def test_issue_3460():
assert (-12 * x + y).subs(-x, 1) == 12 + y
# though this involves cse it generated a failure in Mul._eval_subs
x0, x1 = symbols("x0 x1")
e = -log(-12 * sqrt(2) + 17) / 24 - log(-2 * sqrt(2) + 3) / 12 + sqrt(2) / 3
# XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
assert cse(e) == ([(x0, sqrt(2))], [x0 / 3 - log(-12 * x0 + 17) / 24 - log(-2 * x0 + 3) / 12])
def test_getn():
# other lines are tested incidentally by the suite
assert O(x).getn() == 1
assert O(x / log(x)).getn() == 1
assert O(x ** 2 / log(x) ** 2).getn() == 2
assert O(x * log(x)).getn() == 1
raises(NotImplementedError, lambda: (O(x) + O(y)).getn())
def test_issue_1364():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
assert solve((a/x + exp(x/2)).diff(x), x) == [4*LambertW(sqrt(2)*sqrt(a)/4)]
def _display_expression(self, keys, user_substitutes={}):
"""Helper function for human friendly display of the symbolic components."""
# Create some pretty maths symbols for the display.
sigma, alpha, nu, omega, l, variance = sym.var('\sigma, \alpha, \nu, \omega, \ell, \sigma^2')
substitutes = {'scale': sigma, 'shape': alpha, 'lengthscale': l, 'variance': variance}
substitutes.update(user_substitutes)
function_substitutes = {normcdfln : lambda arg : sym.log(normcdf(arg)),
logisticln : lambda arg : -sym.log(1+sym.exp(-arg)),
logistic : lambda arg : 1/(1+sym.exp(-arg)),
erfcx : lambda arg : erfc(arg)/sym.exp(arg*arg),
gammaln : lambda arg : sym.log(sym.gamma(arg))}
expr = getFromDict(self.expressions, keys)
for var_name, sub in self.variable_sort(self.expressions['update_cache'], reverse=True):
for var in self.variables['cache']:
if var_name == var.name:
expr = expr.subs(var, sub)
break
for var_name, sub in self.variable_sort(self.expressions['parameters_changed'], reverse=True):
for var in self.variables['sub']:
if var_name == var.name:
expr = expr.subs(var, sub)
break
for var_name, sub in self.variable_sort(substitutes, reverse=True):
for var in self.variables['theta']:
if var_name == var.name:
expr = expr.subs(var, sub)
break
for m, r in function_substitutes.iteritems():
expr = expr.replace(m, r)#normcdfln, lambda arg : sym.log(normcdf(arg)))
return expr.simplify()
def test_acceleration():
e = (1 + 1/n)**n
assert round(richardson(e, n, 10, 20).evalf(), 10) == round(E.evalf(), 10)
A = Sum(Integer(-1)**(k + 1) / k, (k, 1, n))
assert round(shanks(A, n, 25).evalf(), 4) == round(log(2).evalf(), 4)
assert round(shanks(A, n, 25, 5).evalf(), 10) == round(log(2).evalf(), 10)
def test_entropy():
up = JzKet(S(1)/2, S(1)/2)
down = JzKet(S(1)/2, -S(1)/2)
d = Density((up, 0.5), (down, 0.5))
# test for density object
ent = entropy(d)
assert entropy(d) == 0.5*log(2)
assert d.entropy() == 0.5*log(2)
np = import_module('numpy', min_module_version='1.4.0')
if np:
#do this test only if 'numpy' is available on test machine
np_mat = represent(d, format='numpy')
ent = entropy(np_mat)
assert isinstance(np_mat, np.matrixlib.defmatrix.matrix)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
scipy = import_module('scipy', __import__kwargs={'fromlist': ['sparse']})
if scipy and np:
#do this test only if numpy and scipy are available
mat = represent(d, format="scipy.sparse")
assert isinstance(mat, scipy_sparse_matrix)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
def test_issue_1572_1364_1368():
assert solve((sqrt(x**2 - 1) - 2)) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == (
[-sqrt(x)*sqrt(log((log(2) + I*pi)/log(2))),
sqrt(x)*sqrt(log((log(2) + I*pi)/log(2)))]
)
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = symbols('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert solve((a**2 + 1) * (sin(a*x) + cos(a*x)), x) == [-pi/(4*a), 3*pi/(4*a)]
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [2*atanh(S.Half)/a]
assert solve(3-(sinh(a*x) + cosh(a*x)**2), x) == \
[
2*atanh(-1 + sqrt(2))/a,
2*atanh(S(1)/2 + sqrt(5)/2)/a,
2*atanh(-sqrt(2) - 1)/a,
2*atanh(-sqrt(5)/2 + S(1)/2)/a
]
assert solve(atan(x) - 1) == [tan(1)]
def test_manual_subs():
x, y = symbols('x y')
expr = log(x) + exp(x)
# if log(x) is y, then exp(y) is x
assert manual_subs(expr, log(x), y) == y + exp(exp(y))
# if exp(x) is y, then log(y) need not be x
assert manual_subs(expr, exp(x), y) == log(x) + y
def test_ci():
m1 = exp_polar(I*pi)
m1_ = exp_polar(-I*pi)
pI = exp_polar(I*pi/2)
mI = exp_polar(-I*pi/2)
assert Ci(m1*x) == Ci(x) + I*pi
assert Ci(m1_*x) == Ci(x) - I*pi
assert Ci(pI*x) == Chi(x) + I*pi/2
assert Ci(mI*x) == Chi(x) - I*pi/2
assert Chi(m1*x) == Chi(x) + I*pi
assert Chi(m1_*x) == Chi(x) - I*pi
assert Chi(pI*x) == Ci(x) + I*pi/2
assert Chi(mI*x) == Ci(x) - I*pi/2
assert Ci(exp_polar(2*I*pi)*x) == Ci(x) + 2*I*pi
assert Chi(exp_polar(-2*I*pi)*x) == Chi(x) - 2*I*pi
assert Chi(exp_polar(2*I*pi)*x) == Chi(x) + 2*I*pi
assert Ci(exp_polar(-2*I*pi)*x) == Ci(x) - 2*I*pi
assert mytd(Ci(x), cos(x)/x, x)
assert mytd(Chi(x), cosh(x)/x, x)
assert mytn(Ci(x), Ci(x).rewrite(Ei),
Ei(x*exp_polar(-I*pi/2))/2 + Ei(x*exp_polar(I*pi/2))/2, x)
assert mytn(Chi(x), Chi(x).rewrite(Ei),
Ei(x)/2 + Ei(x*exp_polar(I*pi))/2 - I*pi/2, x)
assert tn_arg(Ci)
assert tn_arg(Chi)
from sympy import O, EulerGamma, log, limit
assert Ci(x).nseries(x, n=4) == EulerGamma + log(x) - x**2/4 + x**4/96 + O(x**5)
assert Chi(x).nseries(x, n=4) == EulerGamma + log(x) + x**2/4 + x**4/96 + O(x**5)
