def test_factorial_simplify_fail():
# simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
from sympy.abc import x
assert (
simplify(x * polygamma(0, x + 1) - x * polygamma(0, x + 2) + polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
)
def test_polygamma_expansion():
# A. & S., pa. 259 and 260
assert polygamma(0, 1 / x).nseries(x, n=3) == -log(x) - x / 2 - x ** 2 / 12 + O(x ** 4)
assert polygamma(1, 1 / x).series(x, n=5) == x + x ** 2 / 2 + x ** 3 / 6 + O(x ** 5)
assert polygamma(3, 1 / x).nseries(x, n=11) == 2 * x ** 3 + 3 * x ** 4 + 2 * x ** 5 - x ** 7 + 4 * x ** 9 / 3 + O(
x ** 11
)
def test_loggamma():
raises(TypeError, lambda: loggamma(2, 3))
raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
assert loggamma(x).diff(x) == polygamma(0, x)
s1 = loggamma(1/(x + sin(x)) + cos(x)).nseries(x, n=4)
s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
log(x)*x**2/2
assert (s1 - s2).expand(force=True).removeO() == 0
s1 = loggamma(1/x).series(x)
s2 = (1/x - S(1)/2)*log(1/x) - 1/x + log(2*pi)/2 + \
x/12 - x**3/360 + x**5/1260 + O(x**7)
assert ((s1 - s2).expand(force=True)).removeO() == 0
assert loggamma(x).rewrite('intractable') == log(gamma(x))
s1 = loggamma(x).series(x)
assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
assert s1 == loggamma(x).rewrite('intractable').series(x)
assert loggamma(x).is_real is None
y, z = Symbol('y', real=True), Symbol('z', imaginary=True)
assert loggamma(y).is_real
assert loggamma(z).is_real is False
def tN(N, M):
assert loggamma(1/x)._eval_nseries(x, n=N, logx=None).getn() == M
tN(0, 0)
tN(1, 1)
tN(2, 3)
tN(3, 3)
tN(4, 5)
tN(5, 5)
def test_factorial_diff():
n = Symbol('n', integer=True)
assert factorial(n).diff(n) == \
gamma(1 + n)*polygamma(0, 1 + n)
assert factorial(n**2).diff(n) == \
2*n*gamma(1 + n**2)*polygamma(0, 1 + n**2)
def fdiff(self, argindex=1):
from sympy import polygamma
if argindex == 1:
# http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/
n, k = self.args
return binomial(n, k) * (polygamma(0, n + 1) - 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) * (polygamma(0, n - k + 1) - polygamma(0, k + 1))
else:
raise ArgumentIndexError(self, argindex)
def test_function__eval_nseries():
n = Symbol("n")
assert sin(x)._eval_nseries(x, 2, None) == x + O(x ** 2)
assert sin(x + 1)._eval_nseries(x, 2, None) == x * cos(1) + sin(1) + O(x ** 2)
assert sin(pi * (1 - x))._eval_nseries(x, 2, None) == pi * x + O(x ** 2)
assert acos(1 - x ** 2)._eval_nseries(x, 2, None) == sqrt(2) * x + O(x ** 2)
assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == polygamma(n, 1) + polygamma(n + 1, 1) * x + O(x ** 2)
raises(PoleError, "sin(1/x)._eval_nseries(x,2,None)")
raises(PoleError, "acos(1-x)._eval_nseries(x,2,None)")
raises(PoleError, "acos(1+x)._eval_nseries(x,2,None)")
assert loggamma(1 / x)._eval_nseries(x, 0, None) == log(x) / 2 - log(x) / x - 1 / x + O(1, x)
assert loggamma(log(1 / x)).nseries(x, n=1, logx=y) == loggamma(-y)
def test_catalan():
assert catalan(1) == 1
assert catalan(2) == 2
assert catalan(3) == 5
assert catalan(4) == 14
assert catalan(x) == catalan(x)
assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
assert catalan(Rational(1,2)).rewrite(gamma) == 8/(3*pi)
assert catalan(3*x).rewrite(gamma) == 4**(3*x)*gamma(3*x + Rational(1,2))/(sqrt(pi)*gamma(3*x + 2))
assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
assert diff(catalan(x),x) == (polygamma(0, x + Rational(1,2)) - polygamma(0, x + 2) + 2*log(2))*catalan(x)
c = catalan(0.5).evalf()
assert str(c) == '0.848826363156775'
def test_gamma_series():
assert gamma(x + 1).series(x, 0, 3) == \
1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3)
assert gamma(x).series(x, -1, 3) == \
-1/x + EulerGamma - 1 + x*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma) \
+ x**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 -
polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O(x**3)
def fdiff(self, argindex=1):
from sympy import gamma, polygamma
if argindex == 1:
return gamma(self.args[0] + 1) * polygamma(0, self.args[0] + 1)
else:
raise ArgumentIndexError(self, argindex)
def test_lowergamma():
from sympy import meijerg, exp_polar, I, expint
assert lowergamma(x, y).diff(y) == y**(x-1)*exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
+ meijerg([], [1, 1], [0, 0, x], [], y)
assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False),
lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False),
lowergamma(S.Half - 3, x), x)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(S(1)/3, lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert lowergamma(x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
def test_gamma_series():
assert gamma(x + 1).series(x, 0, 3) == \
1 - EulerGamma*x + x**2*(EulerGamma**2/2 + pi**2/12) + O(x**3)
assert gamma(x).series(x, -1, 3) == \
-1/(x + 1) + EulerGamma - 1 + (x + 1)*(-1 - pi**2/12 - EulerGamma**2/2 + \
EulerGamma) + (x + 1)**2*(-1 - pi**2/12 - EulerGamma**2/2 + EulerGamma**3/6 - \
polygamma(2, 1)/6 + EulerGamma*pi**2/12 + EulerGamma) + O((x + 1)**3, (x, -1))
def test_FunctionWrapper():
import sympy
n, m, theta, phi = sympy.symbols("n, m, theta, phi")
r = sympy.Ynm(n, m, theta, phi)
s = Integer(2)*r
assert isinstance(s, Mul)
assert isinstance(s.args[1]._sympy_(), sympy.Ynm)
x = symbols("x")
e = x + sympy.loggamma(x)
assert str(e) == "x + loggamma(x)"
assert isinstance(e, Add)
assert e + sympy.loggamma(x) == x + 2*sympy.loggamma(x)
f = e.subs({x : 10})
assert f == 10 + log(362880)
f = e.subs({x : 2})
assert f == 2
f = e.subs({x : 100});
v = f.n(53, real=True);
assert abs(float(v) - 459.13420537) < 1e-7
f = e.diff(x)
assert f == 1 + sympy.polygamma(0, x)
def test_gamma():
assert gamma(nan) == nan
assert gamma(oo) == oo
assert gamma(-100) == zoo
assert gamma(0) == zoo
assert gamma(1) == 1
assert gamma(2) == 1
assert gamma(3) == 2
assert gamma(102) == factorial(101)
assert gamma(Rational(1,2)) == sqrt(pi)
assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi)
assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi)
assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi)
assert gamma(Rational(-1, 2)) == -2*sqrt(pi)
assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi)
assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi)
assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi)
assert gamma(x).diff(x) == gamma(x)*polygamma(0, x)
assert gamma(x - 1).expand(func=True) == gamma(x)/(x-1)
assert gamma(x + 2).expand(func=True, mul=False) == x*(x+1)*gamma(x)
assert expand_func(gamma(x + Rational(3, 2))) ==\
(x + Rational(1, 2))*gamma(x + Rational(1, 2))
def test_gamma():
assert gamma(nan) == nan
assert gamma(oo) == oo
assert gamma(-100) == zoo
assert gamma(0) == zoo
assert gamma(1) == 1
assert gamma(2) == 1
assert gamma(3) == 2
assert gamma(102) == factorial(101)
assert gamma(Rational(1, 2)) == sqrt(pi)
assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi)
assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi)
assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi)
assert gamma(Rational(-1, 2)) == -2*sqrt(pi)
assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi)
assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi)
assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi)
assert gamma(Rational(
-11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8))
assert gamma(Rational(
-10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3))
assert gamma(Rational(
14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3))
assert gamma(Rational(
17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7))
assert gamma(Rational(
19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8))
assert gamma(x).diff(x) == gamma(x)*polygamma(0, x)
assert gamma(x - 1).expand(func=True) == gamma(x)/(x - 1)
assert gamma(x + 2).expand(func=True, mul=False) == x*(x + 1)*gamma(x)
assert conjugate(gamma(x)) == gamma(conjugate(x))
assert expand_func(gamma(x + Rational(3, 2))) == \
(x + Rational(1, 2))*gamma(x + Rational(1, 2))
assert expand_func(gamma(x - Rational(1, 2))) == \
gamma(Rational(1, 2) + x)/(x - Rational(1, 2))
# Test a bug:
assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
assert gamma(3*exp_polar(I*pi)/4).is_nonnegative is False
assert gamma(3*exp_polar(I*pi)/4).is_nonpositive is True
