def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a 2-tuple representing a complex number.
Examples
========
>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import exp
>>> exp(x).as_real_imag()
(exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))
>>> exp(1).as_real_imag()
(E, 0)
>>> exp(I).as_real_imag()
(cos(1), sin(1))
>>> exp(1+I).as_real_imag()
(E*cos(1), E*sin(1))
See Also
========
sympy.functions.elementary.complexes.re
sympy.functions.elementary.complexes.im
"""
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return (exp(re)*cos, exp(re)*sin)
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * C.atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * C.atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * C.atan(i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def _eval_expand_complex(self, deep=True, **hints):
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return exp(re) * cos + S.ImaginaryUnit * exp(re) * sin
def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
Examples
========
>>> from sympy import I
>>> from sympy.abc import x
>>> from sympy.functions import log
>>> log(x).as_real_imag()
(log(Abs(x)), arg(x))
>>> log(I).as_real_imag()
(0, pi/2)
>>> log(1+I).as_real_imag()
(log(sqrt(2)), pi/4)
>>> log(I*x).as_real_imag()
(log(Abs(x)), arg(I*x))
"""
if deep:
abs = C.Abs(self.args[0].expand(deep, **hints))
arg = C.arg(self.args[0].expand(deep, **hints))
else:
abs = C.Abs(self.args[0])
arg = C.arg(self.args[0])
if hints.get('log', False): # Expand the log
hints['complex'] = False
return (log(abs).expand(deep, **hints), arg)
else:
return (log(abs), arg)
def vertices(self):
points = []
c, r, n = self
v = 2*S.Pi/n
for k in xrange(0, n):
points.append( Point(c[0] + r*C.cos(k*v), c[1] + r*C.sin(k*v)) )
return points
def vertices(self):
"""The vertices of the regular polygon.
Returns
-------
vertices : list
Each vertex is a Point.
See Also
--------
Point
Examples
--------
>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point(5, 0), Point(0, 5), Point(-5, 0), Point(0, -5)]
"""
points = []
c, r, n = self
v = 2*S.Pi/n
for k in xrange(0, n):
points.append(Point(c[0] + r*C.cos(k*v), c[1] + r*C.sin(k*v)))
return points
def arbitrary_point(self, parameter_name='t'):
"""A parametric point on the ellipse.
Parameters
----------
parameter_name : str, optional
Default value is 't'.
Returns
-------
arbitrary_point : Point
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point(3*cos(t), 2*sin(t))
"""
t = C.Symbol(parameter_name, real=True)
return Point(self.center[0] + self.hradius*C.cos(t),
self.center[1] + self.vradius*C.sin(t))
def monomial_count(V, N):
r"""
Computes the number of monomials.
The number of monomials is given by the following formula:
.. math::
\frac{(\#V + N)!}{\#V! N!}
where `N` is a total degree and `V` is a set of variables.
Examples
========
>>> from sympy.polys.monomials import itermonomials, monomial_count
>>> from sympy.polys.orderings import monomial_key
>>> from sympy.abc import x, y
>>> monomial_count(2, 2)
6
>>> M = itermonomials([x, y], 2)
>>> sorted(M, key=monomial_key('grlex', [y, x]))
[1, x, y, x**2, x*y, y**2]
>>> len(M)
6
"""
return C.factorial(V + N) / C.factorial(V) / C.factorial(N)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
----------
parameter : str, optional
Default value is 't'.
Returns
-------
arbitrary_point : Point
Raises
------
ValueError
When `parameter` already appears in the functions.
See Also
--------
Point
Examples
--------
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
return Point(self.center[0] + self.hradius*C.cos(t),
self.center[1] + self.vradius*C.sin(t))
def monomial_count(V, N):
r"""
Computes the number of monomials.
The number of monomials is given by the following formula:
.. math::
\frac{(\#V + N)!}{\#V! N!}
where `N` is a total degree and `V` is a set of variables.