assert limit(log(x) - Ci(2*x), x, 0) == -log(2) - EulerGamma
def test_lognormal():
mean = Symbol('mu', real=True, finite=True)
std = Symbol('sigma', positive=True, real=True, finite=True)
X = LogNormal('x', mean, std)
# The sympy integrator can't do this too well
#assert E(X) == exp(mean+std**2/2)
#assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)
# Right now, only density function and sampling works
# Test sampling: Only e^mean in sample std of 0
for i in range(3):
X = LogNormal('x', i, 0)
assert S(sample(X)) == N(exp(i))
# The sympy integrator can't do this too well
#assert E(X) ==
mu = Symbol("mu", real=True)
sigma = Symbol("sigma", positive=True)
X = LogNormal('x', mu, sigma)
assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
/(2*sigma**2))/(2*x*sqrt(pi)*sigma))
X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
def test_manualintegrate_exponentials():
assert manualintegrate(exp(2*x), x) == exp(2*x) / 2
assert manualintegrate(2**x, x) == (2 ** x) / log(2)
assert manualintegrate(1 / x, x) == log(x)
assert manualintegrate(1 / (2*x + 3), x) == log(2*x + 3) / 2
assert manualintegrate(log(x)**2 / x, x) == log(x)**3 / 3
def test_derivative_by_array():
from sympy.abc import a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z
bexpr = x*y**2*exp(z)*log(t)
sexpr = sin(bexpr)
cexpr = cos(bexpr)
a = Array([sexpr])
assert derive_by_array(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert derive_by_array(sexpr, [x, y, z]) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert derive_by_array(a, [x, y, z]) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert derive_by_array(sexpr, [[x, y], [z, t]]) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert derive_by_array(a, [[x, y], [z, t]]) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert derive_by_array([[x, y], [z, t]], [x, y]) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert derive_by_array([[x, y], [z, t]], [[x, y], [z, t]]) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
assert diff(sexpr, t) == x*y**2*exp(z)*cos(x*y**2*exp(z)*log(t))/t
assert diff(sexpr, Array([x, y, z])) == Array([bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr, bexpr*cexpr])
assert diff(a, Array([x, y, z])) == Array([[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr], [bexpr*cexpr]])
assert diff(sexpr, Array([[x, y], [z, t]])) == Array([[bexpr/x*cexpr, 2*y*bexpr/y**2*cexpr], [bexpr*cexpr, bexpr/log(t)/t*cexpr]])
assert diff(a, Array([[x, y], [z, t]])) == Array([[[bexpr/x*cexpr], [2*y*bexpr/y**2*cexpr]], [[bexpr*cexpr], [bexpr/log(t)/t*cexpr]]])
assert diff(Array([[x, y], [z, t]]), Array([x, y])) == Array([[[1, 0], [0, 0]], [[0, 1], [0, 0]]])
assert diff(Array([[x, y], [z, t]]), Array([[x, y], [z, t]])) == Array([[[[1, 0], [0, 0]], [[0, 1], [0, 0]]],
[[[0, 0], [1, 0]], [[0, 0], [0, 1]]]])
def test_fraction():
x, y, z = map(Symbol, 'xyz')
A = Symbol('A', commutative=False)
assert fraction(Rational(1, 2)) == (1, 2)
assert fraction(x) == (x, 1)
assert fraction(1/x) == (1, x)
assert fraction(x/y) == (x, y)
assert fraction(x/2) == (x, 2)
assert fraction(x*y/z) == (x*y, z)
assert fraction(x/(y*z)) == (x, y*z)
assert fraction(1/y**2) == (1, y**2)
assert fraction(x/y**2) == (x, y**2)
assert fraction((x**2 + 1)/y) == (x**2 + 1, y)
assert fraction(x*(y + 1)/y**7) == (x*(y + 1), y**7)
assert fraction(exp(-x), exact=True) == (exp(-x), 1)
assert fraction((1/(x + y))/2, exact=True) == (1, Mul(2,(x + y), evaluate=False))
assert fraction(x*A/y) == (x*A, y)
assert fraction(x*A**-1/y) == (x*A**-1, y)
n = symbols('n', negative=True)
assert fraction(exp(n)) == (1, exp(-n))
assert fraction(exp(-n)) == (exp(-n), 1)
p = symbols('p', positive=True)
assert fraction(exp(-p)*log(p), exact=True) == (exp(-p)*log(p), 1)
def test_acosh():
# TODO please write more tests -- see issue 3751
# From http://functions.wolfram.com/ElementaryFunctions/ArcCosh/03/01/
# at specific points
assert acosh(1) == 0
assert acosh(-1) == pi*I
assert acosh(0) == I*pi/2
assert acosh(Rational(1, 2)) == I*pi/3
assert acosh(Rational(-1, 2)) == 2*pi*I/3
assert acosh(zoo) == oo
assert acosh(I) == log(I*(1 + sqrt(2)))
assert acosh(-I) == log(-I*(1 + sqrt(2)))
assert acosh((sqrt(3) - 1)/(2*sqrt(2))) == 5*pi*I/12
assert acosh(-(sqrt(3) - 1)/(2*sqrt(2))) == 7*pi*I/12
assert acosh(sqrt(2)/2) == I*pi/4
assert acosh(-sqrt(2)/2) == 3*I*pi/4
assert acosh(sqrt(3)/2) == I*pi/6
assert acosh(-sqrt(3)/2) == 5*I*pi/6
assert acosh(sqrt(2 + sqrt(2))/2) == I*pi/8
assert acosh(-sqrt(2 + sqrt(2))/2) == 7*I*pi/8
assert acosh(sqrt(2 - sqrt(2))/2) == 3*I*pi/8
assert acosh(-sqrt(2 - sqrt(2))/2) == 5*I*pi/8
assert acosh((1 + sqrt(3))/(2*sqrt(2))) == I*pi/12
assert acosh(-(1 + sqrt(3))/(2*sqrt(2))) == 11*I*pi/12
assert acosh((sqrt(5) + 1)/4) == I*pi/5
assert acosh(-(sqrt(5) + 1)/4) == 4*I*pi/5
def test_messy():
from sympy import (laplace_transform, Si, Shi, Chi, atan, Piecewise,
acoth, E1, besselj, acosh, asin, And, re,
fourier_transform, sqrt)
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi/2)/s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s)/s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(S(0) < re(a/2) + S(1)/2, S(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(S(1)/2 + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_issue_2314():
