 0.001
self.beta = 0.07
else:
self.m = 0.001
self.beta = 0.07
# calculate infinite product
num = abs(np.prod(
[1 + np.tan(math.pi * (z_real + 1j * z_imag) * n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
FUNC_B = abs(np.prod(
[(1 * z_real / n) * ((z_real * math.pi + 1j * z_imag * math.pi) * np.prod(
[1 - ( (z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2)
for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-q)
norm = num / scipy.special.gamma(num)
FUNC_B_norm = FUNC_B / scipy.special.gamma(FUNC_B)
# num_exp = FUNC_B_norm ** norm
# num_exp = norm ** FUNC_B_norm
# num_exp = norm/FUNC_B_norm
num_exp = np.cos(FUNC_B_norm/norm)
return num_exp
# -----------------------------------------------------------------------------------------------------------------
def Nested_roots_product_for_2(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
# m = 0.26
# q = 0.08
m = -1
if Normalize_type == 'Y':
self.m = 0.001
self.beta = 0.07
else:
self.m = 0.001
self.beta = 0.07
# #TODO Sqrt Sum
# sum = abs(np.prod(
# [ (z_real + 1j * z_imag) ** 2 ** (-n)
# for n in range(2, int(z_real) + 1)])) ** (-m)
# #TODO Sqrt Prod
# prod = abs(np.sum(
# [ (z_real + 1j * z_imag) ** 2 ** (-n)
# for n in range(2, int(z_real) + 1)])) ** (-m)
# #TODO Viete's paradox nest roots sum product representation
# paradox = abs(np.prod(
# [np.sum([( k ** (z_real + 1j * z_imag) ** (-n)) / 2
# for k in range(2, int(z_real) + 1)]) for n in range(2, int(z_real) + 1)]))
#TODO Sqrt Prod
prod = abs(np.sum(
[ (z_real + 1j * z_imag) ** 2 ** (-n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
paradox = abs(np.prod(
[np.prod([(z_real + 1j * z_imag) ** k ** (-n)
for k in range(2, int(z_real) + 1)]) for n in range(2, int(z_real) + 1)]))
# norm = num / scipy.special.gamma(num)
# FUNC_B_norm = FUNC_B / scipy.special.gamma(FUNC_B)
# num_exp = FUNC_B_norm ** norm
# num_exp = norm ** FUNC_B_norm
# num_exp = norm/FUNC_B_norm
# num_exp = np.cos(FUNC_B_norm/norm)
return paradox
# -----------------------------------------------------------------------------------------------------------------
def natural_logarithm_of_product_of_product_representation_for_sin(self, z, Normalize_type):
"""
Computes the natural logarithm of the normalized infinite product of product representation of sin(z).
f(x) = ln ( ∏_(n=2)^x (pi*x) ∏_(n=2)^x (1-(x^2)/(i^2)(n^2)) )
Args:
z (complex): A complex number to evaluate.
m (float): A constant value for the result.
Returns:
(complex): The natural logarithm of the normalized infinite product of product representation of sin(z).
"""
z_real = np.real(z)
z_imag = np.imag(z)
# check for imaginary magnification
if self.im_mag is True:
imaginary_magnification = z_imag
else:
imaginary_magnification = 1
# calculate the infinite product
result = abs(np.prod(
[self.beta * (z_real * imaginary_magnification / n) * ((z_real * math.pi + 1j * z_imag * math.pi) * np.prod(
[1 - ((z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2)
for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
# take the logarithm
log_result = cmath.log(result)
#log_result=cmath.log(result) # scipy.special.gamma(
#return result / np.log(z)
# if Normalize_type == 'Y':
# result = result / scipy.special.gamma(result)
# else:
# result = result
return log_result
# -----------------------------------------------------------------------------------------------------------------
def gamma_of_product_of_product_representation_for_sin(self, z, Normalize_type):
"""
Args:
Returns:
"""
z_real = np.real(z)
z_imag = np.imag(z)
# check for imaginary magnification
if self.im_mag is True:
imaginary_magnification = z_imag
else:
imaginary_magnification = 1
result = abs(np.prod(
[self.beta * (z_real * imaginary_magnification / n) * ((z_real * math.pi + 1j * z_imag * math.pi) * np.prod(
[1 - ((z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2)
for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
#TODO FUNCTION TRANSFORMATION SELECTOR
# right now there are multiple methods for what is a graphing technique of transforming the product with logs and factorials.
gamma_result = scipy.special.gamma(result)
#log_result=cmath.log(result) # scipy.special.gamma(
# if Normalize_type == 'Y':
# result = result / scipy.special.gamma(result)
# else:
# result = result
return gamma_result
# -----------------------------------------------------------------------------------------------------------------
def gamma_form_product_of_product_representation_for_sin(self, z, Normalize_type):
"""
Computes the product of the product representation for sin(z).
z / [Γ(z) * Γ(1-z)] * e^(-γz^2) * ∏_(n=2)^(∞) e^(z^2 / n^2)
Args:
z (complex): A complex number to evaluate.
m (float): A constant value for the result.
