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 eval(cls, arg):
from sympy import exp_polar, pi, I, arg as argument
if arg.is_number:
ar = argument(arg)
#if not ar.has(argument) and not ar.has(atan):
if ar in (0, pi/2, -pi/2, pi):
return exp_polar(I*ar)*abs(arg)
if arg.is_Mul:
args = arg.args
else:
args = [arg]
included = []
excluded = []
positive = []
for arg in args:
if arg.is_polar:
included += [arg]
elif arg.is_positive:
positive += [arg]
else:
excluded += [arg]
if len(excluded) < len(args):
if excluded:
return Mul(*(included + positive))*polar_lift(Mul(*excluded))
elif included:
return Mul(*(included + positive))
else:
return Mul(*positive)*exp_polar(0)
def t(func, hyp, z):
""" Test that func is a valid representation of hyp. """
# First test that func agrees with hyp for small z
if not tn(func.rewrite('nonrepsmall'), hyp, z,
a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
return False
# Next check that the two small representations agree.
if not tn(
func.rewrite('nonrepsmall').subs(
z, exp_polar(I*pi)*z).replace(exp_polar, exp),
func.subs(z, exp_polar(I*pi)*z).rewrite('nonrepsmall'),
z, a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
return False
# Next check continuity along exp_polar(I*pi)*t
expr = func.subs(z, exp_polar(I*pi)*z).rewrite('nonrep')
if abs(expr.subs(z, 1 + 1e-15).n() - expr.subs(z, 1 - 1e-15).n()) > 1e-10:
return False
# Finally check continuity of the big reps.
def dosubs(func, a, b):
rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep')
return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp)
for n in [0, 1, 2, 3, 4, -1, -2, -3, -4]:
expr1 = dosubs(func, 2*I*pi*n, I*pi/2)
expr2 = dosubs(func, 2*I*pi*n + I*pi, -I*pi/2)
if not tn(expr1, expr2, z):
return False
expr1 = dosubs(func, 2*I*pi*(n + 1), -I*pi/2)
expr2 = dosubs(func, 2*I*pi*n + I*pi, I*pi/2)
if not tn(expr1, expr2, z):
return False
return True
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 eval(cls, arg):
from sympy import exp_polar, pi, I, arg as argument
if arg.is_number:
ar = argument(arg)
# In general we want to affirm that something is known,
# e.g. `not ar.has(argument) and not ar.has(atan)`
# but for now we will just be more restrictive and
# see that it has evaluated to one of the known values.
if ar in (0, pi/2, -pi/2, pi):
return exp_polar(I*ar)*abs(arg)
if arg.is_Mul:
args = arg.args
else:
args = [arg]
included = []
excluded = []
positive = []
for arg in args:
if arg.is_polar:
included += [arg]
elif arg.is_positive:
positive += [arg]
else:
excluded += [arg]
if len(excluded) < len(args):
if excluded:
return Mul(*(included + positive))*polar_lift(Mul(*excluded))
elif included:
return Mul(*(included + positive))
else:
return Mul(*positive)*exp_polar(0)
def tn(func, s):
from random import uniform
c = uniform(1, 5)
expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
return abs(expr.n() - expr2.n()).n() < 1e-10
def test_uppergamma():
from sympy import meijerg, exp_polar, I, expint
assert uppergamma(4, 0) == 6
assert uppergamma(x, y).diff(y) == -y**(x-1)*exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
assert uppergamma(S.Half, x) == sqrt(pi)*(1 - erf(sqrt(x)))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(S(1)/3, uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4*pi*I)*x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + gamma(y)*(1-exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x)/x**2
assert uppergamma(x, y).rewrite(expint) == y**x*expint(-x + 1, y)
def test_si():
assert Si(I*x) == I*Shi(x)
assert Shi(I*x) == I*Si(x)
assert Si(-I*x) == -I*Shi(x)
assert Shi(-I*x) == -I*Si(x)
assert Si(-x) == -Si(x)
assert Shi(-x) == -Shi(x)
assert Si(exp_polar(2*pi*I)*x) == Si(x)
assert Si(exp_polar(-2*pi*I)*x) == Si(x)
assert Shi(exp_polar(2*pi*I)*x) == Shi(x)
assert Shi(exp_polar(-2*pi*I)*x) == Shi(x)
assert mytd(Si(x), sin(x)/x, x)
assert mytd(Shi(x), sinh(x)/x, x)
assert mytn(Si(x), Si(x).rewrite(Ei),
-I*(-Ei(x*exp_polar(-I*pi/2))/2 \
+ Ei(x*exp_polar(I*pi/2))/2 - I*pi) + pi/2, x)
assert mytn(Si(x), Si(x).rewrite(expint),
-I*(-expint(1, x*exp_polar(-I*pi/2))/2 + \
expint(1, x*exp_polar(I*pi/2))/2) + pi/2, x)
assert mytn(Shi(x), Shi(x).rewrite(Ei),
Ei(x)/2 - Ei(x*exp_polar(I*pi))/2 + I*pi/2, x)
assert mytn(Shi(x), Shi(x).rewrite(expint),
expint(1, x)/2 - expint(1, x*exp_polar(I*pi))/2 - I*pi/2, x)
assert tn_arg(Si)
assert tn_arg(Shi)
from sympy import O
assert Si(x).nseries(x, n=8) == x - x**3/18 + x**5/600 - x**7/35280 + O(x**9)
assert Shi(x).nseries(x, n=8) == x + x**3/18 + x**5/600 + x**7/35280 + O(x**9)
assert Si(sin(x)).nseries(x, n=5) == x - 2*x**3/9 + 17*x**5/450 + O(x**6)
assert Si(x).nseries(x, 1, n=3) == \
Si(1) + x*sin(1) + x**2/2*cos(1) - x**2/2*sin(1) + O(x**3)
def test_meijerg_eval():
from sympy import besseli, exp_polar
from sympy.abc import l
a = randcplx()
arg = x*exp_polar(k*pi*I)
expr1 = pi*meijerg([[], [(a + 1)/2]], [[a/2], [-a/2, (a + 1)/2]], arg**2/4)
expr2 = besseli(a, arg)