# Issue 8526
k = Symbol('k', integer=True, nonnegative=True)
assert isinstance(gamma(k), gamma)
assert gamma(-k) == zoo
def test_gamma_series():
assert gamma(x + 1).series(x, 0, 3) == \
1 - x*EulerGamma + x**2*EulerGamma**2/2 + pi**2*x**2/12 + O(x**3)
assert gamma(x).series(x, -1, 3) == \
-1/x + EulerGamma - 1 + EulerGamma*x - EulerGamma**2*x/2 - pi**2*x/12 \
- x + EulerGamma*x**2 + EulerGamma*pi**2*x**2/12 - \
x**2*polygamma(2, 1)/6 + EulerGamma**3*x**2/6 - EulerGamma**2*x**2/2 \
- pi**2*x**2/12 - x**2 + O(x**3)
def test_evalf_default():
from sympy.functions.special.gamma_functions import polygamma
assert type(sin(4.0)) == Real
assert type(re(sin(I + 1.0))) == Real
assert type(im(sin(I + 1.0))) == Real
assert type(sin(4)) == sin
assert type(polygamma(2,4.0)) == Real
assert type(sin(Rational(1,4))) == sin
def test_function__eval_nseries():
n = Symbol("n")
assert sin(x)._eval_nseries(x, 2, None) == x + O(x ** 2)
assert sin(x + 1)._eval_nseries(x, 2, None) == x * cos(1) + sin(1) + O(x ** 2)
assert sin(pi * (1 - x))._eval_nseries(x, 2, None) == pi * x + O(x ** 2)
assert acos(1 - x ** 2)._eval_nseries(x, 2, None) == sqrt(2) * x + O(x ** 2)
assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == polygamma(n, 1) + polygamma(n + 1, 1) * x + O(x ** 2)
raises(PoleError, lambda: sin(1 / x)._eval_nseries(x, 2, None))
raises(PoleError, lambda: acos(1 - x)._eval_nseries(x, 2, None))
raises(PoleError, lambda: acos(1 + x)._eval_nseries(x, 2, None))
assert loggamma(1 / x)._eval_nseries(x, 0, None) == log(x) / 2 - log(x) / x - 1 / x + O(1, x)
assert loggamma(log(1 / x)).nseries(x, n=1, logx=y) == loggamma(-y)
# issue 6725:
assert expint(S(3) / 2, -x)._eval_nseries(x, 5, None) == 2 - 2 * sqrt(pi) * sqrt(
-x
) - 2 * x - x ** 2 / 3 - x ** 3 / 15 - x ** 4 / 84 + O(x ** 5)
assert sin(sqrt(x))._eval_nseries(x, 3, None) == sqrt(x) - x ** (S(3) / 2) / 6 + x ** (S(5) / 2) / 120 + O(x ** 3)
def test_binomial_diff():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert binomial(n, k).diff(n) == \
(-polygamma(0, 1 + n - k) + polygamma(0, 1 + n))*binomial(n, k)
assert binomial(n**2, k**3).diff(n) == \
2*n*(-polygamma(0, 1 + n**2 - k**3) + polygamma(0, 1 + n**2))*binomial(n**2, k**3)
assert binomial(n, k).diff(k) == \
(-polygamma(0, 1 + k) + polygamma(0, 1 + n - k))*binomial(n, k)
assert binomial(n**2, k**3).diff(k) == \
3*k**2*(-polygamma(0, 1 + k**3) + polygamma(0, 1 + n**2 - k**3))*binomial(n**2, k**3)
def test_lowergamma():
from sympy import meijerg
assert lowergamma(x, y).diff(y) == y ** (x - 1) * exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert lowergamma(x, y).diff(x) == gamma(x) * polygamma(0, x) - uppergamma(x, y) * log(y) + meijerg(
[], [1, 1], [0, 0, x], [], y
)
assert lowergamma(S.Half, x) == sqrt(pi) * erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False), lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False), lowergamma(S.Half - 3, x), x)
def test_harmonic_rewrite_polygamma():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
def test_lowergamma():
from sympy import meijerg, exp_polar, I, expint
assert lowergamma(x, 0) == 0
assert lowergamma(x, y).diff(y) == y**(x - 1)*exp(-y)
assert td(lowergamma(randcplx(), y), y)
assert td(lowergamma(x, randcplx()), x)
assert lowergamma(x, y).diff(x) == \
gamma(x)*polygamma(0, x) - uppergamma(x, y)*log(y) \
- meijerg([], [1, 1], [0, 0, x], [], y)
assert lowergamma(S.Half, x) == sqrt(pi)*erf(sqrt(x))
assert not lowergamma(S.Half - 3, x).has(lowergamma)
assert not lowergamma(S.Half + 3, x).has(lowergamma)
assert lowergamma(S.Half, x, evaluate=False).has(lowergamma)
assert tn(lowergamma(S.Half + 3, x, evaluate=False),
lowergamma(S.Half + 3, x), x)
assert tn(lowergamma(S.Half - 3, x, evaluate=False),
lowergamma(S.Half - 3, x), x)
assert tn_branch(-3, lowergamma)
assert tn_branch(-4, lowergamma)
assert tn_branch(S(1)/3, lowergamma)
assert tn_branch(pi, lowergamma)
assert lowergamma(3, exp_polar(4*pi*I)*x) == lowergamma(3, x)
assert lowergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*lowergamma(y, x*exp_polar(pi*I))
assert lowergamma(-2, exp_polar(5*pi*I)*x) == \
lowergamma(-2, x*exp_polar(I*pi)) + 2*pi*I
assert conjugate(lowergamma(x, y)) == lowergamma(conjugate(x), conjugate(y))
assert conjugate(lowergamma(x, 0)) == conjugate(lowergamma(x, 0))
assert conjugate(lowergamma(x, -oo)) == conjugate(lowergamma(x, -oo))
assert lowergamma(
x, y).rewrite(expint) == -y**x*expint(-x + 1, y) + gamma(x)
k = Symbol('k', integer=True)
assert lowergamma(
k, y).rewrite(expint) == -y**k*expint(-k + 1, y) + gamma(k)
k = Symbol('k', integer=True, positive=False)
assert lowergamma(k, y).rewrite(expint) == lowergamma(k, y)
assert lowergamma(x, y).rewrite(uppergamma) == gamma(x) - uppergamma(x, y)
assert lowergamma(70, 6) == factorial(69) - 69035724522603011058660187038367026272747334489677105069435923032634389419656200387949342530805432320 * exp(-6)
assert (lowergamma(S(77) / 2, 6) - lowergamma(S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
assert (lowergamma(-S(77) / 2, 6) - lowergamma(-S(77) / 2, 6, evaluate=False)).evalf() < 1e-16
def test_gamma():
assert gamma(nan) == nan
assert gamma(oo) == oo
assert gamma(-100) == zoo
assert gamma(0) == zoo
assert gamma(1) == 1
assert gamma(2) == 1
assert gamma(3) == 2
assert gamma(102) == factorial(101)
assert gamma(Rational(1,2)) == sqrt(pi)
assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi)
assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi)
assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi)
assert gamma(Rational(-1, 2)) == -2*sqrt(pi)
assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi)
assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi)
assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi)
assert gamma(Rational(-11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8))
assert gamma(Rational(-10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3))
assert gamma(Rational(14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3))
assert gamma(Rational(17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7))
assert gamma(Rational(19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8))
assert gamma(x).diff(x) == gamma(x)*polygamma(0, x)
assert gamma(x - 1).expand(func=True) == gamma(x)/(x-1)
assert gamma(x + 2).expand(func=True, mul=False) == x*(x+1)*gamma(x)
assert expand_func(gamma(x + Rational(3, 2))) == \
(x + Rational(1, 2))*gamma(x + Rational(1, 2))
assert expand_func(gamma(x - Rational(1, 2))) == \
gamma(Rational(1, 2) + x)/(x - Rational(1, 2))
# Test a bug:
assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
def test_harmonic_rewrite_sum_fail():
n = Symbol("n")
m = Symbol("m")
assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1)
assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
_k = Dummy("k")
assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n))
assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))
def test_polygamma():
from sympy import I
assert polygamma(n, nan) == nan
assert polygamma(0, oo) == oo
assert polygamma(0, -oo) == oo
assert polygamma(0, I * oo) == oo
assert polygamma(0, -I * oo) == oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) == zoo
assert polygamma(0, -9) == zoo
assert polygamma(0, -1) == zoo
assert polygamma(0, 0) == zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2 / 6
assert polygamma(1, 2) == pi**2 / 6 - 1
assert polygamma(1, 3) == pi**2 / 6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6 * (Rational(-22369, 20736) + pi**4 / 90)
assert polygamma(5, 1) == 8 * pi**6 / 63
def t(m, n):
x = S(m) / n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
assert polygamma(2, x).rewrite(zeta) == -2 * zeta(3, x)
assert polygamma(3, 7 * x).diff(x) == 7 * polygamma(4, 7 * x)
assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert polygamma(
2, x).rewrite(harmonic) == 2 * harmonic(x - 1, 3) - 2 * zeta(3)
ni = Symbol("n", integer=True)
assert polygamma(
ni,
x).rewrite(harmonic) == (-1)**(ni + 1) * (-harmonic(x - 1, ni + 1) +
zeta(ni + 1)) * factorial(ni)
# Polygamma of non-negative integer order is unbranched:
from sympy import exp_polar
k = Symbol('n', integer=True, nonnegative=True)
assert polygamma(k, exp_polar(2 * I * pi) * x) == polygamma(k, x)