**Examples**
>>> from sympy import monomials, monomial_count
>>> from sympy.abc import x, y
>>> monomial_count(2, 2)
6
>>> M = monomials([x, y], 2)
>>> sorted(M)
[1, x, y, x**2, y**2, x*y]
>>> len(M)
6
"""
return C.factorial(V + N) / C.factorial(V) / C.factorial(N)
def eval(cls, z, a=S.One):
z, a = map(sympify, (z, a))
if a.is_Number:
if a is S.NaN:
return S.NaN
elif a is S.Zero:
return cls(z)
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
return S.One
elif z is S.Zero:
if a.is_negative:
return S.Half - a - 1
else:
return S.Half - a
elif z is S.One:
return S.ComplexInfinity
elif z.is_Integer:
if a.is_Integer:
if z.is_negative:
zeta = (-1)**z * C.bernoulli(-z+1)/(-z+1)
elif z.is_even:
B, F = C.bernoulli(z), C.factorial(z)
zeta = 2**(z-1) * abs(B) * pi**z / F
else:
return
if a.is_negative:
return zeta + C.harmonic(abs(a), z)
else:
return zeta - C.harmonic(a-1, z)
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg is S.Zero:
return S.Zero
elif arg is S.One:
return C.log(2**S.Half + 1)
elif arg is S.NegativeOne:
return C.log(2**S.Half - 1)
elif arg.is_negative:
return -cls(-arg)
else:
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
return S.ImaginaryUnit * C.asin(i_coeff)
else:
if arg.as_coeff_mul()[0].is_negative:
return -cls(-arg)
def as_real_imag(self, deep=True, **hints):
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = C.cos(im), C.sin(im)
return (exp(re) * cos, exp(re) * sin)
def eval(cls, arg):
from sympy.simplify.simplify import signsimp
if hasattr(arg, '_eval_Abs'):
obj = arg._eval_Abs()
if obj is not None:
return obj
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if tnew.func is cls:
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else S.One
return known*unk
if arg is S.NaN:
return S.NaN
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if base.is_real:
if exponent.is_integer:
if exponent.is_even:
return arg
if base is S.NegativeOne:
return S.One
if base.func is cls and exponent is S.NegativeOne:
return arg
return Abs(base)**exponent
if base.is_positive == True:
return base**re(exponent)
return (-base)**re(exponent)*C.exp(-S.Pi*im(exponent))
if isinstance(arg, C.exp):
return C.exp(re(arg.args[0]))
if arg.is_zero: # it may be an Expr that is zero
return S.Zero
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -S.ImaginaryUnit * arg
if arg2.is_nonnegative:
return arg2
if arg.is_Add:
if arg.has(S.Infinity, S.NegativeInfinity):
if any(a.is_infinite for a in arg.as_real_imag()):
return S.Infinity
if arg.is_real is None and arg.is_imaginary is None:
if all(a.is_real or a.is_imaginary or (S.ImaginaryUnit*a).is_real for a in arg.args):
from sympy import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
if arg.is_real is False and arg.is_imaginary is False:
from sympy import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
def real_root(arg, n=None):
"""Return the real nth-root of arg if possible. If n is omitted then
all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this
will only create a real root of a principle root -- the presence of
other factors may cause the result to not be real.