# Note that this is not the same as testing ratint() becuase integrate()
# pulls out the coefficient.
a = Symbol('a')
assert integrate(-a/(a**2+x**2), x) == \
-a*(sqrt(-1/a**2)*log(x + a**2*sqrt(-1/a**2))/2 - sqrt(-1/a**2)*log(x -
a**2*sqrt(-1/a**2))/2)
def test_asinh():
x, y = symbols('x,y')
assert asinh(x) == asinh(x)
assert asinh(-x) == -asinh(x)
assert asinh(nan) == nan
assert asinh( 0) == 0
assert asinh(+1) == log(sqrt(2) + 1)
assert asinh(-1) == log(sqrt(2) - 1)
assert asinh(I) == pi*I/2
assert asinh(-I) == -pi*I/2
assert asinh(I/2) == pi*I/6
assert asinh(-I/2) == -pi*I/6
assert asinh(oo) == oo
assert asinh(-oo) == -oo
assert asinh(I*oo) == oo
assert asinh(-I *oo) == -oo
assert asinh(zoo) == zoo
assert asinh(I *(sqrt(3) - 1)/(2**(S(3)/2))) == pi*I/12
assert asinh(-I *(sqrt(3) - 1)/(2**(S(3)/2))) == -pi*I/12
assert asinh(I*(sqrt(5) - 1)/4) == pi*I/10
assert asinh(-I*(sqrt(5) - 1)/4) == -pi*I/10
assert asinh(I*(sqrt(5) + 1)/4) == 3*pi*I/10
assert asinh(-I*(sqrt(5) + 1)/4) == -3*pi*I/10
def test_imageset_intersect_interval():
from sympy.abc import n
f1 = ImageSet(Lambda(n, n*pi), S.Integers)
f2 = ImageSet(Lambda(n, 2*n), Interval(0, pi))
f3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
# complex expressions
f4 = ImageSet(Lambda(n, n*I*pi), S.Integers)
f5 = ImageSet(Lambda(n, 2*I*n*pi + pi/2), S.Integers)
# non-linear expressions
f6 = ImageSet(Lambda(n, log(n)), S.Integers)
f7 = ImageSet(Lambda(n, n**2), S.Integers)
f8 = ImageSet(Lambda(n, Abs(n)), S.Integers)
f9 = ImageSet(Lambda(n, exp(n)), S.Naturals0)
assert f1.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f2.intersect(Interval(1, 2)) == Interval(1, 2)
assert f3.intersect(Interval(-1, 1)) == S.EmptySet
assert f3.intersect(Interval(-5, 5)) == FiniteSet(-3*pi/2, pi/2)
assert f4.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f4.intersect(Interval(1, 2)) == S.EmptySet
assert f5.intersect(Interval(0, 1)) == S.EmptySet
assert f6.intersect(Interval(0, 1)) == FiniteSet(S.Zero, log(2))
assert f7.intersect(Interval(0, 10)) == Intersection(f7, Interval(0, 10))
assert f8.intersect(Interval(0, 2)) == Intersection(f8, Interval(0, 2))
assert f9.intersect(Interval(1, 2)) == Intersection(f9, Interval(1, 2))
def test_H():
X = Normal('X', 0, 1)
D = Die('D', sides = 4)
G = Geometric('G', 0.5)
assert H(X, X > 0) == -log(2)/2 + S(1)/2 + log(pi)/2
assert H(D, D > 2) == log(2)
assert H(G).evalf().round(2) == 1.39
def test_trigsimp1():
x, y = symbols('x,y')
assert trigsimp(1 - sin(x)**2) == cos(x)**2
assert trigsimp(1 - cos(x)**2) == sin(x)**2
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
assert trigsimp(1/sin(x)**2 - 1) == cot(x)**2
assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) in \
[2 + 3*cos(x/2)**2, 5 - 3*sin(x/2)**2]
assert trigsimp(sin(x)/cos(x)) == tan(x)
assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
e = 2*sin(x)**2 + 2*cos(x)**2
assert trigsimp(log(e), deep=True) == log(2)
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1+x) == 1+x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1-GoldenRatio) == (1-sqrt(5))/2
assert nsimplify((1+sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2+I), [pi]) == sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == 2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi,log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi,log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
def test_seriesbug2c():
w = Symbol("w", real=True)
#more complicated case, but sin(x)~x, so the result is the same as in (1)
e=(sin(2*w)/w)**(1+w)
assert e.nseries(w,0,1) == 2 + O(w)
assert e.nseries(w,0,3) == 2-Rational(4,3)*w**2+w**2*log(2)**2+2*w*log(2)+O(w**3, w)
assert e.nseries(w,0,2).subs(w,0) == 2
def test_residue_reduce():
a = Poly(2*t**2 - t - x**2, t)
d = Poly(t**3 - x**2*t, t)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)], 'Tfuncs': [log]})
assert residue_reduce(a, d, DE, z, invert=False) == \
([(Poly(z**2 - S(1)/4, z), Poly((1 + 3*x*z - 6*z**2 -
2*x**2 + 4*x**2*z**2)*t - x*z + x**2 + 2*x**2*z**2 - 2*z*x**3, t))], False)
assert residue_reduce(a, d, DE, z, invert=True) == \
([(Poly(z**2 - S(1)/4, z), Poly(t + 2*x*z, t))], False)
assert residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t,), DE, z, invert=False) == \
([(Poly(z**2 - 1, z), Poly(-2*z*t/x - 2/x, t))], True)