# but negative integers are branched!
k = Symbol('n', integer=True)
assert polygamma(k,
exp_polar(2 * I * pi) *
x).args == (k, exp_polar(2 * I * pi) * x)
# Polygamma of order -1 is loggamma:
assert polygamma(-1, x) == loggamma(x)
# But smaller orders are iterated integrals and don't have a special name
assert polygamma(-2, x).func is polygamma
# Test a bug
assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
def t(m, n):
x = S(m) / n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
def test_loggamma():
raises(TypeError, lambda: loggamma(2, 3))
raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
assert loggamma(-1) is oo
assert loggamma(-2) is oo
assert loggamma(0) is oo
assert loggamma(1) == 0
assert loggamma(2) == 0
assert loggamma(3) == log(2)
assert loggamma(4) == log(6)
n = Symbol("n", integer=True, positive=True)
assert loggamma(n) == log(gamma(n))
assert loggamma(-n) is oo
assert loggamma(n / 2) == log(2**(-n + 1) * sqrt(pi) * gamma(n) /
gamma(n / 2 + S.Half))
assert loggamma(oo) is oo
assert loggamma(-oo) is zoo
assert loggamma(I * oo) is zoo
assert loggamma(-I * oo) is zoo
assert loggamma(zoo) is zoo
assert loggamma(nan) is nan
L = loggamma(Rational(16, 3))
E = -5 * log(3) + loggamma(Rational(
1, 3)) + log(4) + log(7) + log(10) + log(13)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(Rational(19, 4))
E = -4 * log(4) + loggamma(Rational(
3, 4)) + log(3) + log(7) + log(11) + log(15)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(Rational(23, 7))
E = -3 * log(7) + log(2) + loggamma(Rational(2, 7)) + log(9) + log(16)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(Rational(19, 4) - 7)
E = -log(9) - log(5) + loggamma(Rational(3, 4)) + 3 * log(4) - 3 * I * pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(Rational(23, 7) - 6)
E = -log(19) - log(12) - log(5) + loggamma(Rational(
2, 7)) + 3 * log(7) - 3 * I * pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
assert loggamma(x).diff(x) == polygamma(0, x)
s1 = loggamma(1 / (x + sin(x)) + cos(x)).nseries(x, n=4)
s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
log(x)*x**2/2
assert (s1 - s2).expand(force=True).removeO() == 0
s1 = loggamma(1 / x).series(x)
s2 = (1/x - S.Half)*log(1/x) - 1/x + log(2*pi)/2 + \
x/12 - x**3/360 + x**5/1260 + O(x**7)
assert ((s1 - s2).expand(force=True)).removeO() == 0
assert loggamma(x).rewrite('intractable') == log(gamma(x))
s1 = loggamma(x).series(x).cancel()
assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
assert s1 == loggamma(x).rewrite('intractable').series(x).cancel()
assert conjugate(loggamma(x)) == loggamma(conjugate(x))
assert conjugate(loggamma(0)) is oo
assert conjugate(loggamma(1)) == loggamma(conjugate(1))
assert conjugate(loggamma(-oo)) == conjugate(zoo)
assert loggamma(Symbol('v', positive=True)).is_real is True
assert loggamma(Symbol('v', zero=True)).is_real is False
assert loggamma(Symbol('v', negative=True)).is_real is False
assert loggamma(Symbol('v', nonpositive=True)).is_real is False
assert loggamma(Symbol('v', nonnegative=True)).is_real is None
assert loggamma(Symbol('v', imaginary=True)).is_real is None
assert loggamma(Symbol('v', real=True)).is_real is None
assert loggamma(Symbol('v')).is_real is None
assert loggamma(S.Half).is_real is True
assert loggamma(0).is_real is False
assert loggamma(Rational(-1, 2)).is_real is False
assert loggamma(I).is_real is None
assert loggamma(2 + 3 * I).is_real is None
def tN(N, M):
assert loggamma(1 / x)._eval_nseries(x, n=N).getn() == M
tN(0, 0)
tN(1, 1)
tN(2, 2)
tN(3, 3)
tN(4, 4)
tN(5, 5)
def test_issue1893():
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_polygamma_expand_func():
assert polygamma(0, x).expand(func=True) == polygamma(0, x)
assert polygamma(0,
2 * x).expand(func=True) == log(2) + polygamma(0, 2 * x)
assert polygamma(2, x).expand(func=True, basic=False) == polygamma(2, x)
#assert polygamma(2, 3*x).expand(func=True) == polygamma(2, 3*x)/9
#assert polygamma(3, 4*x).expand(func=True,basic=False) == polygamma(3, 4*x)/64
assert polygamma(0, 1 + x).expand(func=True,
basic=False) == 1 + x + polygamma(0, x)
assert polygamma(0,
2 + x).expand(func=True,
basic=False) == 5 + 2 * x + polygamma(0, x)
assert polygamma(0, 3 + x).expand(
func=True, basic=False) == 12 + 3 * x + polygamma(0, x)
assert polygamma(0, 4 + x).expand(
func=True, basic=False) == 22 + 4 * x + polygamma(0, x)
assert polygamma(1, 1 + x).expand(func=True,
basic=False) == -1 - x + polygamma(1, x)
assert polygamma(1, 2 + x).expand(
func=True, basic=False) == -5 - 2 * x + polygamma(1, x)
assert polygamma(1, 3 + x).expand(
func=True, basic=False) == -12 - 3 * x + polygamma(1, x)
assert polygamma(1, 4 + x).expand(
func=True, basic=False) == -22 - 4 * x + polygamma(1, x)
assert polygamma(0, x + y).expand(func=True,
basic=False) == polygamma(0, x + y)
assert polygamma(1, x + y).expand(func=True,
basic=False) == polygamma(1, x + y)
assert polygamma(1, 3 + 4 * x + y).expand(
func=True,
basic=False) == -12 - 12 * x - 3 * y + polygamma(1, y + 4 * x)
assert polygamma(3, 3 + 4 * x + y).expand(
func=True,
basic=False) == -72 - 72 * x - 18 * y + polygamma(3, y + 4 * x)
assert polygamma(3, 4 * x + y + 1).expand(
func=True,
basic=False) == -6 - 24 * x - 6 * y + polygamma(3, y + 4 * x)
assert polygamma(3, 4 * x + y + S(3) / 2).expand(func=True,
basic=False) == polygamma(
3,
S(3) / 2 + y + 4 * x)
assert polygamma(3, x + y + S(3) / 4).expand(func=True,
basic=False) == polygamma(
3,
S(3) / 4 + x + y)
def test_gamma():
assert gamma(nan) is nan
assert gamma(oo) is oo
assert gamma(-100) is zoo
assert gamma(0) is zoo
assert gamma(-100.0) is zoo
assert gamma(1) == 1
assert gamma(2) == 1
assert gamma(3) == 2
assert gamma(102) == factorial(101)
assert gamma(S.Half) == sqrt(pi)
assert gamma(Rational(3, 2)) == sqrt(pi) * S.Half
assert gamma(Rational(5, 2)) == sqrt(pi) * Rational(3, 4)
assert gamma(Rational(7, 2)) == sqrt(pi) * Rational(15, 8)
assert gamma(Rational(-1, 2)) == -2 * sqrt(pi)
assert gamma(Rational(-3, 2)) == sqrt(pi) * Rational(4, 3)
assert gamma(Rational(-5, 2)) == sqrt(pi) * Rational(-8, 15)
assert gamma(Rational(-15, 2)) == sqrt(pi) * Rational(256, 2027025)
assert gamma(Rational(
-11, 8)).expand(func=True) == Rational(64, 33) * gamma(Rational(5, 8))
assert gamma(Rational(
-10, 3)).expand(func=True) == Rational(81, 280) * gamma(Rational(2, 3))
assert gamma(Rational(
14, 3)).expand(func=True) == Rational(880, 81) * gamma(Rational(2, 3))
assert gamma(Rational(
17, 7)).expand(func=True) == Rational(30, 49) * gamma(Rational(3, 7))
assert gamma(Rational(
19, 8)).expand(func=True) == Rational(33, 64) * gamma(Rational(3, 8))
assert gamma(x).diff(x) == gamma(x) * polygamma(0, x)
assert gamma(x - 1).expand(func=True) == gamma(x) / (x - 1)
assert gamma(x + 2).expand(func=True, mul=False) == x * (x + 1) * gamma(x)
assert conjugate(gamma(x)) == gamma(conjugate(x))
assert expand_func(gamma(x + Rational(3, 2))) == \
(x + S.Half)*gamma(x + S.Half)
assert expand_func(gamma(x - S.Half)) == \
gamma(S.Half + x)/(x - S.Half)
# Test a bug:
assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
# XXX: Not sure about these tests. I can fix them by defining e.g.
# exp_polar.is_integer but I'm not sure if that makes sense.
assert gamma(3 * exp_polar(I * pi) / 4).is_nonnegative is False
assert gamma(3 * exp_polar(I * pi) / 4).is_extended_nonpositive is True
y = Symbol('y', nonpositive=True, integer=True)
assert gamma(y).is_real == False
y = Symbol('y', positive=True, noninteger=True)
assert gamma(y).is_real == True
assert gamma(-1.0, evaluate=False).is_real == False
assert gamma(0, evaluate=False).is_real == False
assert gamma(-2, evaluate=False).is_real == False
def series_small_a_small_b():
"""Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5.
Be aware of cancellation of poles in b=0 of digamma(b)/Gamma(b) and
polygamma functions.
digamma(b)/Gamma(b) = -1 - 2*M_EG*b + O(b^2)
digamma(b)^2/Gamma(b) = 1/b + 3*M_EG + b*(-5/12*PI^2+7/2*M_EG^2) + O(b^2)
polygamma(1, b)/Gamma(b) = 1/b + M_EG + b*(1/12*PI^2 + 1/2*M_EG^2) + O(b^2)
and so on.
"""
order = 5
a, b, x, k = symbols("a b x k")
M_PI, M_EG, M_Z3 = symbols("M_PI M_EG M_Z3")
c_subs = {pi: M_PI, EulerGamma: M_EG, zeta(3): M_Z3}
A = [] # terms with a
X = [] # terms with x
B = [] # terms with b (polygammas expanded)
C = [] # terms that generate B
# Phi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i])
# B[0] = 1
# B[k] = sum(C[k] * b**k/k!, k=0..)
# Note: C[k] can be obtained from a series expansion of 1/gamma(b).
expression = gamma(b)/sympy.exp(x) * \
Sum(x**k/factorial(k)/gamma(a*k+b), (k, 0, S.Infinity))
# nth term of taylor series in a=0: a^n/n! * (d^n Phi(a, b, x)/da^n at a=0)
for n in range(0, order+1):
term = expression.diff(a, n).subs(a, 0).simplify().doit()
# set the whole bracket involving polygammas to 1
x_part = (term.subs(polygamma(0, b), 1)
.replace(polygamma, lambda *args: 0))
# sign convetion: x part always positive
x_part *= (-1)**n
# expansion of polygamma part with 1/gamma(b)
pg_part = term/x_part/gamma(b)
if n >= 1:
# Note: highest term is digamma^n
pg_part = pg_part.replace(polygamma,
lambda k, x: pg_series(k, x, order+1+n))
pg_part = (pg_part.series(b, 0, n=order+1-n)
.removeO()
.subs(polygamma(2, 1), -2*zeta(3))
.simplify()
)
A.append(a**n/factorial(n))
X.append(horner(x_part))
B.append(pg_part)
# Calculate C and put in the k!
C = sympy.Poly(B[1].subs(c_subs), b).coeffs()
C.reverse()
for i in range(len(C)):
C[i] = (C[i] * factorial(i)).simplify()
s = "Tylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5."
s += "\nPhi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i], i=0..5)\n"
s += "B[0] = 1\n"
s += "B[i] = sum(C[k+i-1] * b**k/k!, k=0..)\n"
s += "\nM_PI = pi"
s += "\nM_EG = EulerGamma"
s += "\nM_Z3 = zeta(3)"
for name, c in zip(['A', 'X'], [A, X]):
for i in range(len(c)):
s += f"\n{name}[{i}] = "
s += str(c[i])
# For C, do also compute the values numerically
for i in range(len(C)):
s += f"\n# C[{i}] = "
s += str(C[i])
s += f"\nC[{i}] = "
s += str(C[i].subs({M_EG: EulerGamma, M_PI: pi, M_Z3: zeta(3)})
.evalf(17))
# Does B have the assumed structure?
s += "\n\nTest if B[i] does have the assumed structure."
s += "\nC[i] are derived from B[1] allone."
s += "\nTest B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + .."
