477 lines
16 KiB
Python
477 lines
16 KiB
Python
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from sympy.core import EulerGamma
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from sympy.core.numbers import (E, I, Integer, Rational, oo, pi)
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (acot, atan, cos, sin)
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from sympy.functions.elementary.complexes import sign as _sign
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from sympy.functions.special.error_functions import (Ei, erf)
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from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma)
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from sympy.functions.special.zeta_functions import zeta
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from sympy.polys.polytools import cancel
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from sympy.functions.elementary.hyperbolic import cosh, coth, sinh, tanh
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from sympy.series.gruntz import compare, mrv, rewrite, mrv_leadterm, gruntz, \
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sign
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from sympy.testing.pytest import XFAIL, skip, slow
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"""
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This test suite is testing the limit algorithm using the bottom up approach.
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See the documentation in limits2.py. The algorithm itself is highly recursive
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by nature, so "compare" is logically the lowest part of the algorithm, yet in
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some sense it's the most complex part, because it needs to calculate a limit
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to return the result.
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Nevertheless, the rest of the algorithm depends on compare working correctly.
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"""
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x = Symbol('x', real=True)
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m = Symbol('m', real=True)
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runslow = False
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def _sskip():
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if not runslow:
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skip("slow")
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@slow
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def test_gruntz_evaluation():
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# Gruntz' thesis pp. 122 to 123
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# 8.1
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assert gruntz(exp(x)*(exp(1/x - exp(-x)) - exp(1/x)), x, oo) == -1
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# 8.2
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assert gruntz(exp(x)*(exp(1/x + exp(-x) + exp(-x**2))
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- exp(1/x - exp(-exp(x)))), x, oo) == 1
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# 8.3
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assert gruntz(exp(exp(x - exp(-x))/(1 - 1/x)) - exp(exp(x)), x, oo) is oo
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# 8.5
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assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(exp(x))), x, oo) is oo
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# 8.6
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assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(x))))),
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x, oo) is oo
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# 8.7
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assert gruntz(exp(exp(exp(x))) / exp(exp(exp(x - exp(-exp(exp(x)))))),
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x, oo) == 1
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# 8.8
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assert gruntz(exp(exp(x)) / exp(exp(x - exp(-exp(exp(x))))), x, oo) == 1
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# 8.9
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assert gruntz(log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2
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* exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x),
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x, oo) == 0
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# 8.10
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assert gruntz((x*log(x)*(log(x*exp(x) - x**2))**2)
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/ (log(log(x**2 + 2*exp(exp(3*x**3*log(x)))))), x, oo) == Rational(1, 3)
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# 8.11
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assert gruntz((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1)))) - exp(x))/x,
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x, oo) == -exp(2)
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# 8.12
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assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5
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# 8.13
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assert gruntz(x/log(x**(log(x**(log(2)/log(x))))), x, oo) is oo
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# 8.14
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assert gruntz(exp(exp(2*log(x**5 + x)*log(log(x))))
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/ exp(exp(10*log(x)*log(log(x)))), x, oo) is oo
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# 8.15
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assert gruntz(exp(exp(Rational(5, 2)*x**Rational(-5, 7) + Rational(21, 8)*x**Rational(6, 11)
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+ 2*x**(-8) + Rational(54, 17)*x**Rational(49, 45)))**8
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/ log(log(-log(Rational(4, 3)*x**Rational(-5, 14))))**Rational(7, 6), x, oo) is oo
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# 8.16
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assert gruntz((exp(4*x*exp(-x)/(1/exp(x) + 1/exp(2*x**2/(x + 1)))) - exp(x))
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/ exp(x)**4, x, oo) == 1
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# 8.17
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assert gruntz(exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))/exp(x), x, oo) \
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== 1
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# 8.19
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assert gruntz(log(x)*(log(log(x) + log(log(x))) - log(log(x)))
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/ (log(log(x) + log(log(log(x))))), x, oo) == 1
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# 8.20
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assert gruntz(exp((log(log(x + exp(log(x)*log(log(x))))))
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/ (log(log(log(exp(x) + x + log(x)))))), x, oo) == E
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# Another
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assert gruntz(exp(exp(exp(x + exp(-x)))) / exp(exp(x)), x, oo) is oo
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def test_gruntz_evaluation_slow():