ans = residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t), DE, z, invert=True)
assert ans == ([(Poly(z**2 - 1, z), Poly(t + z, t))], True)
assert residue_reduce_to_basic(ans[0], DE, z) == -log(-1 + log(x)) + log(1 + log(x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
# TODO: Skip or make faster
assert residue_reduce(Poly((-2*nu**2 - x**4)/(2*x**2)*t - (1 + x**2)/x, t),
Poly(t**2 + 1 + x**2/2, t), DE, z) == \
([(Poly(z + S(1)/2, z, domain='QQ'), Poly(t**2 + 1 + x**2/2, t, domain='EX'))], True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
assert residue_reduce(Poly(-2*x*t + 1 - x**2, t),
Poly(t**2 + 2*x*t + 1 + x**2, t), DE, z) == \
([(Poly(z**2 + S(1)/4, z), Poly(t + x + 2*z, t))], True)
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert residue_reduce(Poly(t, t), Poly(t + sqrt(2), t), DE, z) == \
([(Poly(z - 1, z), Poly(t + sqrt(2), t))], True)
def test_gruntz_evaluation_slow():
sskip()
assert gruntz((exp(exp(-x/(1+exp(-x))))*exp(-x/(1+exp(-x/(1+exp(-x)))))
*exp(exp(-x+exp(-x/(1+exp(-x))))))
/ (exp(-x/(1+exp(-x))))**2 - exp(x) + x, x, oo) == 2
assert gruntz(exp(exp(exp(x)/(1-1/x)))
- exp(exp(exp(x)/(1-1/x-log(x)**(-log(x))))), x, oo) == -oo
def test_ei():
pos = Symbol('p', positive=True)
neg = Symbol('n', negative=True)
assert Ei(-pos) == Ei(polar_lift(-1)*pos) - I*pi
assert Ei(neg) == Ei(polar_lift(neg)) - I*pi
assert tn_branch(Ei)
assert mytd(Ei(x), exp(x)/x, x)
assert mytn(Ei(x), Ei(x).rewrite(uppergamma),
-uppergamma(0, x*polar_lift(-1)) - I*pi, x)
assert mytn(Ei(x), Ei(x).rewrite(expint),
-expint(1, x*polar_lift(-1)) - I*pi, x)
assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
assert Ei(x*exp_polar(2*I*pi)) == Ei(x) + 2*I*pi
assert Ei(x*exp_polar(-2*I*pi)) == Ei(x) - 2*I*pi
assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
assert mytn(Ei(x*polar_lift(I)), Ei(x*polar_lift(I)).rewrite(Si),
Ci(x) + I*Si(x) + I*pi/2, x)
assert Ei(log(x)).rewrite(li) == li(x)
assert Ei(2*log(x)).rewrite(li) == li(x**2)
assert gruntz(Ei(x+exp(-x))*exp(-x)*x, x, oo) == 1
assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \
x**3/18 + x**4/96 + x**5/600 + O(x**6)
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2)/2,
log(tan(x) + I) + I*x,
log(tan(x) - I) - I*x,
]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723
assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3
assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x)
- 1) - atan(sqrt(2)*sin(x) + 1)
assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
def ideal_mixing_energy(self, phase, symbols, param_search):
#pylint: disable=W0613
"""
Returns the ideal mixing energy in symbolic form.
"""
# Normalize site ratios
site_ratio_normalization = self._site_ratio_normalization(phase)
site_ratios = phase.sublattices
site_ratios = [c/site_ratio_normalization for c in site_ratios]
ideal_mixing_term = S.Zero
for subl_index, sublattice in enumerate(phase.constituents):
active_comps = set(sublattice).intersection(self.components)
if len(active_comps) == 1:
continue # no mixing if only one species in sublattice
ratio = site_ratios[subl_index]
for comp in active_comps:
sitefrac = \
v.SiteFraction(phase.name, subl_index, comp)
# -35.8413*x term is there to keep derivative continuous
mixing_term = Piecewise((sitefrac * log(sitefrac), \
sitefrac > 1e-12), ((3.0*log(10.0)/(2.5e11))+ \
(sitefrac-1e-12)*(1.0-log(1e-12))+(5e-11) * \
(sitefrac-1e-12)**2, True))
ideal_mixing_term += (mixing_term*ratio)
ideal_mixing_term *= (v.R * v.T)
return ideal_mixing_term
def test_DifferentialExtension_all_attrs():
# Test 'unimportant' attributes
DE = DifferentialExtension(exp(x)*log(x), x, dummy=False, handle_first='exp')
assert DE.f == exp(x)*log(x)
assert DE.newf == t0*t1
assert DE.x == x
assert DE.cases == ['base', 'exp', 'primitive']
assert DE.case == 'primitive'
assert DE.level == -1
assert DE.t == t1 == DE.T[DE.level]
assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
raises(ValueError, lambda: DE.increment_level())
DE.decrement_level()
assert DE.level == -2
assert DE.t == t0 == DE.T[DE.level]
assert DE.d == Poly(t0, t0) == DE.D[DE.level]
assert DE.case == 'exp'
DE.decrement_level()
assert DE.level == -3
assert DE.t == x == DE.T[DE.level] == DE.x
assert DE.d == Poly(1, x) == DE.D[DE.level]
assert DE.case == 'base'