# Test that the two expressions agree for all arguments.
for x_ in [0.5, 1.5]:
for k_ in [0.0, 0.1, 0.3, 0.5, 0.8, 1, 5.751, 15.3]:
assert abs((expr1 - expr2).n(subs={x: x_, k: k_})) < 1e-10
assert abs((expr1 - expr2).n(subs={x: x_, k: -k_})) < 1e-10
# Test continuity independently
eps = 1e-13
expr2 = expr1.subs(k, l)
for x_ in [0.5, 1.5]:
for k_ in [0.5, S(1)/3, 0.25, 0.75, S(2)/3, 1.0, 1.5]:
assert abs((expr1 - expr2).n(
subs={x: x_, k: k_ + eps, l: k_ - eps})) < 1e-10
assert abs((expr1 - expr2).n(
subs={x: x_, k: -k_ + eps, l: -k_ - eps})) < 1e-10
expr = (meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(-I*pi)/4)
+ meijerg(((0.5,), ()), ((0.5, 0, 0.5), ()), exp_polar(I*pi)/4)) \
/(2*sqrt(pi))
assert (expr - pi/exp(1)).n(chop=True) == 0
def test_branching():
from sympy import exp_polar, polar_lift, Symbol, I, exp
assert besselj(polar_lift(k), x) == besselj(k, x)
assert besseli(polar_lift(k), x) == besseli(k, x)
n = Symbol('n', integer=True)
assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
assert besselj(n, polar_lift(x)) == besselj(n, x)
assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
assert besseli(n, polar_lift(x)) == besseli(n, x)
def tn(func, s):
from random import uniform
c = uniform(1, 5)
expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
return abs(expr.n() - expr2.n()).n() < 1e-10
nu = Symbol('nu')
assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
assert tn(besselj, 2)
assert tn(besselj, pi)
assert tn(besselj, I)
assert tn(besseli, 2)
assert tn(besseli, pi)
assert tn(besseli, I)
def test_issue_7173():
assert laplace_transform(sinh(a*x)*cosh(a*x), x, s) == \
(a/(s**2 - 4*a**2), 0,
And(Or(Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <
pi/2, Abs(periodic_argument(exp_polar(I*pi)*polar_lift(a), oo)) <=
pi/2), Or(Abs(periodic_argument(a, oo)) < pi/2,
Abs(periodic_argument(a, oo)) <= pi/2)))
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_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(-S(3) / 2, x) == exp(-x) / x + 3 * exp(-x) / (2 * x ** 2) - 3 * sqrt(pi) * erf(sqrt(x)) / (
4 * x ** S("5/2")
) + 3 * sqrt(pi) / (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 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)
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 tn_branch(s, func):
from sympy import I, pi, exp_polar
from random import uniform
c = uniform(1, 5)
expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
return abs(expr.n() - expr2.n()).n() < 1e-10
def test_hyper_unpolarify():
from sympy import exp_polar
a = exp_polar(2*pi*I)*x
b = x
assert hyper([], [], a).argument == b
assert hyper([0], [], a).argument == a
assert hyper([0], [0], a).argument == b
assert hyper([0, 1], [0], a).argument == a
assert hyper([0, 1], [0], exp_polar(2*pi*I)).argument == 1
def test_polar():
x, y = symbols('x y', polar=True)
z = Symbol('z')
assert abs(exp_polar(I*4)) == 1
assert exp_polar(I*10).n() == exp_polar(I*10)
assert log(exp_polar(z)) == z
assert log(x*y).expand() == log(x) + log(y)
assert log(x**z).expand() == z*log(x)
def test_principal_branch():
from sympy import principal_branch, polar_lift, exp_polar
p = Symbol('p', positive=True)
x = Symbol('x')
neg = Symbol('x', negative=True)
assert principal_branch(polar_lift(x), p) == principal_branch(x, p)
assert principal_branch(polar_lift(2 + I), p) == principal_branch(2 + I, p)
assert principal_branch(2*x, p) == 2*principal_branch(x, p)
assert principal_branch(1, pi) == exp_polar(0)
assert principal_branch(-1, 2*pi) == exp_polar(I*pi)
assert principal_branch(-1, pi) == exp_polar(0)
assert principal_branch(exp_polar(3*pi*I)*x, 2*pi) == \
