def kepler_trig_expand(s):
"""
Transform trigonometric part of series s [sin(nM) and cos(nM)]
to sum of items like cos(M)^k1 * sin(M)^k2, where n=k1+k2.
- s is series of e and M,
- n is number of terms.
"""
from math import ceil
from pyranha.math import binomial, cos, sin, degree
temp, e, M, ecosM, esinM = 0, epst('e'), epst('M'), epst('ecosM'), epst(
'esinM')
n = degree(s).numerator
s_list = s.list
for i in range(len(s_list)):
for j in range(n + 1):
trig_cos, trig_sin = 0, 0
# calculate cos(nM):
if s_list[i][1] == cos(j * M):
for k in range(int(ceil(j / 2)) + 1):
trig_cos = trig_cos + (-1)**k * binomial(
j, 2 * k) * ecosM**(j - 2 * k) * esinM**(2 * k) * e**-j
temp = temp + s_list[i][0] * trig_cos
# calculate sin(nM):
if s_list[i][1] == sin(j * M):
for k in range(int(ceil((j - 1) / 2)) + 1):
trig_sin = trig_sin + (-1)**k * binomial(
j, 2 * k + 1) * ecosM**(j - 2 * k -
1) * esinM**(2 * k + 1) * e**-j
temp = temp + s_list[i][0] * trig_sin
if type(temp) != type(1): temp = temp.trim()
return temp
def binomial_exp(x, y, r, order):
"""
Binomial exponentiations.
"""
from pyranha.math import binomial
temp = 0
for k in range(0, order + 1):
temp = temp + binomial(r, k) * x**(r - k) * y**k
return temp
# Import the pyranha.math submodule.
from pyranha import math
# Various ways of constructing an int.
print(42)
print(int("42"))
print(int(42.123))
# Arbitrarily large ints are supported directly,
# no need to go through a string conversion.
print(12345678987654321234567898765432)
# Interoperability with float.
print(43. - 1)
# Truncated division in Python 2.x, floating-point division in Python 3.x.
print(85 / 2)
# Exponentiation via the '**' operator.
print(42**2)
# Calculate (42 choose 21) using the binomial() function from the math
# submodule.
print(math.binomial(42, 21))
# Import the pyranha.math submodule.
from pyranha import math
# Import the standard Fraction class.
from fractions import Fraction as F
# Various ways of constructing a rational.
print(F(42))
print(F("42"))
print(F(1.5))
print(F(42, 12))
print(F("42/12"))
# Arithmetics and interoperability with float and int.
print(F(42, 13) * 2)
print(1 + F(42, 13))
print(43. - F(1, 2))
print(F(85) / 13)
print(F(84, 2) == 42)
print(42 >= F(84, 3))
# Exponentiation is provided by the standard '**' operator.
print(F(42, 13)**2)
print(F(42, 13)**-3)
# Conversion to int yields the truncated value.
print(int(F(10, 3)))
# Calculate (42/13 choose 21) via the math::binomial() function.
print(math.binomial(F(42, 13), 21))