raises(ValueError, lambda: DE.decrement_level())
DE.increment_level()
DE.increment_level()
assert DE.level == -1
assert DE.t == t1 == DE.T[DE.level]
assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
assert DE.case == 'primitive'
def test_to_expr():
x = symbols('x')
R, Dx = DifferentialOperators(ZZ.old_poly_ring(x), 'Dx')
p = HolonomicFunction(Dx - 1, x, 0, [1]).to_expr()
q = exp(x)
assert p == q
p = HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).to_expr()
q = cos(x)
assert p == q
p = HolonomicFunction(Dx**2 - 1, x, 0, [1, 0]).to_expr()
q = cosh(x)
assert p == q
p = HolonomicFunction(2 + (4*x - 1)*Dx + \
(x**2 - x)*Dx**2, x, 0, [1, 2]).to_expr().expand()
q = 1/(x**2 - 2*x + 1)
assert p == q
p = expr_to_holonomic(sin(x)**2/x).integrate((x, 0, x)).to_expr()
q = (sin(x)**2/x).integrate((x, 0, x))
assert p == q
C_0, C_1, C_2, C_3 = symbols('C_0, C_1, C_2, C_3')
p = expr_to_holonomic(log(1+x**2)).to_expr()
q = C_2*log(x**2 + 1)
assert p == q
p = expr_to_holonomic(log(1+x**2)).diff().to_expr()
q = C_0*x/(x**2 + 1)
assert p == q
p = expr_to_holonomic(erf(x) + x).to_expr()
q = 3*C_3*x - 3*sqrt(pi)*C_3*erf(x)/2 + x + 2*x/sqrt(pi)
assert p == q
p = expr_to_holonomic(sqrt(x), x0=1).to_expr()
assert p == sqrt(x)
assert expr_to_holonomic(sqrt(x)).to_expr() == sqrt(x)
p = expr_to_holonomic(sqrt(1 + x**2)).to_expr()
assert p == sqrt(1+x**2)
p = expr_to_holonomic((2*x**2 + 1)**(S(2)/3)).to_expr()
assert p == (2*x**2 + 1)**(S(2)/3)
p = expr_to_holonomic(sqrt(-x**2+2*x)).to_expr()
assert p == sqrt(x)*sqrt(-x + 2)
p = expr_to_holonomic((-2*x**3+7*x)**(S(2)/3)).to_expr()
q = x**(S(2)/3)*(-2*x**2 + 7)**(S(2)/3)
assert p == q
p = from_hyper(hyper((-2, -3), (S(1)/2, ), x))
s = hyperexpand(hyper((-2, -3), (S(1)/2, ), x))
D_0 = Symbol('D_0')
C_0 = Symbol('C_0')
assert (p.to_expr().subs({C_0:1, D_0:0}) - s).simplify() == 0
p.y0 = {0: [1], S(1)/2: [0]}
assert p.to_expr() == s
assert expr_to_holonomic(x**5).to_expr() == x**5
assert expr_to_holonomic(2*x**3-3*x**2).to_expr().expand() == \
2*x**3-3*x**2
a = symbols("a")
p = (expr_to_holonomic(1.4*x)*expr_to_holonomic(a*x, x)).to_expr()
q = 1.4*a*x**2
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(a*x, x)).to_expr()
q = x*(a + 1.4)
assert p == q
p = (expr_to_holonomic(1.4*x)+expr_to_holonomic(x)).to_expr()
assert p == 2.4*x
def test_ratsimp():
f, g = 1/x + 1/y, (x + y)/(x*y)
assert f != g and ratsimp(f) == g
f, g = 1/(1 + 1/x), 1 - 1/(x + 1)
assert f != g and ratsimp(f) == g
f, g = x/(x + y) + y/(x + y), 1
assert f != g and ratsimp(f) == g
f, g = -x - y - y**2/(x + y) + x**2/(x + y), -2*y
assert f != g and ratsimp(f) == g
f = (a*c*x*y + a*c*z - b*d*x*y - b*d*z - b*t*x*y - b*t*x - b*t*z + e*x)/(x*y + z)
G = [a*c - b*d - b*t + (-b*t*x + e*x)/(x*y + z),
a*c - b*d - b*t - ( b*t*x - e*x)/(x*y + z)]
assert f != g and ratsimp(f) in G
A = sqrt(pi)
B = log(erf(x) - 1)
C = log(erf(x) + 1)
D = 8 - 8*erf(x)
f = A*B/D - A*C/D + A*C*erf(x)/D - A*B*erf(x)/D + 2*A/D
assert ratsimp(f) == A*B/8 - A*C/8 - A/(4*erf(x) - 4)
def test_issue_4153():
assert integrate(1 / (1 + x + y + z), (x, 0, 1), (y, 0, 1), (z, 0, 1)) in [
-12 * log(3) - 3 * log(6) / 2 + 3 * log(8) / 2 + 5 * log(2) +
7 * log(4), 6 * log(2) + 8 * log(4) - 27 * log(3) / 2,
22 * log(2) - 27 * log(3) / 2,
-12 * log(3) - 3 * log(6) / 2 + 47 * log(2) / 2
]
def test_issue_3740():
f = 4 * log(x) - 2 * log(x)**2
fid = diff(integrate(f, x), x)
assert abs(f.subs(x, 42).evalf() - fid.subs(x, 42).evalf()) < 1e-10
def test_log_polylog():
assert integrate(log(1 - x) / x, (x, 0, 1)) == -pi**2 / 6
def test_multiple_integration():
assert integrate((x**2) * (y**2), (x, 0, 1), (y, -1, 2)) == Rational(1)
assert integrate((y**2) * (x**2), x, y) == Rational(1, 9) * (x**3) * (y**3)
assert integrate(1/(x + 3)/(1 + x)**3, x) == \
-S(1)/8*log(3 + x) + S(1)/8*log(1 + x) + x/(4 + 8*x + 4*x**2)
assert integrate(sin(x * y) * y, (x, 0, 1), (y, 0, 1)) == -sin(1) + 1
def test_issue_4968():
assert integrate(sin(log(
x**2))) == x * sin(2 * log(x)) / 5 - 2 * x * cos(2 * log(x)) / 5
def test_issue_8901():
assert integrate(sinh(1.0 * x)) == 1.0 * cosh(1.0 * x)
assert integrate(tanh(1.0 * x)) == 1.0 * x - 1.0 * log(tanh(1.0 * x) + 1)
assert integrate(tanh(x)) == x - log(tanh(x) + 1)
def test_powers():
assert integrate(2**x + 3**x, x) == 2**x / log(2) + 3**x / log(3)
def test_issue_4992():