Examples
========
>>> from sympy import root, real_root, Rational
>>> from sympy.abc import x, n
>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*(-1)**(1/3)
>>> real_root(_)
-2
If one creates a non-principle root and applies real_root, the
result will not be real (so use with caution):
>>> root(-8, 3, 2)
-2*(-1)**(2/3)
>>> real_root(_)
-2*(-1)**(2/3)
See Also
========
sympy.polys.rootoftools.RootOf
sympy.core.power.integer_nthroot
root, sqrt
"""
if n is not None:
try:
n = as_int(n)
arg = sympify(arg)
if arg.is_positive or arg.is_negative:
rv = root(arg, n)
else:
raise ValueError
except ValueError:
return root(arg, n)*C.Piecewise(
(S.One, ~C.Equality(C.im(arg), 0)),
(C.Pow(S.NegativeOne, S.One/n)**(2*C.floor(n/2)), C.And(
C.Equality(n % 2, 1),
arg < 0)),
(S.One, True))
else:
rv = sympify(arg)
n1pow = Transform(lambda x: -(-x.base)**x.exp,
lambda x:
x.is_Pow and
x.base.is_negative and
x.exp.is_Rational and
x.exp.p == 1 and x.exp.q % 2)
return rv.xreplace(n1pow)
def eval(cls, n, z):
n, z = list(map(sympify, (n, z)))
from sympy import unpolarify
if n.is_integer:
if n.is_nonnegative:
nz = unpolarify(z)
if z != nz:
return polygamma(n, nz)
if n == -1:
return loggamma(z)
else:
if z.is_Number:
if z is S.NaN:
return S.NaN
elif z is S.Infinity:
if n.is_Number:
if n is S.Zero:
return S.Infinity
else:
return S.Zero
elif z.is_Integer:
if z.is_nonpositive:
return S.ComplexInfinity
else:
if n is S.Zero:
return -S.EulerGamma + C.harmonic(z - 1, 1)
elif n.is_odd:
return (-1) ** (n + 1) * C.factorial(n) * zeta(n + 1, z)
if n == 0:
if z is S.NaN:
return S.NaN
elif z.is_Rational:
# TODO actually *any* n/m can be done, but that is messy
lookup = {
S(1) / 2: -2 * log(2) - S.EulerGamma,
S(1) / 3: -S.Pi / 2 / sqrt(3) - 3 * log(3) / 2 - S.EulerGamma,
S(1) / 4: -S.Pi / 2 - 3 * log(2) - S.EulerGamma,
S(3) / 4: -3 * log(2) - S.EulerGamma + S.Pi / 2,
S(2) / 3: -3 * log(3) / 2 + S.Pi / 2 / sqrt(3) - S.EulerGamma,
}
if z > 0:
n = floor(z)
z0 = z - n
if z0 in lookup:
return lookup[z0] + Add(*[1 / (z0 + k) for k in range(n)])
elif z < 0:
n = floor(1 - z)
z0 = z + n
if z0 in lookup:
return lookup[z0] - Add(*[1 / (z0 - 1 - k) for k in range(n)])
elif z in (S.Infinity, S.NegativeInfinity):
return S.Infinity
else:
t = z.extract_multiplicatively(S.ImaginaryUnit)
if t in (S.Infinity, S.NegativeInfinity):
return S.Infinity
def eval(cls, arg):
arg = sympify(arg)
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.Zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = {
S.ImaginaryUnit : C.log(S.ImaginaryUnit*(1+sqrt(2))),
-S.ImaginaryUnit : C.log(-S.ImaginaryUnit*(1+sqrt(2))),
S.Half : S.Pi/3,
-S.Half : 2*S.Pi/3,
sqrt(2)/2 : S.Pi/4,
-sqrt(2)/2 : 3*S.Pi/4,
1/sqrt(2) : S.Pi/4,
-1/sqrt(2) : 3*S.Pi/4,
sqrt(3)/2 : S.Pi/6,