principal_branch(exp_polar(I*pi)*x, 2*pi)
assert principal_branch(neg*exp_polar(pi*I), 2*pi) == neg*exp_polar(-I*pi)
# related to issue #14692
assert principal_branch(exp_polar(-I*pi/2)/polar_lift(neg), 2*pi) == \
exp_polar(-I*pi/2)/neg
assert N_equals(principal_branch((1 + I)**2, 2*pi), 2*I)
assert N_equals(principal_branch((1 + I)**2, 3*pi), 2*I)
assert N_equals(principal_branch((1 + I)**2, 1*pi), 2*I)
# test argument sanitization
assert principal_branch(x, I).func is principal_branch
assert principal_branch(x, -4).func is principal_branch
assert principal_branch(x, -oo).func is principal_branch
assert principal_branch(x, zoo).func is principal_branch
def tn_arg(func):
def test(arg, e1, e2):
from random import uniform
v = uniform(1, 5)
v1 = func(arg*x).subs(x, v).n()
v2 = func(e1*v + e2*1e-15).n()
return abs(v1 - v2).n() < 1e-10
return test(exp_polar(I*pi/2), I, 1) and \
test(exp_polar(-I*pi/2), -I, 1) and \
test(exp_polar(I*pi), -1, I) and \
test(exp_polar(-I*pi), -1, -I)
def test_exp_log():
x = Symbol("x", real=True)
assert log(exp(x)) == x
assert exp(log(x)) == x
if not global_parameters.exp_is_pow:
assert log(x).inverse() == exp
assert exp(x).inverse() == log
y = Symbol("y", polar=True)
assert log(exp_polar(z)) == z
assert exp(log(y)) == y
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)
def _eval_expand_func(self, **hints):
from sympy import log, expand_mul, Dummy, exp_polar, I
s, z = self.args
if s == 1:
return -log(1 + exp_polar(-I*pi)*z)
if s.is_Integer and s <= 0:
u = Dummy('u')
start = u/(1 - u)
for _ in range(-s):
start = u*start.diff(u)
return expand_mul(start).subs(u, z)
return polylog(s, z)
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 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)
def test_rewrite_single():
def t(expr, c, m):
e = _rewrite_single(meijerg([a], [b], [c], [d], expr), x)
assert e is not None
assert isinstance(e[0][0][2], meijerg)
assert e[0][0][2].argument.as_coeff_mul(x) == (c, (m,))
def tn(expr):
assert _rewrite_single(meijerg([a], [b], [c], [d], expr), x) is None
t(x, 1, x)
t(x ** 2, 1, x ** 2)
t(x ** 2 + y * x ** 2, y + 1, x ** 2)
tn(x ** 2 + x)
tn(x ** y)
def u(expr, x):
from sympy import Add, exp, exp_polar
r = _rewrite_single(expr, x)
e = Add(*[res[0] * res[2] for res in r[0]]).replace(exp_polar, exp) # XXX Hack?
assert verify_numerically(e, expr, x)
u(exp(-x) * sin(x), x)
# The following has stopped working because hyperexpand changed slightly.
# It is probably not worth fixing
# u(exp(-x)*sin(x)*cos(x), x)
# This one cannot be done numerically, since it comes out as a g-function
# of argument 4*pi
# NOTE This also tests a bug in inverse mellin transform (which used to
# turn exp(4*pi*I*t) into a factor of exp(4*pi*I)**t instead of
# exp_polar).
# u(exp(x)*sin(x), x)
assert _rewrite_single(exp(x) * sin(x), x) == (
[
(
-sqrt(2) / (2 * sqrt(pi)),
0,
meijerg(
(
(Rational(-1, 2), 0, Rational(1, 4), S.Half, Rational(3, 4)),
(1,),
),
((), (Rational(-1, 2), 0)),
64 * exp_polar(-4 * I * pi) / x ** 4,
),
)
],
True,
)
def test_polar():
x, y = symbols('x y', polar=True)
assert abs(exp_polar(I * 4)) == 1
assert abs(exp_polar(0)) == 1
assert abs(exp_polar(2 + 3 * I)) == exp(2)
assert exp_polar(I * 10).n() == exp_polar(I * 10)
assert log(exp_polar(z)) == z
assert log(x * y).expand() == log(x) + log(y)
assert log(x**z).expand() == z * log(x)
assert exp_polar(3).exp == 3
# Compare exp(1.0*pi*I).
assert (exp_polar(1.0 * pi * I).n(n=5)).as_real_imag()[1] >= 0
assert exp_polar(0).is_rational is True # issue 8008
def plot_and_save_6(name):
tmp_file = TmpFileManager.tmp_file
x = Symbol('x')