# Note: psi in _check_antecedents becomes NaN.
from sympy import simplify, expand_func, polygamma, gamma
a = Symbol('a', positive=True)
assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \
(a*polygamma(0, a) + 1)*gamma(a)
def test_is_convergent():
# divergence tests --
assert Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() is S.false
assert Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(3**(-2*n - 1)*n**n, (n, 1, oo)).is_convergent() is S.false
assert Sum((-1)**n*n, (n, 3, oo)).is_convergent() is S.false
assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false
# root test --
assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false
# integral test --
# p-series test --
assert Sum(1/(n**2 + 1), (n, 1, oo)).is_convergent() is S.true
assert Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent() is S.true
assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
# comparison test --
assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
assert Sum(1/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)), (n, 2, oo)).is_convergent() is S.false
assert Sum(2/(n*log(n)*log(log(n))**2), (n, 5, oo)).is_convergent() is S.true
assert Sum(2/(n*log(n)**2), (n, 2, oo)).is_convergent() is S.true
assert Sum((n - 1)/(n**2*log(n)**3), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(n*log(n)*log(log(n))), (n, 5, oo)).is_convergent() is S.false
assert Sum((n - 1)/(n*log(n)**3), (n, 3, oo)).is_convergent() is S.false
assert Sum(2/(n**2*log(n)), (n, 2, oo)).is_convergent() is S.true
# alternating series tests --
assert Sum((-1)**(n - 1)/(n**2 - 1), (n, 3, oo)).is_convergent() is S.true
# with -negativeInfinite Limits
assert Sum(1/(n**2 + 1), (n, -oo, 1)).is_convergent() is S.true
assert Sum(1/(n - 1), (n, -oo, -1)).is_convergent() is S.false
assert Sum(1/(n**2 - 1), (n, -oo, -5)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, 2)).is_convergent() is S.true
assert Sum(1/(n**2 - 1), (n, -oo, oo)).is_convergent() is S.true
# piecewise functions
f = Piecewise((n**(-2), n <= 1), (n**2, n > 1))
assert Sum(f, (n, 1, oo)).is_convergent() is S.false
assert Sum(f, (n, -oo, oo)).is_convergent() is S.false
#assert Sum(f, (n, -oo, 1)).is_convergent() is S.true
# integral test
assert Sum(log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
assert Sum(-log(n)/n**3, (n, 1, oo)).is_convergent() is S.true
# the following function has maxima located at (x, y) =
# (1.2, 0.43), (3.0, -0.25) and (6.8, 0.050)
eq = (x - 2)*(x**2 - 6*x + 4)*exp(-x)
assert Sum(eq, (x, 1, oo)).is_convergent() is S.true
def test_issue_4400():
n = Symbol('n', integer=True, positive=True)
assert integrate((x**n)*log(x), x) == \
n*x*x**n*log(x)/(n**2 + 2*n + 1) + x*x**n*log(x)/(n**2 + 2*n + 1) - \
x*x**n/(n**2 + 2*n + 1)
def test_issue_14111():
assert Sum(1/log(log(n)), (n, 22, oo)).is_convergent() is S.false
def test_limit_bug():
z = Symbol('z', zero=False)
assert integrate(sin(x*y*z), (x, 0, pi), (y, 0, pi)) == \
(log(z**2) + 2*EulerGamma + 2*log(pi))/(2*z) - \
(-log(pi*z) + log(pi**2*z**2)/2 + Ci(pi**2*z))/z + log(pi)/z
def test_acot_rewrite():
assert acot(x).rewrite(log) == I * log((x - I) / (x + I)) / 2
def test_issue_14484():
raises(NotImplementedError, lambda: Sum(sin(n)/log(log(n)), (n, 22, oo)).is_convergent())
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi / 2 + I * log(I * x + sqrt(1 - x**2))
assert acos(0).rewrite(atan) == S.Pi / 2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == S.Pi / 2 - asin(x)
def test_sin_times_absolutely_convergent():
assert Sum(sin(n) / n**3, (n, 1, oo)).is_convergent() is S.true
assert Sum(sin(n) * log(n) / n**3, (n, 1, oo)).is_convergent() is S.true
def test_logexppow(): # no eval()
x = Symbol('x', real=True)
w = Symbol('w')
e = (3**(1 + x) + 2**(1 + x)) / (3**x + 2**x)
assert e.subs(2**x, w) != e
assert e.subs(exp(x * log(Rational(2))), w) != e
def test_atan_rewrite():
assert atan(x).rewrite(log) == I * log((1 - I * x) / (1 + I * x)) / 2
def issue_1785():
assert integrate(
sqrt(x) *
(1 + x)) == 2 * x**Rational(3, 2) / 3 + 2 * x**Rational(5, 2) / 5
assert integrate(x**x * (1 + log(x))) is not None
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I * log(I * x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2 * atan(x / (1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == S.Pi / 2 - acos(x)
def test_issue1417():
assert integrate(2**x - 2 * x, x) == 2**x / log(2) - x**2
def issue_1785_fail():
assert integrate(x**x * (1 + log(x)).expand(mul=True)) is None
def test_li():
z = Symbol("z")
zr = Symbol("z", real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) == 0
assert li(1) is -oo
assert li(oo) is oo
assert isinstance(li(z), li)
assert unchanged(li, -zp)
assert unchanged(li, zn)
assert diff(li(z), z) == 1 / log(z)
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert unchanged(conjugate, li(-zp))
assert unchanged(conjugate, li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1 / log(z)) / 2 - log(-log(z)) +
log(log(z)) / 2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I * log(z)) - log(1 / log(z)) / 2 +
log(log(z)) / 2 + Ci(I * log(z)) +
Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I * log(z)) - log(1 / log(z)) / 2 +
log(log(z)) / 2 + Ci(I * log(z)) +
Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1 / log(z)) / 2 + log(log(z)) / 2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1 / log(z)) / 2 + log(log(z)) / 2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) == (log(z) * hyper(
(1, 1),
(2, 2), log(z)) - log(1 / log(z)) / 2 + log(log(z)) / 2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1 / log(z)) / 2 - log(-log(z)) +
log(log(z)) / 2 - meijerg(
((), (1, )), ((0, 0), ()), -log(z)))
assert gruntz(1 / li(z), z, oo) == 0
assert li(z).series(z) == log(z)**5/600 + log(z)**4/96 + log(z)**3/18 + log(z)**2/4 + \
log(z) + log(log(z)) + EulerGamma
raises(ArgumentIndexError, lambda: li(z).fdiff(2))
def test_integrate_functions():
assert integrate(f(x), x) == Integral(f(x), x)
assert integrate(f(x), (x, 0, 1)) == Integral(f(x), (x, 0, 1))
assert integrate(f(x) * diff(f(x), x), x) == f(x)**2 / 2
assert integrate(diff(f(x), x) / f(x), x) == log(f(x))
def test_intrinsic_math_codegen():
# not included: log10
from sympy import (acos, asin, atan, ceiling, cos, cosh, floor, log, ln,
sin, sinh, sqrt, tan, tanh, N, Abs)
x = symbols('x')
name_expr = [
("test_abs", Abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
# ("test_ceil", ceiling(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
# ("test_floor", floor(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
result = codegen(name_expr, "F95", "file", header=False, empty=False)
assert result[0][0] == "file.f90"
expected = ('REAL*8 function test_abs(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_abs = Abs(x)\n'
'end function\n'
'REAL*8 function test_acos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_acos = acos(x)\n'
'end function\n'
'REAL*8 function test_asin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_asin = asin(x)\n'
'end function\n'
'REAL*8 function test_atan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_atan = atan(x)\n'
'end function\n'
'REAL*8 function test_cos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_cos = cos(x)\n'
'end function\n'
'REAL*8 function test_cosh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_cosh = cosh(x)\n'
'end function\n'
'REAL*8 function test_log(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_log = log(x)\n'
'end function\n'
'REAL*8 function test_ln(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_ln = log(x)\n'
'end function\n'
'REAL*8 function test_sin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sin = sin(x)\n'
'end function\n'
'REAL*8 function test_sinh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sinh = sinh(x)\n'
'end function\n'
'REAL*8 function test_sqrt(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_sqrt = sqrt(x)\n'
'end function\n'
'REAL*8 function test_tan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_tan = tan(x)\n'
'end function\n'
'REAL*8 function test_tanh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'test_tanh = tanh(x)\n'
'end function\n')
assert result[0][1] == expected
assert result[1][0] == "file.h"
expected = ('interface\n'
'REAL*8 function test_abs(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_acos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_asin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_atan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_cos(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_cosh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_log(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_ln(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sin(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sinh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_sqrt(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_tan(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n'
'interface\n'
'REAL*8 function test_tanh(x)\n'
'implicit none\n'
'REAL*8, intent(in) :: x\n'
'end function\n'
'end interface\n')
assert result[1][1] == expected
def test_improper_integral():
assert integrate(log(x), (x, 0, 1)) == -1
assert integrate(x**(-2), (x, 1, oo)) == 1
def log_to_real(h, q, x, t):
"""
Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
Examples
========
>>> from sympy.integrals.rationaltools import log_to_real
>>> from sympy.abc import x, y
>>> from sympy import Poly, sqrt, S
>>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'),
... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
>>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
... Poly(-2*y + 1, y, domain='ZZ'), x, y)
log(x**2 - 1)/2
See Also
========
log_to_atan
"""
u, v = symbols('u,v', cls=Dummy)
H = h.as_expr().subs({t: u + I * v}).expand()
Q = q.as_expr().subs({t: u + I * v}).expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(S(1), S(0)), H_map.get(I, S(0))
c, d = Q_map.get(S(1), S(0)), Q_map.get(I, S(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, filter='R')
if len(R_u) != R.count_roots():
return None
result = S(0)
for r_u in R_u.iterkeys():
C = Poly(c.subs({u: r_u}), v)
R_v = roots(C, filter='R')
if len(R_v) != C.count_roots():
return None
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u: r_u, v: r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u: r_u, v: r_v}), x)
B = Poly(b.subs({u: r_u, v: r_v}), x)
AB = (A**2 + B**2).as_expr()
result += r_u * log(AB) + r_v * log_to_atan(A, B)
R_q = roots(q, filter='R')