test = sum([b**k/factorial(k) * C[k+1] for k in range(order-1)])
test = (test - B[2].subs(c_subs)).simplify()
s += f"\ntest successful = {test==S(0)}"
s += "\nTest B[3] == C[2] + b*C[3] + b^2/2*C[4] + .."
test = sum([b**k/factorial(k) * C[k+2] for k in range(order-2)])
test = (test - B[3].subs(c_subs)).simplify()
s += f"\ntest successful = {test==S(0)}"
return s
def test_polygamma_expand_func():
assert polygamma(0, x).expand(func=True) == polygamma(0, x)
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols("s t mu", real=True)
assert integrate(
meijerg([], [], [0], [], s * t) * meijerg([], [], [mu / 2], [-mu / 2], t ** 2 / 4), (t, 0, oo)
).is_Piecewise
s = symbols("s", positive=True)
assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo)) == gamma(s + 1)
assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols("a b", positive=True)
assert simplify(meijerint_definite(x ** a, x, 0, b)[0]) == b ** (a + 1) / (a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1) ** 3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols("sigma mu", positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma)) ** 2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (2 - erfc(mu / (2 * sigma)))
assert c == True
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == 1 - exp(-exp(I * arg(x)) * abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x ** 2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2 * x - 3) ** 2), x, -oo, oo) == (sqrt(pi) / 2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x - mu) / sigma) ** 2 / 2) / sqrt(2 * pi * sigma ** 2), x, -oo, oo) == (1, True)
assert meijerint_definite(sinc(x) ** 2, x, -oo, oo) == (pi, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S(1) / 2, True)
# Test a bug
def res(n):
return (1 / (1 + x ** 2)).diff(x, n).subs(x, 1) * (-1) ** n
for n in range(6):
assert integrate(exp(-x) * sin(x) * x ** n, (x, 0, oo), meijerg=True) == res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo), meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols("a b s")
from sympy import And, re
assert meijerint_definite(
meijerg([], [], [a / 2], [-a / 2], x / 4) * meijerg([], [], [b / 2], [-b / 2], x / 4) * x ** (s - 1), x, 0, oo
) == (
4
* 2 ** (2 * s - 2)
* gamma(-2 * s + 1)
* gamma(a / 2 + b / 2 + s)
/ (gamma(-a / 2 + b / 2 - s + 1) * gamma(a / 2 - b / 2 - s + 1) * gamma(a / 2 + b / 2 - s + 1)),
And(0 < -2 * re(4 * s) + 8, 0 < re(a / 2 + b / 2 + s), re(2 * s) < 1),
)
# test a bug
assert integrate(sin(x ** a) * sin(x ** b), (x, 0, oo), meijerg=True) == Integral(
sin(x ** a) * sin(x ** b), (x, 0, oo)
)
# test better hyperexpand
assert (
integrate(exp(-x ** 2) * log(x), (x, 0, oo), meijerg=True) == (sqrt(pi) * polygamma(0, S(1) / 2) / 4).expand()
)
# Test hyperexpand bug.
from sympy import lowergamma
n = symbols("n", integer=True)
assert simplify(integrate(exp(-x) * x ** n, x, meijerg=True)) == lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols("alpha", positive=True)
assert meijerint_definite((2 - x) ** alpha * sin(alpha / x), x, 0, 2) == (
sqrt(pi)
* alpha
* gamma(alpha + 1)
* meijerg(((), (alpha / 2 + S(1) / 2, alpha / 2 + 1)), ((0, 0, S(1) / 2), (-S(1) / 2,)), alpha ** S(2) / 16)
/ 4,
True,
)
# test a bug related to 3016
a, s = symbols("a s", positive=True)
assert (
simplify(integrate(x ** s * exp(-a * x ** 2), (x, -oo, oo)))
== a ** (-s / 2 - S(1) / 2) * ((-1) ** s + 1) * gamma(s / 2 + S(1) / 2) / 2
)
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_loggamma():
raises(TypeError, lambda: loggamma(2, 3))
raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
assert loggamma(-1) == oo
assert loggamma(-2) == oo
assert loggamma(0) == oo
assert loggamma(1) == 0
assert loggamma(2) == 0
assert loggamma(3) == log(2)
assert loggamma(4) == log(6)
n = Symbol("n", integer=True, positive=True)
assert loggamma(n) == log(gamma(n))
assert loggamma(-n) == oo
assert loggamma(n / 2) == log(2**(-n + 1) * sqrt(pi) * gamma(n) /
gamma(n / 2 + S.Half))
from sympy import I
assert loggamma(oo) == oo
assert loggamma(-oo) == zoo
assert loggamma(I * oo) == zoo
assert loggamma(-I * oo) == zoo
assert loggamma(zoo) == zoo
assert loggamma(nan) == nan
L = loggamma(S(16) / 3)
E = -5 * log(3) + loggamma(S(1) / 3) + log(4) + log(7) + log(10) + log(13)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(19 / S(4))
E = -4 * log(4) + loggamma(S(3) / 4) + log(3) + log(7) + log(11) + log(15)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(S(23) / 7)
E = -3 * log(7) + log(2) + loggamma(S(2) / 7) + log(9) + log(16)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(19 / S(4) - 7)
E = -log(9) - log(5) + loggamma(S(3) / 4) + 3 * log(4) - 3 * I * pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(23 / S(7) - 6)
E = -log(19) - log(12) - log(5) + loggamma(
S(2) / 7) + 3 * log(7) - 3 * I * pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
assert loggamma(x).diff(x) == polygamma(0, x)
s1 = loggamma(1 / (x + sin(x)) + cos(x)).nseries(x, n=4)
s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
log(x)*x**2/2
assert (s1 - s2).expand(force=True).removeO() == 0
s1 = loggamma(1 / x).series(x)
s2 = (1/x - S(1)/2)*log(1/x) - 1/x + log(2*pi)/2 + \
x/12 - x**3/360 + x**5/1260 + O(x**7)
assert ((s1 - s2).expand(force=True)).removeO() == 0
assert loggamma(x).rewrite('intractable') == log(gamma(x))
s1 = loggamma(x).series(x)
assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
assert s1 == loggamma(x).rewrite('intractable').series(x)
assert conjugate(loggamma(x)) == loggamma(conjugate(x))
assert conjugate(loggamma(0)) == conjugate(loggamma(0))
assert conjugate(loggamma(1)) == loggamma(conjugate(1))
assert conjugate(loggamma(-oo)) == conjugate(loggamma(-oo))
assert loggamma(x).is_real is None
y, z = Symbol('y', real=True), Symbol('z', imaginary=True)
assert loggamma(y).is_real
assert loggamma(z).is_real is False
def tN(N, M):
assert loggamma(1 / x)._eval_nseries(x, n=N).getn() == M
tN(0, 0)
tN(1, 1)
tN(2, 3)
tN(3, 3)
tN(4, 5)
tN(5, 5)
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols("s t mu", real=True)
assert integrate(
meijerg([], [], [0], [], s * t)
* meijerg([], [], [mu / 2], [-mu / 2], t ** 2 / 4),
(t, 0, oo),
).is_Piecewise
s = symbols("s", positive=True)
assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo)) == gamma(
s + 1
)
assert integrate(
x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=True
) == gamma(s + 1)
assert isinstance(
integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=False),
Integral,
)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols("a b", positive=True)
assert simplify(meijerint_definite(x ** a, x, 0, b)[0]) == b ** (a + 1) / (a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1) ** 3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols("sigma mu", positive=True)
i, c = meijerint_definite(exp(-(((x - mu) / (2 * sigma)) ** 2)), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (2 - erfc(mu / (2 * sigma)))
assert c == True
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == 1 - exp(
-exp(I * arg(x)) * abs(x)
)
# Test -oo to oo
assert meijerint_definite(exp(-(x ** 2)), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-((2 * x - 3) ** 2)), x, -oo, oo) == (
sqrt(pi) / 2,
True,
)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(
exp(-(((x - mu) / sigma) ** 2) / 2) / sqrt(2 * pi * sigma ** 2), x, -oo, oo
) == (1, True)
assert meijerint_definite(sinc(x) ** 2, x, -oo, oo) == (pi, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S.Half, True)
# Test a bug
def res(n):
return (1 / (1 + x ** 2)).diff(x, n).subs(x, 1) * (-1) ** n
for n in range(6):
assert integrate(exp(-x) * sin(x) * x ** n, (x, 0, oo), meijerg=True) == res(n)
# This used to test trigexpand... now it is done by linear substitution
assert (
simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo), meijerg=True))
== sqrt(2) * sin(a + pi / 4) / 2
)
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols("a b s")
from sympy import And, re
assert meijerint_definite(
meijerg([], [], [a / 2], [-a / 2], x / 4)
* meijerg([], [], [b / 2], [-b / 2], x / 4)
* x ** (s - 1),
x,
0,
oo,
) == (
4
* 2 ** (2 * s - 2)
* gamma(-2 * s + 1)
* gamma(a / 2 + b / 2 + s)
/ (
gamma(-a / 2 + b / 2 - s + 1)
* gamma(a / 2 - b / 2 - s + 1)
* gamma(a / 2 + b / 2 - s + 1)
),
And(0 < -2 * re(4 * s) + 8, 0 < re(a / 2 + b / 2 + s), re(2 * s) < 1),
)
# test a bug
assert integrate(sin(x ** a) * sin(x ** b), (x, 0, oo), meijerg=True) == Integral(
sin(x ** a) * sin(x ** b), (x, 0, oo)
)
# test better hyperexpand
assert (
integrate(exp(-(x ** 2)) * log(x), (x, 0, oo), meijerg=True)
== (sqrt(pi) * polygamma(0, S.Half) / 4).expand()
)