###
# Test expressions that can not be translated to np and generate complex
# results.
###
plot(sin(x) + I * cos(x)).save(tmp_file())
plot(sqrt(sqrt(-x))).save(tmp_file())
plot(LambertW(x)).save(tmp_file())
plot(sqrt(LambertW(x))).save(tmp_file())
#Characteristic function of a StudentT distribution with nu=10
plot((meijerg(
((1 / 2, ), ()),
((5, 0, 1 / 2), ()), 5 * x**2 * exp_polar(-I * pi) / 2) + meijerg(
((1 / 2, ), ()),
((5, 0, 1 / 2),
()), 5 * x**2 * exp_polar(I * pi) / 2)) / (48 * pi),
(x, 1e-6, 1e-2)).save(tmp_file())
def test_issue_7173():
from sympy import cse
x0, x1, x2 = symbols('x:3')
ans = laplace_transform(sinh(a * x) * cosh(a * x), x, s)
r, e = cse(ans)
assert r == [(x0, pi / 2), (x1, Abs(periodic_argument(a, oo))),
(x2,
Abs(periodic_argument(exp_polar(I * pi) * polar_lift(a),
oo)))]
assert e == [
a / (-4 * a**2 + s**2), 0,
((x0 >= x1) | (x1 < x0)) & ((x0 >= x2) | (x2 < x0))
]
def test_uppergamma():
from sympy import meijerg, exp_polar, I, expint
assert uppergamma(4, 0) == 6
assert uppergamma(x, y).diff(y) == -y**(x - 1) * exp(-y)
assert td(uppergamma(randcplx(), y), y)
assert uppergamma(x, y).diff(x) == \
uppergamma(x, y)*log(y) + meijerg([], [1, 1], [0, 0, x], [], y)
assert td(uppergamma(x, randcplx()), x)
assert uppergamma(S.Half, x) == sqrt(pi) * (1 - erf(sqrt(x)))
assert not uppergamma(S.Half - 3, x).has(uppergamma)
assert not uppergamma(S.Half + 3, x).has(uppergamma)
assert uppergamma(S.Half, x, evaluate=False).has(uppergamma)
assert tn(uppergamma(S.Half + 3, x, evaluate=False),
uppergamma(S.Half + 3, x), x)
assert tn(uppergamma(S.Half - 3, x, evaluate=False),
uppergamma(S.Half - 3, x), x)
assert tn_branch(-3, uppergamma)
assert tn_branch(-4, uppergamma)
assert tn_branch(S(1) / 3, uppergamma)
assert tn_branch(pi, uppergamma)
assert uppergamma(3, exp_polar(4 * pi * I) * x) == uppergamma(3, x)
assert uppergamma(y, exp_polar(5*pi*I)*x) == \
exp(4*I*pi*y)*uppergamma(y, x*exp_polar(pi*I)) + \
gamma(y)*(1 - exp(4*pi*I*y))
assert uppergamma(-2, exp_polar(5*pi*I)*x) == \
uppergamma(-2, x*exp_polar(I*pi)) - 2*pi*I
assert uppergamma(-2, x) == expint(3, x) / x**2
assert conjugate(uppergamma(x, y)) == uppergamma(conjugate(x),
conjugate(y))
assert conjugate(uppergamma(x, 0)) == gamma(conjugate(x))
assert conjugate(uppergamma(x, -oo)) == conjugate(uppergamma(x, -oo))
assert uppergamma(x, y).rewrite(expint) == y**x * expint(-x + 1, y)
assert uppergamma(x, y).rewrite(lowergamma) == gamma(x) - lowergamma(x, y)
def test_bessel():
from sympy import (besselj, Heaviside, besseli, polar_lift, exp_polar,
powdenest)
assert simplify(integrate(besselj(a, z)*besselj(b, z)/z, (z, 0, oo),
meijerg=True, conds='none')) == \
2*sin(pi*a/2 - pi*b/2)/(pi*(a - b)*(a + b))
assert simplify(integrate(besselj(a, z)*besselj(a, z)/z, (z, 0, oo),
meijerg=True, conds='none')) == 1/(2*a)
# TODO more orthogonality integrals
# TODO there is some improvement possible here:
# - the result can be simplified to besselj(y, z))
assert powdenest(simplify(integrate(sin(z*x)*(x**2-1)**(-(y+S(1)/2)),
(x, 1, oo), meijerg=True, conds='none')
*2/((z/2)**y*sqrt(pi)*gamma(S(1)/2-y))),
polar=True) == \
exp(-I*pi*y/2)*besseli(y, z*exp_polar(I*pi/2))
# Werner Rosenheinrich
# SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS
assert integrate(x*besselj(0, x), x, meijerg=True) == x*besselj(1, x)
assert integrate(x*besseli(0, x), x, meijerg=True) == x*besseli(1, x)
# TODO can do higher powers, but come out as high order ... should they be
# reduced to order 0, 1?
assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x)
assert integrate(besselj(1, x)**2/x, x, meijerg=True) == \
-(besselj(0, x)**2 + besselj(1, x)**2)/2
# TODO more besseli when tables are extended or recursive mellin works
assert integrate(besselj(0, x)**2/x**2, x, meijerg=True) == \
-2*x*besselj(0, x)**2 - 2*x*besselj(1, x)**2 \
+ 2*besselj(0, x)*besselj(1, x) - besselj(0, x)**2/x
assert integrate(besselj(0, x)*besselj(1, x), x, meijerg=True) == \
-besselj(0, x)**2/2
assert integrate(x**2*besselj(0, x)*besselj(1, x), x, meijerg=True) == \
x**2*besselj(1, x)**2/2
assert integrate(besselj(0, x)*besselj(1, x)/x, x, meijerg=True) == \
(x*besselj(0, x)**2 + x*besselj(1, x)**2 - \
besselj(0, x)*besselj(1, x))