-sqrt(3)/2 : 5*S.Pi/6,
(sqrt(3)-1)/sqrt(2**3) : 5*S.Pi/12,
-(sqrt(3)-1)/sqrt(2**3) : 7*S.Pi/12,
sqrt(2+sqrt(2))/2 : S.Pi/8,
-sqrt(2+sqrt(2))/2 : 7*S.Pi/8,
sqrt(2-sqrt(2))/2 : 3*S.Pi/8,
-sqrt(2-sqrt(2))/2 : 5*S.Pi/8,
(1+sqrt(3))/(2*sqrt(2)) : S.Pi/12,
-(1+sqrt(3))/(2*sqrt(2)) : 11*S.Pi/12,
(sqrt(5)+1)/4 : S.Pi/5,
-(sqrt(5)+1)/4 : 4*S.Pi/5
}
if arg in cst_table:
if arg.is_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.Infinity
i_coeff = arg.as_coefficient(S.ImaginaryUnit)
if i_coeff is not None:
if _coeff_isneg(i_coeff):
return S.ImaginaryUnit * C.acos(i_coeff)
return S.ImaginaryUnit * C.acos(-i_coeff)
else:
if _coeff_isneg(arg):
return -cls(-arg)
def _eval_aseries(self, n, args0, x, logx):
if args0[0] != S.Infinity:
return super(_erfs, self)._eval_aseries(n, args0, x, logx)
z = self.args[0]
l = [ 1/sqrt(S.Pi) * C.factorial(2*k)*(-S(4))**(-k)/C.factorial(k) * (1/z)**(2*k+1) for k in xrange(0,n) ]
o = C.Order(1/z**(2*n+1), x)
# It is very inefficient to first add the order and then do the nseries
return (Add(*l))._eval_nseries(x, n, logx) + o
def eval(cls, a, x):
if a.is_Number:
if a is S.One:
return S.One - C.exp(-x)
elif a.is_Integer:
b = a - 1
if b.is_positive:
return b*cls(b, x) - x**b * C.exp(-x)
def eval(cls, n):
if (n.is_Integer and n.is_nonnegative) or \
(n.is_noninteger and n.is_negative):
return 4**n*C.gamma(n + S.Half)/(C.gamma(S.Half)*C.gamma(n + 2))
if (n.is_integer and n.is_negative):
if (n + 1).is_negative:
return S.Zero
if (n + 1).is_zero:
return -S.Half
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = C.floor((n - 1)/S(2))
if len(previous_terms) > 2:
return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
else:
return 2*(-1)**k * x**n/(n*C.factorial(k)*sqrt(S.Pi))
def eval(cls, r, k):
r, k = map(sympify, (r, k))
if k.is_Number:
if k is S.Zero:
return S.One
elif k.is_Integer:
if k.is_negative:
return S.Zero
else:
if r.is_Integer and r.is_nonnegative:
r, k = int(r), int(k)
if k > r:
return S.Zero
elif k > r // 2:
k = r - k
M, result = int(sqrt(r)), 1
for prime in sieve.primerange(2, r + 1):
if prime > r - k:
result *= prime
elif prime > r // 2:
continue
elif prime > M:
if r % prime < k % prime:
result *= prime
else:
R, K = r, k
exp = a = 0
while R > 0:
a = int((R % prime) < (K % prime + a))
R, K = R // prime, K // prime
exp = a + exp
if exp > 0:
result *= prime ** exp
return C.Integer(result)
else:
result = r - k + 1
for i in xrange(2, k + 1):
result *= r - k + i
result /= i
return result
if k.is_integer:
if k.is_negative:
return S.Zero
else:
return C.gamma(r + 1) / (C.gamma(r - k + 1) * C.gamma(k + 1))
def separate(expr, deep=False):
"""Rewrite or separate a power of product to a product of powers
but without any expanding, ie. rewriting products to summations.
