def test_matplotlib_advanced():
"""Examples from the 'advanced' notebook."""
try:
name = 'test'
tmp_file = TmpFileManager.tmp_file
s = summation(1 / x**y, (x, 1, oo))
p = plot(s, (y, 2, 10))
p.save(tmp_file('%s_advanced_inf_sum' % name))
p = plot(summation(1 / x, (x, 1, y)), (y, 2, 10), show=False)
p[0].only_integers = True
p[0].steps = True
p.save(tmp_file('%s_advanced_fin_sum' % name))
###
# 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())
finally:
TmpFileManager.cleanup()
def test_specfun():
for f in [besselj, bessely, besseli, besselk]:
assert octave_code(f(n, x)) == f.__name__ + '(n, x)'
assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)'
assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)'
assert octave_code(airyai(x)) == 'airy(0, x)'
assert octave_code(airyaiprime(x)) == 'airy(1, x)'
assert octave_code(airybi(x)) == 'airy(2, x)'
assert octave_code(airybiprime(x)) == 'airy(3, x)'
assert octave_code(uppergamma(n, x)) == "gammainc(x, n, 'upper')"
assert octave_code(lowergamma(n, x)) == "gammainc(x, n, 'lower')"
assert octave_code(jn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2'
assert octave_code(yn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2'
assert octave_code(Chi(x)) == 'coshint(x)'
assert octave_code(Ci(x)) == 'cosint(x)'
assert octave_code(laguerre(n, x)) == 'laguerreL(n, x)'
assert octave_code(li(x)) == 'logint(x)'
assert octave_code(loggamma(x)) == 'gammaln(x)'
assert octave_code(polygamma(n, x)) == 'psi(n, x)'
assert octave_code(Shi(x)) == 'sinhint(x)'
assert octave_code(Si(x)) == 'sinint(x)'
assert octave_code(LambertW(x)) == 'lambertw(x)'
assert octave_code(LambertW(x, n)) == 'lambertw(n, x)'
assert octave_code(zeta(x)) == 'zeta(x)'
assert octave_code(zeta(
x, y)) == '% Not supported in Octave:\n% zeta\nzeta(x, y)'
def test_sympyissue_5673():
e = LambertW(-1)
assert e.is_comparable is False
assert e.is_positive is not True
e2 = 1 - 1 / (1 - exp(-1000))
assert e.is_positive is not True
e3 = -2 + exp(exp(LambertW(log(2))) * LambertW(log(2)))
assert e3.is_nonzero is not True
def test_pmint_WrightOmega():
def omega(x):
return LambertW(exp(x))
f = (1 + omega(x) * (2 + cos(omega(x)) *
(x + omega(x)))) / (1 + omega(x)) / (x + omega(x))
g = log(x + LambertW(exp(x))) + sin(LambertW(exp(x)))
assert heurisch(f, x) == g
def omega(x):
return LambertW(exp(x))
def test_pmint_LambertW():
f = LambertW(x)
g = x * LambertW(x) - x + x / LambertW(x)
assert heurisch(f, x) == g
def test_lambertw():
k = Symbol('k')
assert LambertW(x, 0) == LambertW(x)
assert LambertW(x, 0, evaluate=False) != LambertW(x)
assert LambertW(0) == 0
assert LambertW(E) == 1
assert LambertW(-1 / E) == -1
assert LambertW(-log(2) / 2) == -log(2)
assert LambertW(oo) == oo
assert LambertW(0, 1) == -oo
assert LambertW(0, 42) == -oo
assert LambertW(-pi / 2, -1) == -I * pi / 2
assert LambertW(-1 / E, -1) == -1
assert LambertW(-2 * exp(-2), -1) == -2
assert LambertW(
x**2).diff(x) == 2 * LambertW(x**2) / x / (1 + LambertW(x**2))
assert LambertW(x, k).diff(x) == LambertW(x, k) / x / (1 + LambertW(x, k))
pytest.raises(ArgumentIndexError, lambda: LambertW(x).fdiff(3))
pytest.raises(ArgumentIndexError, lambda: LambertW(x, k).fdiff(3))
assert LambertW(sqrt(2)).evalf(30).epsilon_eq(
Float("0.701338383413663009202120278965", 30), 1e-29)
assert re(LambertW(2, -1)).evalf().epsilon_eq(Float("-0.834310366631110"))
assert LambertW(-1).is_extended_real is False # issue sympy/sympy#5215
assert LambertW(2, evaluate=False).is_extended_real
p = Symbol('p', positive=True)
assert LambertW(p, evaluate=False).is_extended_real
assert LambertW(p - 1, evaluate=False).is_extended_real is None
assert LambertW(-p - 2 / S.Exp1, evaluate=False).is_extended_real is False
assert LambertW(S.Half, -1, evaluate=False).is_extended_real is False
assert LambertW(-S.One / 10, -1, evaluate=False).is_extended_real
assert LambertW(0, evaluate=False).is_algebraic
na = Symbol('na', nonzero=True, algebraic=True)
assert LambertW(na).is_algebraic is False
assert LambertW(x, -1).is_extended_real is None
assert LambertW(x, 2).is_extended_real is None
# See sympy/sympy#7259:
assert LambertW(x).series(x) == x - x**2 + 3*x**3/2 - 8*x**4/3 + \
125*x**5/24 + O(x**6)
assert LambertW(x).series(x, n=0) == O(1, x)
assert LambertW(x, k).series(x, x0=1,
n=1) == (LambertW(1, k) + O(x - 1, (x, 1)))
def test_transcendental_functions():
assert integrate(LambertW(2*x), x) == \
-x + x*LambertW(2*x) + x/LambertW(2*x)