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_sskip()
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# 8.4
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assert gruntz(exp(exp(exp(x)/(1 - 1/x)))
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- exp(exp(exp(x)/(1 - 1/x - log(x)**(-log(x))))), x, oo) is -oo
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# 8.18
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assert gruntz((exp(exp(-x/(1 + exp(-x))))*exp(-x/(1 + exp(-x/(1 + exp(-x)))))
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*exp(exp(-x + exp(-x/(1 + exp(-x))))))
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/ (exp(-x/(1 + exp(-x))))**2 - exp(x) + x, x, oo) == 2
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@slow
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def test_gruntz_eval_special():
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# Gruntz, p. 126
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assert gruntz(exp(x)*(sin(1/x + exp(-x)) - sin(1/x + exp(-x**2))), x, oo) == 1
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assert gruntz((erf(x - exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2),
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x, oo) == -2/sqrt(pi)
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assert gruntz(exp(exp(x)) * (exp(sin(1/x + exp(-exp(x)))) - exp(sin(1/x))),
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x, oo) == 1
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assert gruntz(exp(x)*(gamma(x + exp(-x)) - gamma(x)), x, oo) is oo
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assert gruntz(exp(exp(digamma(digamma(x))))/x, x, oo) == exp(Rational(-1, 2))
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assert gruntz(exp(exp(digamma(log(x))))/x, x, oo) == exp(Rational(-1, 2))
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assert gruntz(digamma(digamma(digamma(x))), x, oo) is oo
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assert gruntz(loggamma(loggamma(x)), x, oo) is oo
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assert gruntz(((gamma(x + 1/gamma(x)) - gamma(x))/log(x) - cos(1/x))
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* x*log(x), x, oo) == Rational(-1, 2)
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assert gruntz(x * (gamma(x - 1/gamma(x)) - gamma(x) + log(x)), x, oo) \
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== S.Half
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assert gruntz((gamma(x + 1/gamma(x)) - gamma(x)) / log(x), x, oo) == 1
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def test_gruntz_eval_special_slow():
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_sskip()
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assert gruntz(gamma(x + 1)/sqrt(2*pi)
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- exp(-x)*(x**(x + S.Half) + x**(x - S.Half)/12), x, oo) is oo
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assert gruntz(exp(exp(exp(digamma(digamma(digamma(x))))))/x, x, oo) == 0
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@XFAIL
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def test_grunts_eval_special_slow_sometimes_fail():
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_sskip()
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# XXX This sometimes fails!!!
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assert gruntz(exp(gamma(x - exp(-x))*exp(1/x)) - exp(gamma(x)), x, oo) is oo
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def test_gruntz_Ei():
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assert gruntz((Ei(x - exp(-exp(x))) - Ei(x)) *exp(-x)*exp(exp(x))*x, x, oo) == -1
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@XFAIL
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def test_gruntz_eval_special_fail():
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# TODO zeta function series
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assert gruntz(
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exp((log(2) + 1)*x) * (zeta(x + exp(-x)) - zeta(x)), x, oo) == -log(2)
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# TODO 8.35 - 8.37 (bessel, max-min)
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def test_gruntz_hyperbolic():
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assert gruntz(cosh(x), x, oo) is oo
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assert gruntz(cosh(x), x, -oo) is oo
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assert gruntz(sinh(x), x, oo) is oo
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assert gruntz(sinh(x), x, -oo) is -oo
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assert gruntz(2*cosh(x)*exp(x), x, oo) is oo
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assert gruntz(2*cosh(x)*exp(x), x, -oo) == 1
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assert gruntz(2*sinh(x)*exp(x), x, oo) is oo
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assert gruntz(2*sinh(x)*exp(x), x, -oo) == -1
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assert gruntz(tanh(x), x, oo) == 1
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assert gruntz(tanh(x), x, -oo) == -1
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assert gruntz(coth(x), x, oo) == 1
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assert gruntz(coth(x), x, -oo) == -1
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def test_compare1():
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assert compare(2, x, x) == "<"
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assert compare(x, exp(x), x) == "<"
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assert compare(exp(x), exp(x**2), x) == "<"
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assert compare(exp(x**2), exp(exp(x)), x) == "<"
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assert compare(1, exp(exp(x)), x) == "<"
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assert compare(x, 2, x) == ">"
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assert compare(exp(x), x, x) == ">"
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assert compare(exp(x**2), exp(x), x) == ">"
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assert compare(exp(exp(x)), exp(x**2), x) == ">"
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assert compare(exp(exp(x)), 1, x) == ">"
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assert compare(2, 3, x) == "="
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assert compare(3, -5, x) == "="
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assert compare(2, -5, x) == "="
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assert compare(x, x**2, x) == "="
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assert compare(x**2, x**3, x) == "="
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assert compare(x**3, 1/x, x) == "="
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assert compare(1/x, x**m, x) == "="
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assert compare(x**m, -x, x) == "="
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assert compare(exp(x), exp(-x), x) == "="
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assert compare(exp(-x), exp(2*x), x) == "="
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assert compare(exp(2*x), exp(x)**2, x) == "="
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assert compare(exp(x)**2, exp(x + exp(-x)), x) == "="