# Test hyperexpand bug.
from sympy import lowergamma
n = symbols("n", integer=True)
assert simplify(integrate(exp(-x) * x ** n, x, meijerg=True)) == lowergamma(
n + 1, x
)
# Test a bug with argument 1/x
alpha = symbols("alpha", positive=True)
assert meijerint_definite((2 - x) ** alpha * sin(alpha / x), x, 0, 2) == (
sqrt(pi)
* alpha
* gamma(alpha + 1)
* meijerg(
((), (alpha / 2 + S.Half, alpha / 2 + 1)),
((0, 0, S.Half), (Rational(-1, 2),)),
alpha ** 2 / 16,
)
/ 4,
True,
)
# test a bug related to 3016
a, s = symbols("a s", positive=True)
assert (
simplify(integrate(x ** s * exp(-a * x ** 2), (x, -oo, oo)))
== a ** (-s / 2 - S.Half) * ((-1) ** s + 1) * gamma(s / 2 + S.Half) / 2
)
def fdiff(self, argindex=1):
from sympy import gamma, polygamma
if argindex == 1:
return gamma(self.args[0] + 1) * polygamma(0, self.args[0] + 1)
else:
raise ArgumentIndexError(self, argindex)
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols('s t mu', real=True)
assert integrate(meijerg([], [], [0], [], s*t)
*meijerg([], [], [mu/2], [-mu/2], t**2/4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[],[]], [[0],[]], x), (x, 0, oo)) \
== gamma(s + 1)
assert integrate(x**s*meijerg([[],[]], [[0],[]], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x**s*meijerg([[],[]], [[0],[]], x),
(x, 0, oo), meijerg=False),
Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) \
== b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
meijerint_definite((x+1)**3*exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x-mu)/(2*sigma))**2), x, 0, oo)
assert simplify(i) \
== sqrt(pi)*sigma*(erf(mu/(2*sigma)) + 1)
assert c is True
i, _ = meijerint_definite(exp(-mu*x)*exp(sigma*x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1/(mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x-3)**2), x, -oo, oo) == (sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2*x-3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x-mu)/sigma)**2/2)/sqrt(2*pi*sigma**2),
x, -oo, oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x)*sin(x), x, 0, oo) == (S(1)/2, True)
# Test a bug
def res(n): return (1/(1+x**2)).diff(x, n).subs(x,1)*(-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x)*sin(x + a), (x, 0, oo), meijerg=True)).expand().rewrite(sin).expand() == \
sin(a)/2 + cos(a)/2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
from sympy import And, re
assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4) \
*meijerg([], [], [b/2], [-b/2], x/4)*x**(s-1), x, 0, oo) == \
(4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s) \
/(gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1) \
*gamma(a/2 + b/2 - s + 1)),
And(0 < -2*re(4*s) + 8, 0 < re(a/2 + b/2 + s), re(2*s) < 1))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, S(1)/2)/4).expand()
# Test hyperexpand bug.
from sympy import lowergamma
n = symbols('n', integer = True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2-x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*gamma(alpha + 1) \
*meijerg([S(1)/2, 0, S(1)/2], [1], [],
[-alpha/2, -alpha/2 - S(1)/2], 16/alpha**2), True)
def test_polygamma():
assert polygamma(n, nan) is nan
assert polygamma(0, oo) is oo
assert polygamma(0, -oo) is oo
assert polygamma(0, I * oo) is oo
assert polygamma(0, -I * oo) is oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) is zoo
assert polygamma(0, -9) is zoo
assert polygamma(0, -1) is zoo
assert polygamma(0, 0) is zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2 / 6
assert polygamma(1, 2) == pi**2 / 6 - 1
assert polygamma(1, 3) == pi**2 / 6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6 * (Rational(-22369, 20736) + pi**4 / 90)
assert polygamma(5, 1) == 8 * pi**6 / 63
assert polygamma(1, S.Half) == pi**2 / 2
assert polygamma(2, S.Half) == -14 * zeta(3)
assert polygamma(11, S.Half) == 176896 * pi**12
def t(m, n):
x = S(m) / n
r = polygamma(0, x)
if r.has(polygamma):
return False
return abs(polygamma(0, x.n()).n() - r.n()).n() < 1e-10
assert t(1, 2)
assert t(3, 2)
assert t(-1, 2)
assert t(1, 4)
assert t(-3, 4)
assert t(1, 3)
assert t(4, 3)
assert t(3, 4)
assert t(2, 3)
assert t(123, 5)
assert polygamma(0, x).rewrite(zeta) == polygamma(0, x)
assert polygamma(1, x).rewrite(zeta) == zeta(2, x)
assert polygamma(2, x).rewrite(zeta) == -2 * zeta(3, x)
assert polygamma(I, 2).rewrite(zeta) == polygamma(I, 2)
n1 = Symbol('n1')
n2 = Symbol('n2', real=True)
n3 = Symbol('n3', integer=True)
n4 = Symbol('n4', positive=True)
n5 = Symbol('n5', positive=True, integer=True)
assert polygamma(n1, x).rewrite(zeta) == polygamma(n1, x)
assert polygamma(n2, x).rewrite(zeta) == polygamma(n2, x)
assert polygamma(n3, x).rewrite(zeta) == polygamma(n3, x)
assert polygamma(n4, x).rewrite(zeta) == polygamma(n4, x)
assert polygamma(
n5,
x).rewrite(zeta) == (-1)**(n5 + 1) * factorial(n5) * zeta(n5 + 1, x)
assert polygamma(3, 7 * x).diff(x) == 7 * polygamma(4, 7 * x)
assert polygamma(0, x).rewrite(harmonic) == harmonic(x - 1) - EulerGamma
assert polygamma(
2, x).rewrite(harmonic) == 2 * harmonic(x - 1, 3) - 2 * zeta(3)
ni = Symbol("n", integer=True)
assert polygamma(
ni,
x).rewrite(harmonic) == (-1)**(ni + 1) * (-harmonic(x - 1, ni + 1) +
zeta(ni + 1)) * factorial(ni)
# Polygamma of non-negative integer order is unbranched:
k = Symbol('n', integer=True, nonnegative=True)
assert polygamma(k, exp_polar(2 * I * pi) * x) == polygamma(k, x)
# but negative integers are branched!
k = Symbol('n', integer=True)
assert polygamma(k,
exp_polar(2 * I * pi) *
x).args == (k, exp_polar(2 * I * pi) * x)
# Polygamma of order -1 is loggamma:
assert polygamma(-1, x) == loggamma(x)
# But smaller orders are iterated integrals and don't have a special name
assert polygamma(-2, x).func is polygamma
# Test a bug
assert polygamma(0, -x).expand(func=True) == polygamma(0, -x)
assert polygamma(2, 2.5).is_positive == False
assert polygamma(2, -2.5).is_positive == False
assert polygamma(3, 2.5).is_positive == True
assert polygamma(3, -2.5).is_positive is True
assert polygamma(-2, -2.5).is_positive is None
assert polygamma(-3, -2.5).is_positive is None
assert polygamma(2, 2.5).is_negative == True
assert polygamma(3, 2.5).is_negative == False
assert polygamma(3, -2.5).is_negative == False
assert polygamma(2, -2.5).is_negative is True
assert polygamma(-2, -2.5).is_negative is None
assert polygamma(-3, -2.5).is_negative is None
assert polygamma(I, 2).is_positive is None
assert polygamma(I, 3).is_negative is None
# issue 17350
assert polygamma(pi, 3).evalf() == polygamma(pi, 3)
assert (I*polygamma(I, pi)).as_real_imag() == \
(-im(polygamma(I, pi)), re(polygamma(I, pi)))
assert (tanh(polygamma(I, 1))).rewrite(exp) == \
(exp(polygamma(I, 1)) - exp(-polygamma(I, 1)))/(exp(polygamma(I, 1)) + exp(-polygamma(I, 1)))
assert (I / polygamma(I, 4)).rewrite(exp) == \
I*sqrt(re(polygamma(I, 4))**2 + im(polygamma(I, 4))**2)\
/((re(polygamma(I, 4)) + I*im(polygamma(I, 4)))*Abs(polygamma(I, 4)))
assert unchanged(polygamma, 2.3, 1.0)
# issue 12569
assert unchanged(im, polygamma(0, I))
assert polygamma(Symbol('a', positive=True), Symbol(
'b', positive=True)).is_real is True
assert polygamma(0, I).is_real is None
def test_issue1893():
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_loggamma():
raises(TypeError, lambda: loggamma(2, 3))
raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2))
assert loggamma(-1) == oo
assert loggamma(-2) == oo
assert loggamma(0) == oo
assert loggamma(1) == 0
assert loggamma(2) == 0
assert loggamma(3) == log(2)
assert loggamma(4) == log(6)
n = Symbol("n", integer=True, positive=True)
assert loggamma(n) == log(gamma(n))
assert loggamma(-n) == oo
assert loggamma(n/2) == log(2**(-n + 1)*sqrt(pi)*gamma(n)/gamma(n/2 + S.Half))
from sympy import I
assert loggamma(oo) == oo
assert loggamma(-oo) == zoo
assert loggamma(I*oo) == zoo
assert loggamma(-I*oo) == zoo
assert loggamma(zoo) == zoo
assert loggamma(nan) == nan
L = loggamma(S(16)/3)
E = -5*log(3) + loggamma(S(1)/3) + log(4) + log(7) + log(10) + log(13)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(19/S(4))
E = -4*log(4) + loggamma(S(3)/4) + log(3) + log(7) + log(11) + log(15)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(S(23)/7)
E = -3*log(7) + log(2) + loggamma(S(2)/7) + log(9) + log(16)
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(19/S(4)-7)
E = -log(9) - log(5) + loggamma(S(3)/4) + 3*log(4) - 3*I*pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
L = loggamma(23/S(7)-6)
E = -log(19) - log(12) - log(5) + loggamma(S(2)/7) + 3*log(7) - 3*I*pi
assert expand_func(L).doit() == E
assert L.n() == E.n()
assert loggamma(x).diff(x) == polygamma(0, x)
s1 = loggamma(1/(x + sin(x)) + cos(x)).nseries(x, n=4)
s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \
log(x)*x**2/2
assert (s1 - s2).expand(force=True).removeO() == 0
s1 = loggamma(1/x).series(x)
s2 = (1/x - S(1)/2)*log(1/x) - 1/x + log(2*pi)/2 + \
x/12 - x**3/360 + x**5/1260 + O(x**7)
assert ((s1 - s2).expand(force=True)).removeO() == 0
assert loggamma(x).rewrite('intractable') == log(gamma(x))
s1 = loggamma(x).series(x)
assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \
pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6)
assert s1 == loggamma(x).rewrite('intractable').series(x)
assert conjugate(loggamma(x)) == loggamma(conjugate(x))
assert conjugate(loggamma(0)) == conjugate(loggamma(0))
assert conjugate(loggamma(1)) == loggamma(conjugate(1))
assert conjugate(loggamma(-oo)) == conjugate(loggamma(-oo))
assert loggamma(x).is_real is None
y, z = Symbol('y', real=True), Symbol('z', imaginary=True)
assert loggamma(y).is_real
assert loggamma(z).is_real is False
def tN(N, M):
assert loggamma(1/x)._eval_nseries(x, n=N).getn() == M
tN(0, 0)
tN(1, 1)
tN(2, 3)
tN(3, 3)
tN(4, 5)