>>> from sympy import *
>>> x, y, z = symbols('x', 'y', 'z')
>>> separate((x*y)**2)
x**2*y**2
>>> separate((x*(y*z)**3)**2)
x**2*y**6*z**6
>>> separate((x*sin(x))**y + (x*cos(x))**y)
x**y*cos(x)**y + x**y*sin(x)**y
#this does not work because of exp combining
#>>> separate((exp(x)*exp(y))**x)
#exp(x*y)*exp(x**2)
>>> separate((sin(x)*cos(x))**y)
cos(x)**y*sin(x)**y
Notice that summations are left un touched. If this is not the
requested behaviour, apply 'expand' to input expression before:
>>> separate(((x+y)*z)**2)
z**2*(x + y)**2
>>> separate((x*y)**(1+z))
x**(1 + z)*y**(1 + z)
"""
expr = sympify(expr)
if expr.is_Pow:
terms, expo = [], separate(expr.exp, deep)
if expr.base.is_Mul:
t = [ separate(C.Pow(t,expo), deep) for t in expr.base.args ]
return C.Mul(*t)
elif expr.base.func is C.exp:
if deep == True:
return C.exp(separate(expr.base[0], deep)*expo)
else:
return C.exp(expr.base[0]*expo)
else:
return C.Pow(separate(expr.base, deep), expo)
elif expr.is_Add or expr.is_Mul:
return type(expr)(*[ separate(t, deep) for t in expr.args ])
elif expr.is_Function and deep:
return expr.func(*[ separate(t) for t in expr.args])
else:
return expr
def fdiff(self, argindex=1):
if argindex == 1:
# http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/
n, k = self.args
return binomial(n, k)*(C.polygamma(0, n + 1) - C.polygamma(0, n - k + 1))
elif argindex == 2:
# http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/02/
n, k = self.args
return binomial(n, k)*(C.polygamma(0, n - k + 1) - C.polygamma(0, k + 1))
else:
raise ArgumentIndexError(self, argindex)
def as_real_imag(self, deep=True, **hints):
if deep:
abs = C.Abs(self.args[0].expand(deep, **hints))
arg = C.arg(self.args[0].expand(deep, **hints))
else:
abs = C.Abs(self.args[0])
arg = C.arg(self.args[0])
if hints.get("log", False): # Expand the log
hints["complex"] = False
return (log(abs).expand(deep, **hints), arg)
else:
return (log(abs), arg)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + C.sin(im)**2
return (sinh(re)*cosh(re)/denom, -C.sin(im)*C.cos(im)/denom)
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (sinh(re)*C.cos(im), cosh(re)*C.sin(im))
def _eval_expand_complex(self, deep=True, **hints):
if deep:
abs = C.abs(self.args[0].expand(deep, **hints))
arg = C.arg(self.args[0].expand(deep, **hints))
else:
abs = C.abs(self.args[0])
arg = C.arg(self.args[0])
if hints['log']: # Expand the log
hints['complex'] = False
return log(abs).expand(deep, **hints) + S.ImaginaryUnit * arg
else:
return log(abs) + S.ImaginaryUnit * arg
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = C.bernoulli(n+1)
F = C.factorial(n+1)
return 2**(n+1) * B/F * x**n
def as_real_imag(self, deep=True, **hints):
if deep:
abs = C.abs(self.args[0].expand(deep, **hints))
arg = C.arg(self.args[0].expand(deep, **hints))
else:
abs = C.abs(self.args[0])
arg = C.arg(self.args[0])
if hints['log']: # Expand the log
hints['complex'] = False
return (log(abs).expand(deep, **hints), arg)
else:
return (log(abs), arg)
def fdiff(self, argindex=1):
n = self.args[0]
return catalan(n) * (C.polygamma(0, n + Rational(1, 2)) -
C.polygamma(0, n + 2) + C.log(4))
def _eval_rewrite_as_gamma(self, n):
return C.gamma(n + 1)
def _eval_rewrite_as_gamma(self, x, k):
return (-1)**k * C.gamma(-x + k) / C.gamma(-x)
def _eval_rewrite_as_Heaviside(self, arg):
# Note this only holds for real arg (since Heaviside is not defined
# for complex arguments).
if arg.is_real:
return arg*(C.Heaviside(arg) - C.Heaviside(-arg))
def _eval_rewrite_as_exp(self, arg):
return (C.exp(arg) + C.exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg):
neg_exp, pos_exp = C.exp(-arg), C.exp(arg)
return (pos_exp + neg_exp) / (pos_exp - neg_exp)
def _eval_rewrite_as_tanh(self, arg):
return (1 + C.tanh(arg/2))/(1 - C.tanh(arg/2))
def root(arg, n):
"""The n-th root function (a shortcut for ``arg**(1/n)``)
root(x, n) -> Returns the principal n-th root of x.
Examples
========
>>> from sympy import root, Rational
>>> from sympy.abc import x, n
>>> root(x, 2)
sqrt(x)
>>> root(x, 3)
x**(1/3)
>>> root(x, n)
x**(1/n)
>>> root(x, -Rational(2, 3))
x**(-3/2)
To get all n n-th roots you can use the RootOf function.