# TODO how does besselj(0, a*x)*besselj(0, b*x) work?
# TODO how does besselj(0, x)**2*besselj(1, x)**2 work?
# TODO sin(x)*besselj(0, x) etc come out a mess
# TODO can x*log(x)*besselj(0, x) be done?
# TODO how does besselj(1, x)*besselj(0, x+a) work?
# TODO more indefinite integrals when struve functions etc are implemented
# test a substitution
assert integrate(besselj(1, x**2)*x, x, meijerg=True) == \
-besselj(0, x**2)/2
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(-S(3)/2, x) == \
exp(-x)/x + 3*exp(-x)/(2*x**2) - 3*sqrt(pi)*erf(sqrt(x))/(4*x**S('5/2')) \
+ 3*sqrt(pi)/(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 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)
def test_ei():
assert Ei(0) is S.NegativeInfinity
assert Ei(oo) is S.Infinity
assert Ei(-oo) is S.Zero
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)
assert Ei(x).series(x, 1, 3) == Ei(1) + E * (x - 1) + O((x - 1)**3, (x, 1))
assert str(Ei(cos(2)).evalf(n=10)) == "-0.6760647401"
raises(ArgumentIndexError, lambda: Ei(x).fdiff(2))
def test_ei():
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)
assert str(Ei(cos(2)).evalf(n=10)) == '-0.6760647401'
def test_issue_10681():
from sympy import RR
from sympy.abc import R, r
f = integrate(r ** 2 * (R ** 2 - r ** 2) ** 0.5, r, meijerg=True)
g = (
(1.0 / 3)
* R ** 1.0
* r ** 3
* hyper(
(-0.5, Rational(3, 2)),
(Rational(5, 2),),
r ** 2 * exp_polar(2 * I * pi) / R ** 2,
)
)
assert RR.almosteq((f / g).n(), 1.0, 1e-12)
def test_expint():
from sympy import E1, expint, Max, re, lerchphi, Symbol, simplify, Si, Ci, Ei
aneg = Symbol('a', negative=True)
u = Symbol('u', polar=True)
assert mellin_transform(E1(x), x, s) == (gamma(s)/s, (0, oo), True)
assert inverse_mellin_transform(gamma(s)/s, s, x,
(0, oo)).rewrite(expint).expand() == E1(x)
assert mellin_transform(expint(a, x), x, s) == \
(gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
# XXX IMT has hickups with complicated strips ...
assert simplify(unpolarify(
inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
(1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
expint(aneg, x)
assert mellin_transform(Si(x), x, s) == \
(-2**s*sqrt(pi)*gamma(s/2 + S.Half)/(
2*s*gamma(-s/2 + 1)), (-1, 0), True)
assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
== Si(x)
assert mellin_transform(Ci(sqrt(x)), x, s) == \
(-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S.Half)), (0, 1), True)
assert inverse_mellin_transform(
-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S.Half)),
s, u, (0, 1)).expand() == Ci(sqrt(u))
# TODO LT of Si, Shi, Chi is a mess ...
assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2)/2/s, 0, True)
assert laplace_transform(expint(a, x), x, s) == \
(lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero)
assert laplace_transform(expint(1, x), x, s) == (log(s + 1)/s, 0, True)
assert laplace_transform(expint(2, x), x, s) == \
((s - log(s + 1))/s**2, 0, True)
assert inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() == \
Heaviside(u)*Ci(u)
assert inverse_laplace_transform(log(s + 1)/s, s, x).rewrite(expint) == \
Heaviside(x)*E1(x)
assert inverse_laplace_transform((s - log(s + 1))/s**2, s,
x).rewrite(expint).expand() == \
(expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand()
def test_polar():
x, y = symbols('x y', polar=True)
z = Symbol('z')
assert abs(exp_polar(I*4)) == 1
assert exp_polar(I*10).n() == exp_polar(I*10)
assert log(exp_polar(z)) == z
assert log(x*y).expand() == log(x) + log(y)
assert log(x**z).expand() == z*log(x)
assert exp_polar(3).exp == 3
# Compare exp(1.0*pi*I).
assert (exp_polar(1.0*pi*I).n(n=5)).as_real_imag()[1] >= 0
def test_hyperexpand_bases():
assert hyperexpand(hyper([2], [a], z)) == \
a + z**(-a + 1)*(-a**2 + 3*a + z*(a - 1) - 2)*exp(z)*lowergamma(a - 1, z) - 1
# TODO [a+1, a-S.Half], [2*a]
assert hyperexpand(hyper(
[1, 2], [3], z)) == -2 / z - 2 * log(exp_polar(-I * pi) * z + 1) / z**2
assert hyperexpand(hyper([S.Half, 2], [S(3)/2], z)) == \
-1/(2*z - 2) + log((sqrt(z) + 1)/(-sqrt(z) + 1))/(4*sqrt(z))
assert hyperexpand(hyper([S(1)/2, S(1)/2], [S(5)/2], z)) == \
(-3*z + 3)/4/(z*sqrt(-z + 1)) \
+ (6*z - 3)*asin(sqrt(z))/(4*z**(S(3)/2))
assert hyperexpand(hyper([1, 2], [S(3)/2], z)) == -1/(2*z - 2) \
- asin(sqrt(z))/(sqrt(z)*(2*z - 2)*sqrt(-z + 1))
assert hyperexpand(hyper([-S.Half - 1, 1, 2], [S.Half, 3], z)) == \
sqrt(z)*(6*z/7 - S(6)/5)*atanh(sqrt(z)) \
+ (-30*z**2 + 32*z - 6)/35/z - 6*log(-z + 1)/(35*z**2)
assert hyperexpand(hyper([1+S.Half, 1, 1], [2, 2], z)) == \
-4*log(sqrt(-z + 1)/2 + S(1)/2)/z
# TODO hyperexpand(hyper([a], [2*a + 1], z))
# TODO [S.Half, a], [S(3)/2, a+1]
assert hyperexpand(hyper([2], [b, 1], z)) == \
z**(-b/2 + S(1)/2)*besseli(b - 1, 2*sqrt(z))*gamma(b) \
+ z**(-b/2 + 1)*besseli(b, 2*sqrt(z))*gamma(b)