tN(5, 5)
def test_polygamma_expand_func():
assert polygamma(0, x).expand(func=True) == polygamma(0, x)
assert polygamma(0, 2*x).expand(func=True) == \
polygamma(0, x)/2 + polygamma(0, Rational(1, 2) + x)/2 + log(2)
assert polygamma(1, 2*x).expand(func=True) == \
polygamma(1, x)/4 + polygamma(1, Rational(1, 2) + x)/4
assert polygamma(2, x).expand(func=True) == \
polygamma(2, x)
assert polygamma(0, -1 + x).expand(func=True) == \
polygamma(0, x) - 1/(x - 1)
assert polygamma(0, 1 + x).expand(func=True) == \
1/x + polygamma(0, x )
assert polygamma(0, 2 + x).expand(func=True) == \
1/x + 1/(1 + x) + polygamma(0, x)
assert polygamma(0, 3 + x).expand(func=True) == \
polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x)
assert polygamma(0, 4 + x).expand(func=True) == \
polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x)
assert polygamma(1, 1 + x).expand(func=True) == \
polygamma(1, x) - 1/x**2
assert polygamma(1, 2 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2
assert polygamma(1, 3 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2
assert polygamma(1, 4 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \
1/(2 + x)**2 - 1/(3 + x)**2
assert polygamma(0, x + y).expand(func=True) == \
polygamma(0, x + y)
assert polygamma(1, x + y).expand(func=True) == \
polygamma(1, x + y)
assert polygamma(1, 3 + 4*x + y).expand(func=True, multinomial=False) == \
polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \
1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2
assert polygamma(3, 3 + 4*x + y).expand(func=True, multinomial=False) == \
polygamma(3, y + 4*x) - 6/(y + 4*x)**4 - \
6/(1 + y + 4*x)**4 - 6/(2 + y + 4*x)**4
assert polygamma(3, 4*x + y + 1).expand(func=True, multinomial=False) == \
polygamma(3, y + 4*x) - 6/(y + 4*x)**4
e = polygamma(3, 4 * x + y + S(3) / 2)
assert e.expand(func=True) == e
e = polygamma(3, x + y + S(3) / 4)
assert e.expand(func=True, basic=False) == e
def test_polygamma():
assert polygamma(n, nan) == nan
assert polygamma(0, oo) == oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) == zoo
assert polygamma(0, -9) == zoo
assert polygamma(0, -1) == zoo
assert polygamma(0, 0) == zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2/6
assert polygamma(1, 2) == pi**2/6 - 1
assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6*(Rational(-22369,20736) + pi**4/90)
assert polygamma(5, 1) == 8 * pi**6 / 63
assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
def test_J5():
assert polygamma(0, R(1, 3)) == -EulerGamma - pi/2*sqrt(R(1, 3)) - R(3, 2)*log(3)
def test_polygamma_expand_func():
assert polygamma(0, x).expand(func=True) == polygamma(0, x)
assert polygamma(0, 2*x).expand(func=True) == \
polygamma(0, x)/2 + polygamma(0, Rational(1, 2) + x)/2 + log(2)
assert polygamma(1, 2*x).expand(func=True) == \
polygamma(1, x)/4 + polygamma(1, Rational(1, 2) + x)/4
assert polygamma(2, x).expand(func=True) == \
polygamma(2, x)
assert polygamma(0, -1 + x).expand(func=True) == \
polygamma(0, x) + 1/(1 - x)
assert polygamma(0, 1 + x).expand(func=True) == \
1/x + polygamma(0, x )
assert polygamma(0, 2 + x).expand(func=True) == \
1/x + 1/(1 + x) + polygamma(0, x)
assert polygamma(0, 3 + x).expand(func=True) == \
polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x)
assert polygamma(0, 4 + x).expand(func=True) == \
polygamma(0, x) + 1/x + 1/(1 + x) + 1/(2 + x) + 1/(3 + x)
assert polygamma(1, 1 + x).expand(func=True) == \
polygamma(1, x) - 1/x**2
assert polygamma(1, 2 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2
assert polygamma(1, 3 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - 1/(2 + x)**2
assert polygamma(1, 4 + x).expand(func=True, multinomial=False) == \
polygamma(1, x) - 1/x**2 - 1/(1 + x)**2 - \
1/(2 + x)**2 - 1/(3 + x)**2
assert polygamma(0, x + y).expand(func=True) == \
polygamma(0, x + y)
assert polygamma(1, x + y).expand(func=True) == \
polygamma(1, x + y)
assert polygamma(1, 3 + 4*x + y).expand(func=True, multinomial=False) == \
polygamma(1, y + 4*x) - 1/(y + 4*x)**2 - \
1/(1 + y + 4*x)**2 - 1/(2 + y + 4*x)**2
assert polygamma(3, 3 + 4*x + y).expand(func=True, multinomial=False) == \
polygamma(3, y + 4*x) - 6/(y + 4*x)**4 - \
6/(1 + y + 4*x)**4 - 6/(2 + y + 4*x)**4
assert polygamma(3, 4*x + y + 1).expand(func=True, multinomial=False) == \
polygamma(3, y + 4*x) - 6/(y + 4*x)**4
e = polygamma(3, 4*x + y + S(3)/2)
assert e.expand(func=True) == e
e = polygamma(3, x + y + S(3)/4)
assert e.expand(func = True, basic = False) == e
def test_factorial_simplify_fail():
# simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
from sympy.abc import x
assert simplify(x*polygamma(0, x + 1) - x*polygamma(0, x + 2) +
polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp, sin
mu1, mu2 = symbols("mu1 mu2", real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols("sigma1 sigma2", real=True, nonzero=True, finite=True, positive=True)
rate = Symbol("lambda", real=True, positive=True, finite=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma ** 2) * exp(-(x - mu) ** 2 / 2 / sigma ** 2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1
assert integrate(x ** 2 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1 ** 2 + sigma1 ** 2
assert integrate(x ** 3 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1 ** 3 + 3 * mu1 * sigma1 ** 2
assert integrate(normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
assert (
integrate(x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
)
assert (
integrate(y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
)
assert (
integrate(x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
== mu1 * mu2
)
assert (
integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True
)
== 1 + mu1 + mu2
)
assert (
integrate(
(x + y - 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True
)
== -1 + mu1 + mu2
)
i = integrate(x ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1 ** 2 + sigma1 ** 2
assert (
integrate(y ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2), (x, -oo, oo), (y, -oo, oo), meijerg=True)
== sigma2 ** 2 + mu2 ** 2
)
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x * exponential(x, rate), (x, 0, oo), meijerg=True) == 1 / rate
assert integrate(x ** 2 * exponential(x, rate), (x, 0, oo), meijerg=True) == 2 / rate ** 2
def E(expr):
res1 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1), (x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1), (y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y ** 2) == mu1 ** 2 / rate + sigma1 ** 2 / rate
ans = sigma1 ** 2 + 1 / rate ** 2
assert simplify(E((x + y + 1) ** 2) - E(x + y + 1) ** 2) == ans
assert simplify(E((x + y - 1) ** 2) - E(x + y - 1) ** 2) == ans
assert simplify(E((x + y) ** 2) - E(x + y) ** 2) == ans
# Beta' distribution
alpha, beta = symbols("alpha beta", positive=True)
betadist = x ** (alpha - 1) * (1 + x) ** (-alpha - beta) * gamma(alpha + beta) / gamma(alpha) / gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert (combsimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x ** 2 * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0] ** 2) == (alpha + beta - 1) * alpha / (beta - 2) / (beta - 1) ** 2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols("a b", positive=True)
betadist = x ** (a - 1) * (-x + 1) ** (b - 1) * gamma(a + b) / (gamma(a) * gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x * betadist, (x, 0, 1), meijerg=True)) == a / (a + b)
assert simplify(integrate(x ** 2 * betadist, (x, 0, 1), meijerg=True)) == a * (a + 1) / (a + b) / (a + b + 1)
assert simplify(integrate(x ** y * betadist, (x, 0, 1), meijerg=True)) == gamma(a + b) * gamma(a + y) / gamma(
a
) / gamma(a + b + y)
# Chi distribution
k = Symbol("k", integer=True, positive=True)
chi = 2 ** (1 - k / 2) * x ** (k - 1) * exp(-x ** 2 / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chi, (x, 0, oo), meijerg=True)) == sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2)
assert simplify(integrate(x ** 2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2 ** (-k / 2) / gamma(k / 2) * x ** (k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x ** 2 * chisquared, (x, 0, oo), meijerg=True)) == k * (k + 2)
assert combsimp(integrate(((x - k) / sqrt(2 * k)) ** 3 * chisquared, (x, 0, oo), meijerg=True)) == 2 * sqrt(
2
) / sqrt(k)
# Dagum distribution
a, b, p = symbols("a b p", positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b) ** (a * p) / (1 + x ** a / b ** a) ** (p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(integrate(arg, (x, 0, oo), meijerg=True, conds="none")) == a * b * gamma(1 - 1 / a) * gamma(
p + 1 + 1 / a
) / ((a * p + 1) * gamma(p))
assert simplify(integrate(x * arg, (x, 0, oo), meijerg=True, conds="none")) == a * b ** 2 * gamma(
1 - 2 / a
) * gamma(p + 1 + 2 / a) / ((a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols("d1 d2", positive=True)
f = (
sqrt(((d1 * x) ** d1 * d2 ** d2) / (d1 * x + d2) ** (d1 + d2))
/ x
/ gamma(d1 / 2)
/ gamma(d2 / 2)
* gamma((d1 + d2) / 2)
)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True, conds="none")) == d2 / (d2 - 2)
assert simplify(integrate(x ** 2 * f, (x, 0, oo), meijerg=True, conds="none")) == d2 ** 2 * (d1 + 2) / d1 / (
d2 - 4
) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols("lamda mu", positive=True)
dist = sqrt(lamda / 2 / pi) * x ** (-S(3) / 2) * exp(-lamda * (x - mu) ** 2 / x / 2 / mu ** 2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu) ** 2 * dist, (x, 0, oo))) == mu ** 3 / lamda
assert mysimp(integrate((x - mu) ** 3 * dist, (x, 0, oo))) == 3 * mu ** 5 / lamda ** 2
# Levi
c = Symbol("c", positive=True)