if len(R_q) != q.count_roots():
return None
for r in R_q.iterkeys():
result += r * log(h.as_expr().subs(t, r))
return result
def test_expint():
assert mytn(expint(x, y),
expint(x, y).rewrite(uppergamma),
y**(x - 1) * uppergamma(1 - x, y), x)
assert mytd(expint(x, y),
-y**(x - 1) * meijerg([], [1, 1], [0, 0, 1 - x], [], y), x)
assert mytd(expint(x, y), -expint(x - 1, y), y)
assert mytn(expint(1, x),
expint(1, x).rewrite(Ei), -Ei(x * polar_lift(-1)) + I * pi, x)
assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \
+ 24*exp(-x)/x**4 + 24*exp(-x)/x**5
assert expint(Rational(-3, 2), x) == \
exp(-x)/x + 3*exp(-x)/(2*x**2) + 3*sqrt(pi)*erfc(sqrt(x))/(4*x**S('5/2'))
assert tn_branch(expint, 1)
assert tn_branch(expint, 2)
assert tn_branch(expint, 3)
assert tn_branch(expint, 1.7)
assert tn_branch(expint, pi)
assert expint(y, x*exp_polar(2*I*pi)) == \
x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(y, x*exp_polar(-2*I*pi)) == \
x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(2,
x * exp_polar(2 * I * pi)) == 2 * I * pi * x + expint(2, x)
assert expint(2, x *
exp_polar(-2 * I * pi)) == -2 * I * pi * x + expint(2, x)
assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x)
assert expint(x, y).rewrite(Ei) == expint(x, y)
assert expint(x, y).rewrite(Ci) == expint(x, y)
assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x)
assert mytn(E1(polar_lift(I) * x),
E1(polar_lift(I) * x).rewrite(Si),
-Ci(x) + I * Si(x) - I * pi / 2, x)
assert mytn(expint(2, x),
expint(2, x).rewrite(Ei).rewrite(expint), -x * E1(x) + exp(-x),
x)
assert mytn(expint(3, x),
expint(3, x).rewrite(Ei).rewrite(expint),
x**2 * E1(x) / 2 + (1 - x) * exp(-x) / 2, x)
assert expint(Rational(3, 2), z).nseries(z) == \
2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \
2*sqrt(pi)*sqrt(z) + O(z**6)
assert E1(z).series(z) == -EulerGamma - log(z) + z - \
z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6)
assert expint(4, z).series(z) == Rational(1, 3) - z/2 + z**2/2 + \
z**3*(log(z)/6 - Rational(11, 36) + EulerGamma/6 - I*pi/6) - z**4/24 + \
z**5/240 + O(z**6)
assert expint(n, x).series(x, oo, n=3) == \
(n*(n + 1)/x**2 - n/x + 1 + O(x**(-3), (x, oo)))*exp(-x)/x
assert expint(z, y).series(z, 0, 2) == exp(-y) / y - z * meijerg(
((), (1, 1)), ((0, 0, 1), ()), y) / y + O(z**2)
raises(ArgumentIndexError, lambda: expint(x, y).fdiff(3))
neg = Symbol('neg', negative=True)
assert Ei(neg).rewrite(Si) == Shi(neg) + Chi(neg) - I * pi
('$PRED\nCL=KA', S('CL'), S('KA')),
('$PRED\nG = BASE - LESS', S('G'), S('BASE') - S('LESS')),
('$PRED CL = THETA(1) * LEFT', S('CL'), S('THETA(1)') * S('LEFT')),
('$PRED D2 = WGT / SRC', S('D2'), S('WGT') / S('SRC')),
('$PRED D = W * B + C', S('D'), S('W') * S('B') + S('C')),
('$PRED D = W * (B + C)', S('D'), S('W') * (S('B') + S('C'))),
('$PRED D = A+B*C+D', S('D'), S('A') + (S('B') * S('C')) + S('D')),
('$PRED D = A**2', S('D'), S('A')**2),
('$PRED D = A - (-2)', S('D'), S('A') + 2),
('$PRED D = A - (+2)', S('D'), S('A') - 2),
('$PRED D = 2.5', S('D'), 2.5),
('$PRED D = 1D-10', S('D'), 1e-10),
(
'$PK CL = PLOG(THETA(1))',
S('CL'),
sympy.Piecewise((sympy.log(2.8e-103), S('THETA(1)') < 2.8e-103),
(sympy.log(S('THETA(1)')), True)),
),
(
'$PK CL = PLOG10(THETA(1))',
S('CL'),
sympy.Piecewise(
(sympy.log(2.8e-103, 10), S('THETA(1)') < 2.8e-103),
(sympy.log(S('THETA(1)'), 10), True),
),
),
('$PRED CL = EXP(2)', S('CL'), sympy.exp(2)),
('$PRED CL = PEXP(2)', S('CL'), sympy.exp(2)),
('$PRED CL = PEXP(101)', S('CL'), sympy.exp(100)),
('$PRED CL = exp(2)', S('CL'), sympy.exp(2)),
('$PRED cL = eXp(2)', S('CL'), sympy.exp(2)),
def ratint(f, x, **flags):
"""Performs indefinite integration of rational functions.
Given a field :math:`K` and a rational function :math:`f = p/q`,
where :math:`p` and :math:`q` are polynomials in :math:`K[x]`,
returns a function :math:`g` such that :math:`f = g'`.
>>> from sympy.integrals.rationaltools import ratint
>>> from sympy.abc import x
>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
(12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)
References
==========
.. [Bro05] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
See Also
========
sympy.integrals.integrals.Integral.doit
ratint_logpart, ratint_ratpart
"""
if type(f) is not tuple:
p, q = f.as_numer_denom()
else:
p, q = f
p, q = Poly(p, x, composite=False), Poly(q, x, composite=False)
coeff, p, q = p.cancel(q)
poly, p = p.div(q)
result = poly.integrate(x).as_expr()
if p.is_zero:
return coeff * result
g, h = ratint_ratpart(p, q, x)
P, Q = h.as_numer_denom()
P = Poly(P, x)
Q = Poly(Q, x)
q, r = P.div(Q)
result += g + q.integrate(x).as_expr()
if not r.is_zero:
symbol = flags.get('symbol', 't')
if not isinstance(symbol, Symbol):
t = Dummy(symbol)
else:
t = symbol.as_dummy()
L = ratint_logpart(r, Q, x, t)
real = flags.get('real')
if real is None:
if type(f) is not tuple:
atoms = f.atoms()
else:
p, q = f
atoms = p.atoms() \
| q.atoms()
for elt in atoms - set([x]):
if not elt.is_real:
real = False
break
else:
real = True
eps = S(0)
if not real:
for h, q in L:
eps += RootSum(q,
Lambda(t, t * log(h.as_expr())),
quadratic=True)
else:
for h, q in L:
R = log_to_real(h, q, x, t)
if R is not None:
eps += R
else:
eps += RootSum(q,
Lambda(t, t * log(h.as_expr())),
quadratic=True)
result += eps
return coeff * result