Returns:
(float): The product of the product representation for sin(z).
"""
if Normalize_type == 'Y':
self.m = 0.0125
self.beta = 0.054
else:
self.m = 0.0125
self.beta = 0.054
Euler_mascheroni = 0.57721566490153286060651209008240243104215933593992359880
z_real = np.real(z)
z_imag = np.imag(z)
result = abs(np.prod(
[self.beta * (z_imag * z_real / n) * scipy.special.gamma(z_real + 1j * z_imag)
* scipy.special.gamma(1 - (z_real + 1j * z_imag))
* cmath.exp((-((z_real + 1j * z_imag) ** 2))) * np.prod(
[(cmath.exp(((z_real + 1j * z_imag)**2/(n**2)*(k**2))))
for k in range(2, int(z_real) + 1)]) for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
result = result / scipy.special.gamma(result)
else:
result = result
return result
# -----------------------------------------------------------------------------------------------------------------
def Log_power_base_Viete_Product_for_Sin(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.001
self.beta = 0.07
else:
self.m = 0.001
self.beta = 0.07
# check for imaginary magnification
if self.im_mag is True:
imaginary_magnification = z_imag
else:
imaginary_magnification = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
num = 1 + abs(np.prod(
[np.sin(math.pi * (z_real + 1j * z_imag) / (2 ** ((1/n)*np.log((z_real + 1j * z_imag)))))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
den = 1 + abs(np.prod(
[np.sin(math.pi * (z_real + 1j * z_imag) / (2 ** ((1/n)*np.log((z_real + 1j * z_imag)))))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
#THis doesnt really need to start at 2, it could start at 1
norm = num / scipy.special.gamma(den)
#TODO add conditional statement for normalization of the function,
# if user norm yes then return norm else return num
return norm
# -----------------------------------------------------------------------------------------------------------------
def Custom_Riesz_Product_for_Tan(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.001
self.beta = 0.07
else:
self.m = 0.001
self.beta = 0.07
# check for imaginary magnification
if self.im_mag is True:
imaginary_magnification = z_imag
else:
imaginary_magnification = 1
# calculate infinite product
num = abs(np.prod(
[1 + np.tan(math.pi * (z_real + 1j * z_imag) * n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
den = abs(np.prod(
[1 + np.tan(math.pi * (z_real + 1j * z_imag) * n)
for n in range(2, int(z_real) + 1)])) ** (-self.m)
norm = num / scipy.special.gamma(den)
return norm
# -----------------------------------------------------------------------------------------------------------------
def Custom_Viete_Product_for_Cos(self, z, Normalize_type):
""" A method to multiply two infinite products.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.m = 0.001
self.beta = 0.07
else:
self.m = 0.001
self.beta = 0.07
# check for imaginary magnification
if self.im_mag is True:
imaginary_magnification = z_imag
else:
imaginary_magnification = 1
#TODO different values for the base of the denominator try 2, 3, 4, 1/2, phi, pi, e, euler-mascheroni
# scipy.constants.golden
# calculate infinite product
num = abs(np.prod(
[1 + np.cos(math.pi * (z_real + 1j * z_imag) / (z_real ** (n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
den = abs(np.prod(
[1 + np.cos(math.pi * (z_real + 1j * z_imag) / (z_real ** (n)))
for n in range(2, int(z_real) + 1)])) ** (-self.m)
#THis doesnt really need to start at 2, it could start at 1
norm = num / scipy.special.gamma(den)
#TODO add conditional statement for normalization of the function,
# if user norm yes then return norm else return num
return norm
# -----------------------------------------------------------------------------------------------------------------
def Binary_Output_Prime_Indicator_Function_H(self, z, Normalize_type):
""" A method to combine two infinite products to produce a prime binary output.
Args:
z: Complex z which isn't converted to z_real & z_imag in this method
beta: magnification value as the lead coefficient
m: exponential magnification coefficient
"""
z_real = np.real(z)
z_imag = np.imag(z)
if Normalize_type == 'Y':
self.c = 0.13
self.m = 0.29
self.alpha = 0.14
self.beta = 0.25
else:
self.c = 0.83
self.m = 0.029
self.alpha = 0.74
self.beta = 0.025
# calculate infinite product
single_prod = abs(np.prod(
[self.alpha * ((z_real) / k) * np.sin(math.pi * (z_real + 1j * z_imag) / k)
for k in range(2, int(z_real) + 1)])) ** (-self.c)
# calculate the double infinite product via the double for loop
double_prod = abs(np.prod(
[self.beta * ( z_real / n) * (((z_real + 1j * z_imag) * math.pi) * np.prod(
[1 - ( (z_real + 1j * z_imag) ** 2) / (n ** 2 * k ** 2)
for k in range(2, int(z_real) + 1)])) for n in range(2, int(z_real) + 1)])) ** (-self.m)
if Normalize_type == 'Y':
norm1 = single_prod / scipy.special.gamma(single_prod)
norm2 = double_prod / scipy.special.gamma(double_prod)
result = pow(norm1, norm2)
else:
result = pow(single_prod, double_prod)
return result