assert integrate(sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu) ** S("3/2"), (x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta / alpha) * x ** (beta - 1) / alpha ** (beta - 1) / (1 + x ** beta / alpha ** beta) ** 2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x * distn, (x, 0, oo), conds="none")) == pi * alpha / beta / sin(pi / beta)
# (similar comment for conditions applies)
assert simplify(integrate(x ** y * distn, (x, 0, oo), conds="none")) == pi * alpha ** y * y / beta / sin(
pi * y / beta
)
# weibull
k = Symbol("k", positive=True)
n = Symbol("n", positive=True)
distn = k / lamda * (x / lamda) ** (k - 1) * exp(-(x / lamda) ** k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x ** n * distn, (x, 0, oo))) == lamda ** n * gamma(1 + n / k)
# rice distribution
from sympy import besseli
nu, sigma = symbols("nu sigma", positive=True)
rice = x / sigma ** 2 * exp(-(x ** 2 + nu ** 2) / 2 / sigma ** 2) * besseli(0, x * nu / sigma ** 2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol("mu", real=True)
b = Symbol("b", positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x ** 2 * laplace, (x, -oo, oo), meijerg=True) == 2 * b ** 2 + mu ** 2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol("k", positive=True)
assert combsimp(expand_mul(integrate(log(x) * x ** (k - 1) * exp(-x) / gamma(k), (x, 0, oo)))) == polygamma(0, k)
def test_polygamma():
assert polygamma(n, nan) == nan
assert polygamma(0, oo) == oo
assert polygamma(1, oo) == 0
assert polygamma(5, oo) == 0
assert polygamma(0, -9) == zoo
assert polygamma(0, -9) == zoo
assert polygamma(0, -1) == zoo
assert polygamma(0, 0) == zoo
assert polygamma(0, 1) == -EulerGamma
assert polygamma(0, 7) == Rational(49, 20) - EulerGamma
assert polygamma(1, 1) == pi**2/6
assert polygamma(1, 2) == pi**2/6 - 1
assert polygamma(1, 3) == pi**2/6 - Rational(5, 4)
assert polygamma(3, 1) == pi**4 / 15
assert polygamma(3, 5) == 6*(Rational(-22369,20736) + pi**4/90)
assert polygamma(5, 1) == 8 * pi**6 / 63
assert polygamma(3, 7*x).diff(x) == 7*polygamma(4, 7*x)
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_probability():
# various integrals from probability theory
from sympy.abc import x, y, z
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp, sin
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True, bounded=True)
sigma1, sigma2 = symbols('sigma1 sigma2',
real=True,
finite=True,
bounded=True,
positive=True)
rate = Symbol('lambda', real=True, positive=True, bounded=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma**2) * exp(-(x - mu)**2 / 2 / sigma**2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == \
mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1
assert integrate(y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu2
assert integrate(x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == mu1 * mu2
assert integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True) == 1 + mu1 + mu2
assert integrate((x + y - 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
-1 + mu1 + mu2
i = integrate(x**2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo),
meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == \
1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) == \
2/rate**2
def E(expr):
res1 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo),
meijerg=True)
res2 = integrate(expr * exponential(x, rate) * normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo),
meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y**2) == mu1**2 / rate + sigma1**2 / rate
ans = sigma1**2 + 1 / rate**2
assert simplify(E((x + y + 1)**2) - E(x + y + 1)**2) == ans
assert simplify(E((x + y - 1)**2) - E(x + y - 1)**2) == ans
assert simplify(E((x + y)**2) - E(x + y)**2) == ans
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (combsimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x**2 * betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols('a b', positive=True)
betadist = x**(a - 1) * (-x + 1)**(b - 1) * gamma(a + b) / (gamma(a) *
gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x*betadist, (x, 0, 1), meijerg=True)) == \
a/(a + b)
assert simplify(integrate(x**2*betadist, (x, 0, 1), meijerg=True)) == \
a*(a + 1)/(a + b)/(a + b + 1)
assert simplify(integrate(x**y*betadist, (x, 0, 1), meijerg=True)) == \
gamma(a + b)*gamma(a + y)/gamma(a)/gamma(a + b + y)
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1 - k / 2) * x**(k - 1) * exp(-x**2 / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k / 2) / gamma(k / 2) * x**(k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert combsimp(
integrate(((x - k) / sqrt(2 * k))**3 * chisquared, (x, 0, oo),
meijerg=True)) == 2 * sqrt(2) / sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b)**(a * p) / (1 + x**a / b**a)**(p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(integrate(
arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b * gamma(1 - 1 / a) * gamma(p + 1 + 1 / a) / (
(a * p + 1) * gamma(p))
assert simplify(integrate(
x * arg, (x, 0, oo), meijerg=True,
conds='none')) == a * b**2 * gamma(1 -
2 / a) * gamma(p + 1 + 2 / a) / (
(a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True,
conds='none')) == d2 / (d2 - 2)
assert simplify(
integrate(x**2 * f, (x, 0, oo), meijerg=True,
conds='none')) == d2**2 * (d1 + 2) / d1 / (d2 - 4) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols('lamda mu', positive=True)
dist = sqrt(lamda / 2 / pi) * x**(-S(3) / 2) * exp(
-lamda * (x - mu)**2 / x / 2 / mu**2)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu)**2 * dist, (x, 0, oo))) == mu**3 / lamda
assert mysimp(integrate((x - mu)**3 * dist,
(x, 0, oo))) == 3 * mu**5 / lamda**2
# Levi
c = Symbol('c', positive=True)
assert integrate(
sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu)**S('3/2'),
(x, mu, oo)) == 1
# higher moments oo
# log-logistic
distn = (beta/alpha)*x**(beta - 1)/alpha**(beta - 1)/ \
(1 + x**beta/alpha**beta)**2
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(integrate(x*distn, (x, 0, oo), conds='none')) == \
pi*alpha/beta/sin(pi/beta)
# (similar comment for conditions applies)
assert simplify(integrate(x**y*distn, (x, 0, oo), conds='none')) == \
pi*alpha**y*y/beta/sin(pi*y/beta)
# weibull
k = Symbol('k', positive=True)
n = Symbol('n', positive=True)
distn = k / lamda * (x / lamda)**(k - 1) * exp(-(x / lamda)**k)
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x**n*distn, (x, 0, oo))) == \
lamda**n*gamma(1 + n/k)
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x / sigma**2 * exp(-(x**2 + nu**2) / 2 / sigma**2) * besseli(
0, x * nu / sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == \
2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert combsimp(
expand_mul(
integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k),
(x, 0, oo)))) == polygamma(0, k)
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y, z
from sympy import symbols, Symbol, Abs, expand_mul, combsimp, powsimp
mu1, mu2 = symbols('mu1 mu2', real=True, finite=True, bounded=True)
sigma1, sigma2 = symbols('sigma1 sigma2', real=True, finite=True,
bounded=True, positive=True)
rate = Symbol('lambda', real=True, positive=True, bounded=True)
def normal(x, mu, sigma):
return 1/sqrt(2*pi*sigma**2)*exp(-(x-mu)**2/2/sigma**2)
def exponential(x, rate):
return rate*exp(-rate*x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1
assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**2 + sigma1**2
assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
== mu1**3 + 3*mu1*sigma1**2
assert integrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
assert integrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
assert integrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
assert integrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1*mu2
assert integrate((x+y+1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == 1 + mu1 + mu2
assert integrate((x+y-1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == -1 + mu1 + mu2
i = integrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True)
assert not i.has(Abs)
assert simplify(i) == mu1**2 + sigma1**2
assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
(x, -oo, oo), (y, -oo, oo), meijerg=True) == \
sigma2**2 + mu2**2
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == 1/rate
assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) \
== 2/rate**2
def E(expr):
res1 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo), meijerg=True)
res2 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo), meijerg=True)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x*y) == mu1/rate
assert E(x*y**2) == mu1**2/rate + sigma1**2/rate
assert simplify(E((x+y+1)**2) - E(x+y+1)**2) == (rate**2*sigma1**2 + 1)/rate**2
assert simplify(E((x+y-1)**2) - E(x+y-1)**2) == (rate**2*sigma1**2 + 1)/rate**2
assert simplify(E((x+y)**2) - E(x+y)**2) == (rate**2*sigma1**2 + 1)/rate**2
# Beta' distribution
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha-1)*(1+x)**(-alpha - beta)*gamma(alpha+beta) \
/gamma(alpha)/gamma(beta)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert (combsimp(i[0]), i[1]) == (alpha/(beta - 1), 1 < beta)
j = integrate(x**2*betadist, (x, 0, oo), meijerg=True, conds='separate')
assert j[1] == (1 < beta - 1)
assert combsimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
/(beta - 2)/(beta - 1)**2
# Chi distribution
k = Symbol('k', integer=True, positive=True)
chi = 2**(1-k/2)*x**(k-1)*exp(-x**2/2)/gamma(k/2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
assert simplify(integrate(x**2*chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2**(-k/2)/gamma(k/2)*x**(k/2-1)*exp(-x/2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x*chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
k*(k + 2)
assert combsimp(integrate(((x-k)/sqrt(2*k))**3*chisquared, (x, 0, oo),