def test_polarify():
from sympy import polar_lift, polarify
x = Symbol('x')
z = Symbol('z', polar=True)
f = Function('f')
ES = {}
assert polarify(-1) == (polar_lift(-1), ES)
assert polarify(1 + I) == (polar_lift(1 + I), ES)
assert polarify(exp(x), subs=False) == exp(x)
assert polarify(1 + x, subs=False) == 1 + x
assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x
assert polarify(x, lift=True) == polar_lift(x)
assert polarify(z, lift=True) == z
assert polarify(f(x), lift=True) == f(polar_lift(x))
assert polarify(1 + x, lift=True) == polar_lift(1 + x)
assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))
newex, subs = polarify(f(x) + z)
assert newex.subs(subs) == f(x) + z
mu = Symbol("mu")
sigma = Symbol("sigma", positive=True)
# Make sure polarify(lift=True) doesn't try to lift the integration
# variable
assert polarify(
Integral(
sqrt(2) * x * exp(-(-mu + x)**2 / (2 * sigma**2)) /
(2 * sqrt(pi) * sigma), (x, -oo, oo)),
lift=True) == Integral(
sqrt(2) * (sigma * exp_polar(0))**exp_polar(I * pi) * exp(
(sigma * exp_polar(0))**(2 * exp_polar(I * pi)) * exp_polar(
I * pi) * polar_lift(-mu + x)**(2 * exp_polar(0)) / 2) *
exp_polar(0) * polar_lift(x) / (2 * sqrt(pi)), (x, -oo, oo))
def test_powsimp_polar():
from sympy import polar_lift, exp_polar
x, y, z = symbols('x y z')
p, q, r = symbols('p q r', polar=True)
assert (polar_lift(-1))**(2 * x) == exp_polar(2 * pi * I * x)
assert powsimp(p**x * q**x) == (p * q)**x
assert p**x * (1 / p)**x == 1
assert (1 / p)**x == p**(-x)
assert exp_polar(x) * exp_polar(y) == exp_polar(x) * exp_polar(y)
assert powsimp(exp_polar(x) * exp_polar(y)) == exp_polar(x + y)
assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y) == \
(p*exp_polar(1))**(x + y)
assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y, combine='exp') == \
exp_polar(x + y)*p**(x + y)
assert powsimp(
exp_polar(x)*exp_polar(y)*exp_polar(2)*sin(x) + sin(y) + p**x*p**y) \
== p**(x + y) + sin(x)*exp_polar(2 + x + y) + sin(y)
assert powsimp(sin(exp_polar(x)*exp_polar(y))) == \
sin(exp_polar(x)*exp_polar(y))
assert powsimp(sin(exp_polar(x)*exp_polar(y)), deep=True) == \
sin(exp_polar(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 Ci(oo) == 0
assert Ci(-oo) == I * pi
assert Chi(oo) is oo
assert Chi(-oo) is oo
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
assert Ci(x).rewrite(uppergamma) == -expint(1, x*exp_polar(-I*pi/2))/2 -\
expint(1, x*exp_polar(I*pi/2))/2
assert Ci(x).rewrite(expint) == -expint(1, x*exp_polar(-I*pi/2))/2 -\
expint(1, x*exp_polar(I*pi/2))/2
raises(ArgumentIndexError, lambda: Ci(x).fdiff(2))
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3 * I) == I
assert sign(-3 * I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2 * I).doit() == sqrt(2) * (2 + 2 * I) / 4
assert sign(2 + 3 * I).simplify() == sign(2 + 3 * I)
assert sign(2 + 2 * I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2 * x) == sign(x)
assert sign(2 * x) == sign(x)
assert sign(I * x) == I * sign(x)
assert sign(-2 * I * x) == -I * sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2 * p * x) == sign(x)
assert sign(n * x) == -sign(x)
assert sign(n * m * x) == sign(x)
x = Symbol('x', imaginary=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is False
assert sign(x).is_real is False
assert sign(x).is_zero is False
assert sign(x).diff(x) == 2 * DiracDelta(-I * x)
assert sign(x).doit() == x / Abs(x)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', real=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2 * DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', real=True)).is_nonnegative is None
assert sign(Symbol('x', real=True)).is_nonpositive is None
assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2 * Heaviside(x) - 1
assert sign(y).rewrite(Heaviside) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I * pi) * pi) is S.NegativeOne
eq = -sqrt(10 + 6 * sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3 * sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq).func is sign or sign(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2 * sqrt(3) + 1331 * sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert sign(d).func is sign or sign(d) == 0
def _eval_rewrite_as_expint(self, z):
from sympy import exp_polar
# XXX should we polarify z?
return (E1(z) - E1(exp_polar(I*pi)*z))/2 - I*pi/2
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 Ci(oo) == 0
assert Ci(-oo) == I*pi
assert Chi(oo) == oo
assert Chi(-oo) == oo
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_hyperrep():
from sympy.functions.special.hyper import (HyperRep, HyperRep_atanh,
HyperRep_power1, HyperRep_power2, HyperRep_log1, HyperRep_asin1,
HyperRep_asin2, HyperRep_sqrts1, HyperRep_sqrts2, HyperRep_log2,
HyperRep_cosasin, HyperRep_sinasin)