The following examples show the roots of unity for n
equal 2, 3 and 4:
>>> from sympy import RootOf, I
>>> [ RootOf(x**2-1,i) for i in (0,1) ]
[-1, 1]
>>> [ RootOf(x**3-1,i) for i in (0,1,2) ]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]
>>> [ RootOf(x**4-1,i) for i in (0,1,2,3) ]
[-1, 1, -I, I]
SymPy, like other symbolic algebra systems, returns the
complex root of negative numbers. This is the principal
root and differs from the text-book result that one might
be expecting. For example, the cube root of -8 does not
come back as -2:
>>> root(-8, 3)
2*(-1)**(1/3)
The real_root function can be used to either make such a result
real or simply return the real root in the first place:
>>> from sympy import real_root
>>> real_root(_)
-2
>>> real_root(-32, 5)
-2
See Also
========
sympy.polys.rootoftools.RootOf
sympy.core.power.integer_nthroot
sqrt, real_root
References
==========
* http://en.wikipedia.org/wiki/Square_root
* http://en.wikipedia.org/wiki/Real_root
* http://en.wikipedia.org/wiki/Root_of_unity
* http://en.wikipedia.org/wiki/Principal_value
* http://mathworld.wolfram.com/CubeRoot.html
"""
n = sympify(n)
return C.Pow(arg, 1/n)
def fdiff(self, argindex=1):
if argindex == 1:
return C.gamma(self.args[0] + 1) * C.polygamma(0, self.args[0] + 1)
else:
raise ArgumentIndexError(self, argindex)
def sqrt(arg):
"""The square root function
sqrt(x) -> Returns the principal square root of x.
Examples
========
>>> from sympy import sqrt, Symbol
>>> x = Symbol('x')
>>> sqrt(x)
sqrt(x)
>>> sqrt(x)**2
x
Note that sqrt(x**2) does not simplify to x.
>>> sqrt(x**2)
sqrt(x**2)
This is because the two are not equal to each other in general.
For example, consider x == -1:
>>> from sympy import Eq
>>> Eq(sqrt(x**2), x).subs(x, -1)
False
This is because sqrt computes the principal square root, so the square may
put the argument in a different branch. This identity does hold if x is
positive:
>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y
You can force this simplification by using the powdenest() function with
the force option set to True:
>>> from sympy import powdenest
>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x
To get both branches of the square root you can use the RootOf function:
>>> from sympy import RootOf
>>> [ RootOf(x**2-3,i) for i in (0,1) ]
[-sqrt(3), sqrt(3)]
See Also
========
sympy.polys.rootoftools.RootOf, root, real_root
References
==========
* http://en.wikipedia.org/wiki/Square_root
* http://en.wikipedia.org/wiki/Principal_value
"""
# arg = sympify(arg) is handled by Pow
return C.Pow(arg, S.Half)
def _eval_rewrite_as_factorial(self, n, k):
return C.factorial(n) / (C.factorial(k) * C.factorial(n - k))
def _eval_rewrite_as_tractable(self, z):
return S.One - _erfs(z) * C.exp(-z**2)
def eval(cls, n):
if n.is_Integer and n.is_nonnegative:
return 4**n * C.gamma(n + S.Half) / (C.gamma(S.Half) *
C.gamma(n + 2))
def _eval_rewrite_as_cos(self, arg):
I = S.ImaginaryUnit
return C.cos(I*arg) + I*C.cos(I*arg + S.Pi/2)
def _eval_rewrite_as_Sum(self, n, m=None):
k = C.Dummy("k", integer=True)
if m is None:
m = S.One
return C.Sum(k**(-m), (k, 1, n))
def _eval_rewrite_as_sin(self, arg):
I = S.ImaginaryUnit
return C.sin(I*arg + S.Pi/2) - I*C.sin(I*arg)
def _eval_rewrite_as_intractable(self, z):
return (S.One - erf(z)) * C.exp(z**2)
def _eval_rewrite_as_Heaviside(self, arg):
if arg.is_real:
return C.Heaviside(arg)*2-1
def _eval_rewrite_as_sign(self, arg):
return arg/C.sign(arg)
def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with constant coefficients.
