def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
def test1():
R, x = ring('x', QQ)
r = rs_sin(x, x, 15) * x**(-5)
assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \
QQ(1,120) - x**-2/6 + x**-4
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 2, x, 10)
assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \
x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2)
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 7, x, 10)
r = rs_pow(r, 5, x, 10)
assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \
11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7)
r = rs_exp(x**QQ(1, 2), x, 10)
assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \
x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \
x**QQ(15,2)/1307674368000 + x**7/87178291200 + \
x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \
x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \
x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \
x**QQ(1,2) + 1
def test1():
R, x = ring('x', QQ)
r = rs_sin(x, x, 15)*x**(-5)
assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \
QQ(1,120) - x**-2/6 + x**-4
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 2, x, 10)
assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \
x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2)
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 7, x, 10)
r = rs_pow(r, 5, x, 10)
assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \
11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7)
r = rs_exp(x**QQ(1,2), x, 10)
assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \
x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \
x**QQ(15,2)/1307674368000 + x**7/87178291200 + \
x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \
x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \
x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \
x**QQ(1,2) + 1
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369,
353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[exp(a), a])
assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
exp(a)*x**2/2 + exp(a)*x + exp(a)
assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
R, x, y = ring('x, y', EX)
assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
EX(exp(a))*x + EX(exp(a))
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[exp(a), a])
assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
exp(a)*x**2/2 + exp(a)*x + exp(a)
assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
R, x, y = ring('x, y', EX)
assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
EX(exp(a))*x + EX(exp(a))
def test_series_from_list():
R, x = ring('x', QQ)
p = 1 + 2 * x + x**2 + 3 * x**3
c = [1, 2, 0, 4, 4]
r = rs_series_from_list(p, c, x, 5)
pc = R.from_list(list(reversed(c)))
r1 = rs_trunc(pc.compose(x, p), x, 5)
assert r == r1
R, x, y = ring('x, y', QQ)
c = [1, 3, 5, 7]
p1 = rs_series_from_list(x + y, c, x, 3, concur=0)
p2 = rs_trunc((1 + 3 * (x + y) + 5 * (x + y)**2 + 7 * (x + y)**3), x, 3)
assert p1 == p2
R, x = ring('x', QQ)
h = 25
p = rs_exp(x, x, h) - 1
p1 = rs_series_from_list(p, c, x, h)
p2 = 0
for i, cx in enumerate(c):
p2 += cx * rs_pow(p, i, x, h)
assert p1 == p2
def test_series_from_list():
R, x = ring('x', QQ)
p = 1 + 2*x + x**2 + 3*x**3
c = [1, 2, 0, 4, 4]
r = rs_series_from_list(p, c, x, 5)
pc = R.from_list(list(reversed(c)))
r1 = rs_trunc(pc.compose(x, p), x, 5)
assert r == r1
R, x, y = ring('x, y', QQ)
c = [1, 3, 5, 7]
p1 = rs_series_from_list(x + y, c, x, 3, concur=0)
p2 = rs_trunc((1 + 3*(x+y) + 5*(x+y)**2 + 7*(x+y)**3), x, 3)
assert p1 == p2
R, x = ring('x', QQ)
h = 25
p = rs_exp(x, x, h) - 1
p1 = rs_series_from_list(p, c, x, h)
p2 = 0
for i, cx in enumerate(c):
p2 += cx*rs_pow(p, i, x, h)
assert p1 == p2
def test_puiseux():
R, x, y = ring('x, y', QQ)
p = x**QQ(2, 5) + x**QQ(2, 3) + x
r = rs_series_inversion(p, x, 1)
r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \
2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \
+ x**QQ(-2,5)
assert r == r1
r = rs_nth_root(1 + p, 3, x, 1)
assert r == -x**QQ(4, 5) / 9 + x**QQ(2, 3) / 3 + x**QQ(2, 5) / 3 + 1
r = rs_log(1 + p, x, 1)
assert r == -x**QQ(4, 5) / 2 + x**QQ(2, 3) + x**QQ(2, 5)
r = rs_LambertW(p, x, 1)
assert r == -x**QQ(4, 5) + x**QQ(2, 3) + x**QQ(2, 5)
p1 = x + x**QQ(1, 5) * y
r = rs_exp(p1, x, 1)
assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \
x**QQ(1,5)*y + 1
r = rs_atan(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atan(p1, x, 2)
assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \
x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y
r = rs_asin(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cot(p, x, 1)
assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \
2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \
x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5)
r = rs_cos_sin(p, x, 2)
assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \
x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1
assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atanh(p, x, 2)
assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \
x**QQ(2,3) + x**QQ(2,5)
r = rs_sinh(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cosh(p, x, 2)
assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \
x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1
r = rs_tanh(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
def test_puiseux():
R, x, y = ring('x, y', QQ)
p = x**QQ(2,5) + x**QQ(2,3) + x
r = rs_series_inversion(p, x, 1)
r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \
2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \
+ x**QQ(-2,5)
assert r == r1
r = rs_nth_root(1 + p, 3, x, 1)
assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1
r = rs_log(1 + p, x, 1)
assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5)
r = rs_LambertW(p, x, 1)
assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5)
p1 = x + x**QQ(1,5)*y
r = rs_exp(p1, x, 1)
assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \
x**QQ(1,5)*y + 1
r = rs_atan(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atan(p1, x, 2)
assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \
x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y
r = rs_asin(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cot(p, x, 1)
assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \
2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \
x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5)
r = rs_cos_sin(p, x, 2)
assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \
x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1
assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atanh(p, x, 2)
assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \
x**QQ(2,3) + x**QQ(2,5)
r = rs_sinh(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cosh(p, x, 2)
assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \
x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1
r = rs_tanh(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