# First test the base class works.
from sympy import Piecewise, exp_polar
a, b, c, d, z = symbols('a b c d z')
class myrep(HyperRep):
@classmethod
def _expr_small(cls, x):
return a
@classmethod
def _expr_small_minus(cls, x):
return b
@classmethod
def _expr_big(cls, x, n):
return c*n
@classmethod
def _expr_big_minus(cls, x, n):
return d*n
assert myrep(z).rewrite('nonrep') == Piecewise((0, abs(z) > 1), (a, True))
assert myrep(exp_polar(I*pi)*z).rewrite('nonrep') == \
Piecewise((0, abs(z) > 1), (b, True))
assert myrep(exp_polar(2*I*pi)*z).rewrite('nonrep') == \
Piecewise((c, abs(z) > 1), (a, True))
assert myrep(exp_polar(3*I*pi)*z).rewrite('nonrep') == \
Piecewise((d, abs(z) > 1), (b, True))
assert myrep(exp_polar(4*I*pi)*z).rewrite('nonrep') == \
Piecewise((2*c, abs(z) > 1), (a, True))
assert myrep(exp_polar(5*I*pi)*z).rewrite('nonrep') == \
Piecewise((2*d, abs(z) > 1), (b, True))
assert myrep(z).rewrite('nonrepsmall') == a
assert myrep(exp_polar(I*pi)*z).rewrite('nonrepsmall') == b
def t(func, hyp, z):
""" Test that func is a valid representation of hyp. """
# First test that func agrees with hyp for small z
if not tn(func.rewrite('nonrepsmall'), hyp, z,
a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
return False
# Next check that the two small representations agree.
if not tn(
func.rewrite('nonrepsmall').subs(
z, exp_polar(I*pi)*z).replace(exp_polar, exp),
func.subs(z, exp_polar(I*pi)*z).rewrite('nonrepsmall'),
z, a=S(-1)/2, b=S(-1)/2, c=S(1)/2, d=S(1)/2):
return False
# Next check continuity along exp_polar(I*pi)*t
expr = func.subs(z, exp_polar(I*pi)*z).rewrite('nonrep')
if abs(expr.subs(z, 1 + 1e-15).n() - expr.subs(z, 1 - 1e-15).n()) > 1e-10:
return False
# Finally check continuity of the big reps.
def dosubs(func, a, b):
rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep')
return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp)
for n in [0, 1, 2, 3, 4, -1, -2, -3, -4]:
expr1 = dosubs(func, 2*I*pi*n, I*pi/2)
expr2 = dosubs(func, 2*I*pi*n + I*pi, -I*pi/2)
if not tn(expr1, expr2, z):
return False
expr1 = dosubs(func, 2*I*pi*(n + 1), -I*pi/2)
expr2 = dosubs(func, 2*I*pi*n + I*pi, I*pi/2)
if not tn(expr1, expr2, z):
return False
return True
# Now test the various representatives.
a = S(1)/3
assert t(HyperRep_atanh(z), hyper([S(1)/2, 1], [S(3)/2], z), z)
assert t(HyperRep_power1(a, z), hyper([-a], [], z), z)
assert t(HyperRep_power2(a, z), hyper([a, a - S(1)/2], [2*a], z), z)
assert t(HyperRep_log1(z), -z*hyper([1, 1], [2], z), z)
assert t(HyperRep_asin1(z), hyper([S(1)/2, S(1)/2], [S(3)/2], z), z)
assert t(HyperRep_asin2(z), hyper([1, 1], [S(3)/2], z), z)
assert t(HyperRep_sqrts1(a, z), hyper([-a, S(1)/2 - a], [S(1)/2], z), z)
assert t(HyperRep_sqrts2(a, z),
-2*z/(2*a + 1)*hyper([-a - S(1)/2, -a], [S(1)/2], z).diff(z), z)
assert t(HyperRep_log2(z), -z/4*hyper([S(3)/2, 1, 1], [2, 2], z), z)
assert t(HyperRep_cosasin(a, z), hyper([-a, a], [S(1)/2], z), z)
assert t(HyperRep_sinasin(a, z), 2*a*z*hyper([1 - a, 1 + a], [S(3)/2], z), z)
def _eval_rewrite_as_expint(self, z):
from sympy import exp_polar
return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2
def dosubs(func, a, b):
rv = func.subs(z, exp_polar(a)*z).rewrite('nonrep')
return rv.subs(z, exp_polar(b)*z).replace(exp_polar, exp)
def test_branch_bug():
assert hyperexpand(hyper((-S(1)/3, S(1)/2), (S(2)/3, S(3)/2), -z)) == \
-z**S('1/3')*lowergamma(exp_polar(I*pi)/3, z)/5 \
+ sqrt(pi)*erf(sqrt(z))/(5*sqrt(z))
assert hyperexpand(meijerg([S(7)/6, 1], [], [S(2)/3], [S(1)/6, 0], z)) == \
2*z**S('2/3')*(2*sqrt(pi)*erf(sqrt(z))/sqrt(z) - 2*lowergamma(S(2)/3, z)/z**S('2/3'))*gamma(S(2)/3)/gamma(S(5)/3)
def _eval_expand_func(self, **hints):
from sympy import exp, I, floor, Add, Poly, Dummy, exp_polar, unpolarify
z, s, a = self.args
if z == 1:
return zeta(s, a)
if s.is_Integer and s <= 0:
t = Dummy('t')
p = Poly((t + a)**(-s), t)
start = 1 / (1 - t)
res = S(0)
for c in reversed(p.all_coeffs()):
res += c * start
start = t * start.diff(t)
return res.subs(t, z)
if a.is_Rational:
# See section 18 of
# Kelly B. Roach. Hypergeometric Function Representations.
# In: Proceedings of the 1997 International Symposium on Symbolic and
# Algebraic Computation, pages 205-211, New York, 1997. ACM.
# TODO should something be polarified here?
add = S(0)
mul = S(1)
# First reduce a to the interaval (0, 1]
if a > 1:
n = floor(a)
if n == a:
n -= 1
a -= n
mul = z**(-n)
add = Add(*[-z**(k - n) / (a + k)**s for k in range(n)])
elif a <= 0:
n = floor(-a) + 1
a += n
mul = z**n
add = Add(*[z**(n - 1 - k) / (a - k - 1)**s for k in range(n)])
m, n = S([a.p, a.q])
zet = exp_polar(2 * pi * I / n)
root = z**(1 / n)
return add + mul * n**(s - 1) * Add(*[
polylog(s, zet**k * root)._eval_expand_func(**hints) /
(unpolarify(zet)**k * root)**m for k in range(n)
])
# TODO use minpoly instead of ad-hoc methods when issue 5888 is fixed
if isinstance(z, exp) and (z.args[0] /
(pi * I)).is_Rational or z in [-1, I, -I]:
# TODO reference?
if z == -1:
p, q = S([1, 2])
elif z == I:
p, q = S([1, 4])
elif z == -I:
p, q = S([-1, 4])
else:
arg = z.args[0] / (2 * pi * I)
p, q = S([arg.p, arg.q])
return Add(*[
exp(2 * pi * I * k * p / q) / q**s * zeta(s, (k + a) / q)
for k in range(q)
])
return lerchphi(z, s, a)
def test_meijerg_expand():
from sympy import gammasimp, simplify
# from mpmath docs
assert hyperexpand(meijerg([[], []], [[0], []], -z)) == exp(z)
assert hyperexpand(meijerg([[1, 1], []], [[1], [0]], z)) == \
log(z + 1)
assert hyperexpand(meijerg([[1, 1], []], [[1], [1]], z)) == \
z/(z + 1)
assert hyperexpand(meijerg([[], []], [[S.Half], [0]], (z/2)**2)) \
== sin(z)/sqrt(pi)
assert hyperexpand(meijerg([[], []], [[0], [S.Half]], (z/2)**2)) \
== cos(z)/sqrt(pi)
assert can_do_meijer([], [a], [a - 1, a - S.Half], [])
assert can_do_meijer([], [], [a/2], [-a/2], False) # branches...
assert can_do_meijer([a], [b], [a], [b, a - 1])
# wikipedia
assert hyperexpand(meijerg([1], [], [], [0], z)) == \
Piecewise((0, abs(z) < 1), (1, abs(1/z) < 1),
(meijerg([1], [], [], [0], z), True))
assert hyperexpand(meijerg([], [1], [0], [], z)) == \
Piecewise((1, abs(z) < 1), (0, abs(1/z) < 1),
(meijerg([], [1], [0], [], z), True))
# The Special Functions and their Approximations
assert can_do_meijer([], [], [a + b/2], [a, a - b/2, a + S.Half])
assert can_do_meijer(
[], [], [a], [b], False) # branches only agree for small z
assert can_do_meijer([], [S.Half], [a], [-a])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, b], [])
assert can_do_meijer([], [], [a, a + S.Half], [b, b + S.Half])
assert can_do_meijer([], [], [a, -a], [0, S.Half], False) # dito
assert can_do_meijer([], [], [a, a + S.Half, b, b + S.Half], [])
assert can_do_meijer([S.Half], [], [0], [a, -a])
assert can_do_meijer([S.Half], [], [a], [0, -a], False) # dito
assert can_do_meijer([], [a - S.Half], [a, b], [a - S.Half], False)
assert can_do_meijer([], [a + S.Half], [a + b, a - b, a], [], False)
assert can_do_meijer([a + S.Half], [], [b, 2*a - b, a], [], False)
# This for example is actually zero.
assert can_do_meijer([], [], [], [a, b])
# Testing a bug:
assert hyperexpand(meijerg([0, 2], [], [], [-1, 1], z)) == \
Piecewise((0, abs(z) < 1),
(z/2 - 1/(2*z), abs(1/z) < 1),
(meijerg([0, 2], [], [], [-1, 1], z), True))
# Test that the simplest possible answer is returned:
assert gammasimp(simplify(hyperexpand(
meijerg([1], [1 - a], [-a/2, -a/2 + S.Half], [], 1/z)))) == \
-2*sqrt(pi)*(sqrt(z + 1) + 1)**a/a
# Test that hyper is returned
assert hyperexpand(meijerg([1], [], [a], [0, 0], z)) == hyper(
(a,), (a + 1, a + 1), z*exp_polar(I*pi))*z**a*gamma(a)/gamma(a + 1)**2
# Test place option
f = meijerg(((0, 1), ()), ((S.Half,), (0,)), z**2)
assert hyperexpand(f) == sqrt(pi)/sqrt(1 + z**(-2))
assert hyperexpand(f, place=0) == sqrt(pi)*z/sqrt(z**2 + 1)
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)