The general form of this partial differential equation is
.. math:: a \frac{df(x,y)}{dx} + b \frac{df(x,y)}{dy} + c f(x,y) = G(x,y)
where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary
function in `x` and `y`.
The general solution of the PDE is::
>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> G = Function('G')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*u + b*ux + c*uy - G(x,y)
>>> pprint(genform)
d d
a*f(x, y) + b*--(f(x, y)) + c*--(f(x, y)) - G(x, y)
dx dy
>>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral'))
// b*x + c*y \
|| / |
|| | |
|| | a*xi |
|| | ------- |
|| | 2 2 |
|| | /b*xi + c*eta -b*eta + c*xi\ b + c |
|| | G|------------, -------------|*e d(xi)|
|| | | 2 2 2 2 | |
|| | \ b + c b + c / |
|| | |
|| / |
|| |
f(x, y) = ||F(eta) + -------------------------------------------------------|*
|| 2 2 |
\\ b + c /
<BLANKLINE>
\|
||
||
||
||
||
||
||
||
-a*xi ||
-------||
2 2||
b + c ||
e ||
||
/|eta=-b*y + c*x, xi=b*x + c*y
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y)
>>> pdsolve(eq)
f(x, y) == (F(4*x + 2*y) + exp(x/2 + 4*y)/15)*exp(x/2 - y)
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
# TODO : For now homogeneous first order linear PDE's having
# two variables are implemented. Once there is support for
# solving systems of ODE's, this can be extended to n variables.
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
expterm = exp(-S(d) / (b**2 + c**2) * xi)
functerm = solvefun(eta)
solvedict = solve((b * x + c * y - xi, c * x - b * y - eta), x, y)
# Integral should remain as it is in terms of xi,
# doit() should be done in _handle_Integral.
genterm = (1 / S(b**2 + c**2)) * C.Integral(
(1 / expterm * e).subs(solvedict), (xi, b * x + c * y))
return Eq(
f(x, y),
Subs(expterm * (functerm + genterm), (eta, xi),
(c * x - b * y, b * x + c * y)))
def eval(cls, s):
if s == 1:
return C.log(2)
else:
return (1-2**(1-s)) * zeta(s)
def _eval_rewrite_as_gamma(self, x, k):
return C.gamma(x + k) / C.gamma(x)
def _eval_rewrite_as_uppergamma(self, z):
return sqrt(z**
2) / z * (S.One - C.uppergamma(S.Half, z**2) / sqrt(S.Pi))
def limit(e, z, z0, dir="+"):
"""
Compute the limit of e(z) at the point z0.
z0 can be any expression, including oo and -oo.
For dir="+" (default) it calculates the limit from the right
(z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0
(oo or -oo), the dir argument doesn't matter.
Examples:
>>> from sympy import limit, sin, Symbol, oo
>>> from sympy.abc import x
>>> limit(sin(x)/x, x, 0)
1
>>> limit(1/x, x, 0, dir="+")
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, oo)
0
Strategy:
First we try some heuristics for easy and frequent cases like "x", "1/x",
"x**2" and similar, so that it's fast. For all other cases, we use the
Gruntz algorithm (see the gruntz() function).
