def test_arithmetic_sums():
assert sum(1, (n, a, b)) == b - a + 1
assert sum(1, (n, 1, 10)) == 10
assert sum(2 * n, (n, 0, 10**10)) == 100000000010000000000
assert sum(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert sum(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
def test_arithmetic_sums():
assert sum(1, (n, a, b)) == b-a+1
assert sum(1, (n, 1, 10)) == 10
assert sum(2*n, (n, 0, 10**10)) == 100000000010000000000
assert sum(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert sum(cos(n), (n, -2, 1)) == cos(-2)+cos(-1)+cos(0)+cos(1)
def test_polynomial_sums():
assert sum(n**2, (n, 3, 8)) == 199
assert sum(n, (n, a, b)) == \
((a+b)*(b-a+1)/2).expand()
assert sum(n**2, (n, 1, b)) == \
((2*b**3+3*b**2+b)/6).expand()
assert sum(n**3, (n, 1, b)) == \
((b**4+2*b**3+b**2)/4).expand()
assert sum(n**6, (n, 1, b)) == \
((6*b**7+21*b**6+21*b**5-7*b**3+b)/42).expand()
def test_polynomial_sums():
assert sum(n**2, (n, 3, 8)) == 199
assert sum(n, (n, a, b)) == \
((a+b)*(b-a+1)/2).expand()
assert sum(n**2, (n, 1, b)) == \
((2*b**3+3*b**2+b)/6).expand()
assert sum(n**3, (n, 1, b)) == \
((b**4+2*b**3+b**2)/4).expand()
assert sum(n**6, (n, 1, b)) == \
((6*b**7+21*b**6+21*b**5-7*b**3+b)/42).expand()
def test_geometric_sums():
assert sum(pi**n, (n, 0, b)) == (1-pi**(b+1)) / (1-pi)
assert sum(2 * 3**n, (n, 0, b)) == 3**(b+1) - 1
assert sum(Rational(1,2)**n, (n, 1, oo)) == 1
assert sum(2**n, (n, 0, b)) == 2**(b+1) - 1
assert sum(2**n, (n, 1, oo)) == oo
assert sum(2**(-n), (n, 1, oo)) == 1
assert sum(3**(-n), (n, 4, oo)) == Rational(1,54)
assert sum(2**(-4*n+3), (n, 1, oo)) == Rational(8,15)
assert sum(2**(n+1), (n, 1, b)).expand() == 4*(2**b-1)
def test_geometric_sums():
assert sum(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
assert sum(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
assert sum(Rational(1, 2)**n, (n, 1, oo)) == 1
assert sum(2**n, (n, 0, b)) == 2**(b + 1) - 1
assert sum(2**n, (n, 1, oo)) == oo
assert sum(2**(-n), (n, 1, oo)) == 1
assert sum(3**(-n), (n, 4, oo)) == Rational(1, 54)
assert sum(2**(-4 * n + 3), (n, 1, oo)) == Rational(8, 15)
assert sum(2**(n + 1), (n, 1, b)).expand() == 4 * (2**b - 1)
def test_composite_sums():
f = Rational(1,2)*(7 - 6*n + Rational(1,7)*n**3)
s = sum(f, (n, a, b))
assert not isinstance(s, Sum)
A = 0
for i in range(-3, 5):
A += f.subs(n, i)
B = s.subs(a,-3).subs(b,4)
assert A == B
def test_composite_sums():
f = Rational(1, 2) * (7 - 6 * n + Rational(1, 7) * n**3)
s = sum(f, (n, a, b))
assert not isinstance(s, Sum)
A = 0
for i in range(-3, 5):
A += f.subs(n, i)
B = s.subs(a, -3).subs(b, 4)
assert A == B
def _eval_product(self, a, n, term):
from sympy import sum, Sum
k = self.index
if not term.has(k):
return term**(n-a+1)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
C_= poly.LC
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a-r, n-a+1)
Q *= n - r
if len(all_roots) < poly.degree:
B = Product(quo(poly, Q.as_poly(k)), (k, a, n))
return poly.LC**(n-a+1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(a, n, p)
q = self._eval_product(a, n, q)
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(a, n, t)
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
A, B = Mul(*exclude), Mul(*include)
return A * Product(B, (k, a, n))
elif term.is_Pow:
if not term.base.has(k):
s = sum(term.exp, (k, a, n))
if not isinstance(s, Sum):
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(a, n, term.base)
if p is not None:
return p**term.exp
def _eval_product(self, a, n, term):
from sympy import sum, Sum
k = self.index
if not term.has(k):
return term**(n - a + 1)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
C_ = poly.LC()
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r
if len(all_roots) < poly.degree():
B = Product(quo(poly, Q.as_poly(k)), (k, a, n))
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(a, n, p)
q = self._eval_product(a, n, q)
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(a, n, t)
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
A, B = Mul(*exclude), Mul(*include)
return A * Product(B, (k, a, n))
elif term.is_Pow:
if not term.base.has(k):
s = sum(term.exp, (k, a, n))
if not isinstance(s, Sum):
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(a, n, term.base)
if p is not None:
return p**term.exp
def test_Sum():
assert str(sum(cos(3 * z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
def test_Sum():
assert str(sum(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
def test_Sum():
assert str(sum(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
def test_Sum():
assert str(sum(cos(3 * z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
def test_Sum_doit():
assert Sum(n * Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep = False) == \
3*Integral(a**2)
assert sum(n * Integral(a**2), (n, 0, 2)) == 3 * Integral(a**2)
def test_Sum_doit():
assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep = False) == \
3*Integral(a**2)
assert sum(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-69-5b80e95b5264> in <module>()
----> 1 power(x,2)
TypeError: 'module' object is not callable
power
Out[70]: <module 'sympy.core.power' from '/home/alexander/py_virt_envs/base/local/lib/python2.7/site-packages/sympy/core/power.pyc'>
x**(0.5).
File "<ipython-input-73-ad48eaeec398>", line 1;
print(';'.join(get_ipython().Completer.all_completions('f.e'))) #PYTHON-MODE SILENT
f.eval(2)
f
sympy.diff(f, x)
sympy.exp(2)
sympy.exp(x)
sympy.sum(x)
sympy.Sum(x)
sum(x)
ImportError Traceback (most recent call last);ImportError: No module named mpmath;ImportError Traceback (most recent call last);ImportError: No module named mpmath;ImportError Traceback (most recent call last);ImportError: No module named mpmath;ImportError Traceback (most recent call last);ImportError: No module named mpmath;ImportError Traceback (most recent call last);ImportError: No module named mpmath;ImportError Traceback (most recent call last)
<ipython-input-63-8855bbdd7472> in <module>()
----> 1 from sympy import pow
ImportError: cannot import name pow
from sympy import power
power(x,2)
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-65-5b80e95b5264> in <module>()
----> 1 power(x,2)
TypeError: 'module' object is not callable
x
Out[66]: x