meijerg=True)) == 2*sqrt(2)/sqrt(k)
# Dagum distribution
a, b, p = symbols('a b p', positive=True)
# XXX (x/b)**a does not work
dagum = a*p/x*(x/b)**(a*p)/(1 + x**a/b**a)**(p+1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
assert simplify(integrate(x*dagum, (x, 0, oo), meijerg=True, conds='none')) \
== b*gamma(1 - 1/a)*gamma(p + 1/a)/gamma(p)
assert simplify(integrate(x**2*dagum, (x, 0, oo), meijerg=True, conds='none')) \
== b**2*gamma(1 - 2/a)*gamma(p + 2/a)/gamma(p)
# F-distribution
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1+d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x*f, (x, 0, oo), meijerg=True, conds='none')) == \
d2/(d2 - 2)
assert simplify(integrate(x**2*f, (x, 0, oo), meijerg=True, conds='none')) == \
d2**2*(d1 + 2)/d1/(d2 - 4)/(d2 - 2)
# TODO gamma, inverse gaussian, Levi, log-logistic, rayleigh, weibull
# rice distribution
from sympy import besseli
nu, sigma = symbols('nu sigma', positive=True)
rice = x/sigma**2*exp(-(x**2+ nu**2)/2/sigma**2)*besseli(0, x*nu/sigma**2)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol('mu', real=True)
b = Symbol('b', positive=True)
laplace = exp(-abs(x-mu)/b)/2/b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x*laplace, (x, -oo, oo), meijerg=True) == mu
assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == 2*b**2 + mu**2
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol('k', positive=True)
assert combsimp(expand_mul(integrate(log(x) * x**(k-1) * exp(-x) / gamma(k),
(x, 0, oo)))) == polygamma(0, k)
def test_probability():
# various integrals from probability theory
from sympy.abc import x, y
from sympy import symbols, Symbol, Abs, expand_mul, gammasimp, powsimp, sin
mu1, mu2 = symbols("mu1 mu2", nonzero=True)
sigma1, sigma2 = symbols("sigma1 sigma2", positive=True)
rate = Symbol("lambda", positive=True)
def normal(x, mu, sigma):
return 1 / sqrt(2 * pi * sigma ** 2) * exp(-((x - mu) ** 2) / 2 / sigma ** 2)
def exponential(x, rate):
return rate * exp(-rate * x)
assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
assert integrate(x * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == mu1
assert (
integrate(x ** 2 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)
== mu1 ** 2 + sigma1 ** 2
)
assert (
integrate(x ** 3 * normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)
== mu1 ** 3 + 3 * mu1 * sigma1 ** 2
)
assert (
integrate(
normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== 1
)
assert (
integrate(
x * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== mu1
)
assert (
integrate(
y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== mu2
)
assert (
integrate(
x * y * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== mu1 * mu2
)
assert (
integrate(
(x + y + 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== 1 + mu1 + mu2
)
assert (
integrate(
(x + y - 1) * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== -1 + mu1 + mu2
)
i = integrate(
x ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
assert not i.has(Abs)
assert simplify(i) == mu1 ** 2 + sigma1 ** 2
assert (
integrate(
y ** 2 * normal(x, mu1, sigma1) * normal(y, mu2, sigma2),
(x, -oo, oo),
(y, -oo, oo),
meijerg=True,
)
== sigma2 ** 2 + mu2 ** 2
)
assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
assert integrate(x * exponential(x, rate), (x, 0, oo), meijerg=True) == 1 / rate
assert (
integrate(x ** 2 * exponential(x, rate), (x, 0, oo), meijerg=True)
== 2 / rate ** 2
)
def E(expr):
res1 = integrate(
expr * exponential(x, rate) * normal(y, mu1, sigma1),
(x, 0, oo),
(y, -oo, oo),
meijerg=True,
)
res2 = integrate(
expr * exponential(x, rate) * normal(y, mu1, sigma1),
(y, -oo, oo),
(x, 0, oo),
meijerg=True,
)
assert expand_mul(res1) == expand_mul(res2)
return res1
assert E(1) == 1
assert E(x * y) == mu1 / rate
assert E(x * y ** 2) == mu1 ** 2 / rate + sigma1 ** 2 / rate
ans = sigma1 ** 2 + 1 / rate ** 2
assert simplify(E((x + y + 1) ** 2) - E(x + y + 1) ** 2) == ans
assert simplify(E((x + y - 1) ** 2) - E(x + y - 1) ** 2) == ans
assert simplify(E((x + y) ** 2) - E(x + y) ** 2) == ans
# Beta' distribution
alpha, beta = symbols("alpha beta", positive=True)
betadist = (
x ** (alpha - 1)
* (1 + x) ** (-alpha - beta)
* gamma(alpha + beta)
/ gamma(alpha)
/ gamma(beta)
)
assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
i = integrate(x * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert (gammasimp(i[0]), i[1]) == (alpha / (beta - 1), 1 < beta)
j = integrate(x ** 2 * betadist, (x, 0, oo), meijerg=True, conds="separate")
assert j[1] == (1 < beta - 1)
assert (
gammasimp(j[0] - i[0] ** 2)
== (alpha + beta - 1) * alpha / (beta - 2) / (beta - 1) ** 2
)
# Beta distribution
# NOTE: this is evaluated using antiderivatives. It also tests that
# meijerint_indefinite returns the simplest possible answer.
a, b = symbols("a b", positive=True)
betadist = x ** (a - 1) * (-x + 1) ** (b - 1) * gamma(a + b) / (gamma(a) * gamma(b))
assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
assert simplify(integrate(x * betadist, (x, 0, 1), meijerg=True)) == a / (a + b)
assert simplify(integrate(x ** 2 * betadist, (x, 0, 1), meijerg=True)) == a * (
a + 1
) / (a + b) / (a + b + 1)
assert simplify(integrate(x ** y * betadist, (x, 0, 1), meijerg=True)) == gamma(
a + b
) * gamma(a + y) / gamma(a) / gamma(a + b + y)
# Chi distribution
k = Symbol("k", integer=True, positive=True)
chi = 2 ** (1 - k / 2) * x ** (k - 1) * exp(-(x ** 2) / 2) / gamma(k / 2)
assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chi, (x, 0, oo), meijerg=True)) == sqrt(2) * gamma(
(k + 1) / 2
) / gamma(k / 2)
assert simplify(integrate(x ** 2 * chi, (x, 0, oo), meijerg=True)) == k
# Chi^2 distribution
chisquared = 2 ** (-k / 2) / gamma(k / 2) * x ** (k / 2 - 1) * exp(-x / 2)
assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
assert simplify(integrate(x * chisquared, (x, 0, oo), meijerg=True)) == k
assert simplify(integrate(x ** 2 * chisquared, (x, 0, oo), meijerg=True)) == k * (
k + 2
)
assert gammasimp(
integrate(((x - k) / sqrt(2 * k)) ** 3 * chisquared, (x, 0, oo), meijerg=True)
) == 2 * sqrt(2) / sqrt(k)
# Dagum distribution
a, b, p = symbols("a b p", positive=True)
# XXX (x/b)**a does not work
dagum = a * p / x * (x / b) ** (a * p) / (1 + x ** a / b ** a) ** (p + 1)
assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
# XXX conditions are a mess
arg = x * dagum
assert simplify(
integrate(arg, (x, 0, oo), meijerg=True, conds="none")
) == a * b * gamma(1 - 1 / a) * gamma(p + 1 + 1 / a) / ((a * p + 1) * gamma(p))
assert simplify(
integrate(x * arg, (x, 0, oo), meijerg=True, conds="none")
) == a * b ** 2 * gamma(1 - 2 / a) * gamma(p + 1 + 2 / a) / ((a * p + 2) * gamma(p))
# F-distribution
d1, d2 = symbols("d1 d2", positive=True)
f = (
sqrt(((d1 * x) ** d1 * d2 ** d2) / (d1 * x + d2) ** (d1 + d2))
/ x
/ gamma(d1 / 2)
/ gamma(d2 / 2)
* gamma((d1 + d2) / 2)
)
assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
# TODO conditions are a mess
assert simplify(integrate(x * f, (x, 0, oo), meijerg=True, conds="none")) == d2 / (
d2 - 2
)
assert simplify(
integrate(x ** 2 * f, (x, 0, oo), meijerg=True, conds="none")
) == d2 ** 2 * (d1 + 2) / d1 / (d2 - 4) / (d2 - 2)
# TODO gamma, rayleigh
# inverse gaussian
lamda, mu = symbols("lamda mu", positive=True)
dist = (
sqrt(lamda / 2 / pi)
* x ** (Rational(-3, 2))
* exp(-lamda * (x - mu) ** 2 / x / 2 / mu ** 2)
)
mysimp = lambda expr: simplify(expr.rewrite(exp))
assert mysimp(integrate(dist, (x, 0, oo))) == 1
assert mysimp(integrate(x * dist, (x, 0, oo))) == mu
assert mysimp(integrate((x - mu) ** 2 * dist, (x, 0, oo))) == mu ** 3 / lamda
assert (
mysimp(integrate((x - mu) ** 3 * dist, (x, 0, oo))) == 3 * mu ** 5 / lamda ** 2
)
# Levi
c = Symbol("c", positive=True)
assert (
integrate(
sqrt(c / 2 / pi) * exp(-c / 2 / (x - mu)) / (x - mu) ** S("3/2"),
(x, mu, oo),
)
== 1
)
# higher moments oo
# log-logistic
alpha, beta = symbols("alpha beta", positive=True)
distn = (
(beta / alpha)
* x ** (beta - 1)
/ alpha ** (beta - 1)
/ (1 + x ** beta / alpha ** beta) ** 2
)
# FIXME: If alpha, beta are not declared as finite the line below hangs
# after the changes in:
# https://github.com/sympy/sympy/pull/16603
assert simplify(integrate(distn, (x, 0, oo))) == 1
# NOTE the conditions are a mess, but correctly state beta > 1
assert simplify(
integrate(x * distn, (x, 0, oo), conds="none")
) == pi * alpha / beta / sin(pi / beta)
# (similar comment for conditions applies)
assert simplify(
integrate(x ** y * distn, (x, 0, oo), conds="none")
) == pi * alpha ** y * y / beta / sin(pi * y / beta)
# weibull
k = Symbol("k", positive=True)
n = Symbol("n", positive=True)
distn = k / lamda * (x / lamda) ** (k - 1) * exp(-((x / lamda) ** k))
assert simplify(integrate(distn, (x, 0, oo))) == 1
assert simplify(integrate(x ** n * distn, (x, 0, oo))) == lamda ** n * gamma(
1 + n / k
)
# rice distribution
from sympy import besseli
nu, sigma = symbols("nu sigma", positive=True)
rice = (
x
/ sigma ** 2
* exp(-(x ** 2 + nu ** 2) / 2 / sigma ** 2)
* besseli(0, x * nu / sigma ** 2)
)
assert integrate(rice, (x, 0, oo), meijerg=True) == 1
# can someone verify higher moments?
# Laplace distribution
mu = Symbol("mu", real=True)
b = Symbol("b", positive=True)
laplace = exp(-abs(x - mu) / b) / 2 / b
assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
assert integrate(x * laplace, (x, -oo, oo), meijerg=True) == mu
assert (
integrate(x ** 2 * laplace, (x, -oo, oo), meijerg=True) == 2 * b ** 2 + mu ** 2
)
# TODO are there other distributions supported on (-oo, oo) that we can do?
# misc tests
k = Symbol("k", positive=True)
assert gammasimp(
expand_mul(integrate(log(x) * x ** (k - 1) * exp(-x) / gamma(k), (x, 0, oo)))
) == polygamma(0, k)