"""
from sympy import Wild, log
e = sympify(e)
z = sympify(z)
z0 = sympify(z0)
if e == z:
return z0
if e.is_Rational:
return e
if not e.has(z):
return e
if e.func is tan:
# discontinuity at odd multiples of pi/2; 0 at even
disc = S.Pi/2
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if i.is_even:
return S.Zero
elif i.is_odd:
if dir == '+':
return S.NegativeInfinity
else:
return S.Infinity
if e.func is cot:
# discontinuity at multiples of pi; 0 at odd pi/2 multiples
disc = S.Pi
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if dir == '-':
return S.NegativeInfinity
else:
return S.Infinity
elif (2*i).is_integer:
return S.Zero
if e.is_Pow:
b, ex = e.args
c = None # records sign of b if b is +/-z or has a bounded value
if b.is_Mul:
c, b = b.as_two_terms()
if c is S.NegativeOne and b == z:
c = '-'
elif b == z:
c = '+'
if ex.is_number:
if c is None:
base = b.subs(z, z0)
if base.is_bounded and (ex.is_bounded or base is not S.One):
return base**ex
else:
if z0 == 0 and ex < 0:
if dir != c:
# integer
if ex.is_even:
return S.Infinity
elif ex.is_odd:
return S.NegativeInfinity
# rational
elif ex.is_Rational:
return (S.NegativeOne**ex)*S.Infinity
else:
return S.ComplexInfinity
return S.Infinity
return z0**ex
if e.is_Mul or not z0 and e.is_Pow and b.func is log:
if e.is_Mul:
# weed out the z-independent terms
i, d = e.as_independent(z)
if i is not S.One:
return i*limit(d, z, z0, dir)
else:
i, d = S.One, e
if not z0:
# look for log(z)**q or z**p*log(z)**q
p, q = Wild("p"), Wild("q")
r = d.match(z**p * log(z)**q)
if r:
p, q = [r.get(w, w) for w in [p, q]]
if q and q.is_number and p.is_number:
if q > 0:
if p > 0:
return S.Zero
else:
return -oo*i
else:
if p >= 0:
return S.Zero
else:
return -oo*i
if e.is_Add:
if e.is_polynomial() and not z0.is_unbounded:
return Add(*[limit(term, z, z0, dir) for term in e.args])
# this is a case like limit(x*y+x*z, z, 2) == x*y+2*x
# but we need to make sure, that the general gruntz() algorithm is
# executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0"
unbounded = []; unbounded_result=[]
finite = []; unknown = []
ok = True
for term in e.args:
if not term.has(z):
finite.append(term)
continue
result = term.subs(z, z0)
bounded = result.is_bounded
if bounded is False or result is S.NaN:
if unknown:
ok = False
break
unbounded.append(term)
if result != S.NaN:
# take result from direction given
result = limit(term, z, z0, dir)
unbounded_result.append(result)
elif bounded:
finite.append(result)
else:
if unbounded:
ok = False
break
unknown.append(result)
if not ok:
# we won't be able to resolve this with unbounded
# terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -> oo:
# since the Sum is unevaluated it's boundedness is
# unknown and the log(n) is oo so you get Sum - oo
# which is unsatisfactory.
raise NotImplementedError('unknown boundedness for %s' %
(unknown or result))
u = Add(*unknown)
if unbounded:
inf_limit = Add(*unbounded_result)
if inf_limit is not S.NaN:
return inf_limit + u
if finite:
return Add(*finite) + limit(Add(*unbounded), z, z0, dir) + u
else:
return Add(*finite) + u
if e.is_Order:
args = e.args
return C.Order(limit(args[0], z, z0), *args[1:])
try:
r = gruntz(e, z, z0, dir)
if r is S.NaN:
raise PoleError()
except PoleError:
r = heuristics(e, z, z0, dir)
return r
def _eval_rewrite_as_hyper(self, n):
return C.hyper([1 - n, -n], [2], 1)
def fdiff(self, argindex=1):
if argindex == 1:
return 2 * C.exp(-self.args[0]**2) / sqrt(S.Pi)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_gamma(self, n):
# The gamma function allows to generalize Catalan numbers to complex n
return 4**n * C.gamma(n + S.Half) / (C.gamma(S.Half) * C.gamma(n + 2))
def _eval_rewrite_as_binomial(self, n):
return C.binomial(2 * n, n) / (n + 1)
def _eval_rewrite_as_gamma(self, n, k):
return C.gamma(n + 1) / (C.gamma(k + 1) * C.gamma(n - k + 1))
def _eval_rewrite_as_Heaviside(self, *args):
return C.Add(*[j*C.Mul(*[C.Heaviside(i-j) for i in args if i!=j]) \
for j in args])