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assert compare(exp(x), exp(x + exp(-x)), x) == "="
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assert compare(exp(x**2), 1/exp(x**2), x) == "="
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def test_compare2():
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assert compare(exp(x), x**5, x) == ">"
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assert compare(exp(x**2), exp(x)**2, x) == ">"
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assert compare(exp(x), exp(x + exp(-x)), x) == "="
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assert compare(exp(x + exp(-x)), exp(x), x) == "="
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assert compare(exp(x + exp(-x)), exp(-x), x) == "="
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assert compare(exp(-x), x, x) == ">"
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assert compare(x, exp(-x), x) == "<"
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assert compare(exp(x + 1/x), x, x) == ">"
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assert compare(exp(-exp(x)), exp(x), x) == ">"
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assert compare(exp(exp(-exp(x)) + x), exp(-exp(x)), x) == "<"
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def test_compare3():
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assert compare(exp(exp(x)), exp(x + exp(-exp(x))), x) == ">"
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def test_sign1():
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assert sign(Rational(0), x) == 0
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assert sign(Rational(3), x) == 1
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assert sign(Rational(-5), x) == -1
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assert sign(log(x), x) == 1
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assert sign(exp(-x), x) == 1
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assert sign(exp(x), x) == 1
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assert sign(-exp(x), x) == -1
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assert sign(3 - 1/x, x) == 1
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assert sign(-3 - 1/x, x) == -1
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assert sign(sin(1/x), x) == 1
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assert sign((x**Integer(2)), x) == 1
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assert sign(x**2, x) == 1
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assert sign(x**5, x) == 1
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def test_sign2():
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assert sign(x, x) == 1
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assert sign(-x, x) == -1
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y = Symbol("y", positive=True)
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assert sign(y, x) == 1
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assert sign(-y, x) == -1
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assert sign(y*x, x) == 1
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assert sign(-y*x, x) == -1
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def mmrv(a, b):
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return set(mrv(a, b)[0].keys())
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def test_mrv1():
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assert mmrv(x, x) == {x}
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assert mmrv(x + 1/x, x) == {x}
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assert mmrv(x**2, x) == {x}
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assert mmrv(log(x), x) == {x}
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assert mmrv(exp(x), x) == {exp(x)}
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assert mmrv(exp(-x), x) == {exp(-x)}
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assert mmrv(exp(x**2), x) == {exp(x**2)}
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assert mmrv(-exp(1/x), x) == {x}
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assert mmrv(exp(x + 1/x), x) == {exp(x + 1/x)}
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def test_mrv2a():
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assert mmrv(exp(x + exp(-exp(x))), x) == {exp(-exp(x))}
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assert mmrv(exp(x + exp(-x)), x) == {exp(x + exp(-x)), exp(-x)}
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assert mmrv(exp(1/x + exp(-x)), x) == {exp(-x)}
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#sometimes infinite recursion due to log(exp(x**2)) not simplifying
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def test_mrv2b():
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assert mmrv(exp(x + exp(-x**2)), x) == {exp(-x**2)}
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#sometimes infinite recursion due to log(exp(x**2)) not simplifying
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def test_mrv2c():
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assert mmrv(
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exp(-x + 1/x**2) - exp(x + 1/x), x) == {exp(x + 1/x), exp(1/x**2 - x)}
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#sometimes infinite recursion due to log(exp(x**2)) not simplifying
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def test_mrv3():
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assert mmrv(exp(x**2) + x*exp(x) + log(x)**x/x, x) == {exp(x**2)}
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assert mmrv(
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exp(x)*(exp(1/x + exp(-x)) - exp(1/x)), x) == {exp(x), exp(-x)}
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assert mmrv(log(
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x**2 + 2*exp(exp(3*x**3*log(x)))), x) == {exp(exp(3*x**3*log(x)))}
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assert mmrv(log(x - log(x))/log(x), x) == {x}
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assert mmrv(
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(exp(1/x - exp(-x)) - exp(1/x))*exp(x), x) == {exp(x), exp(-x)}
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assert mmrv(
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1/exp(-x + exp(-x)) - exp(x), x) == {exp(x), exp(-x), exp(x - exp(-x))}
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assert mmrv(log(log(x*exp(x*exp(x)) + 1)), x) == {exp(x*exp(x))}
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assert mmrv(exp(exp(log(log(x) + 1/x))), x) == {x}
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def test_mrv4():
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ln = log
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assert mmrv((ln(ln(x) + ln(ln(x))) - ln(ln(x)))/ln(ln(x) + ln(ln(ln(x))))*ln(x),
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x) == {x}
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|
|
assert mmrv(log(log(x*exp(x*exp(x)) + 1)) - exp(exp(log(log(x) + 1/x))), x) == \
|
||
|
|
{exp(x*exp(x))}
|
||
|
|
|
||
|
|
|
||
|
|
def mrewrite(a, b, c):
|
||
|
|
return rewrite(a[1], a[0], b, c)
|
||
|
|
|
||
|
|
|
||
|
|
def test_rewrite1():
|
||
|
|
e = exp(x)
|
||
|
|
assert mrewrite(mrv(e, x), x, m) == (1/m, -x)
|
||
|
|
e = exp(x**2)
|
||
|
|
assert mrewrite(mrv(e, x), x, m) == (1/m, -x**2)
|
||
|
|
e = exp(x + 1/x)
|
||
|
|
assert mrewrite(mrv(e, x), x, m) == (1/m, -x - 1/x)
|
||
|
|
e = 1/exp(-x + exp(-x)) - exp(x)
|
||
|
|
assert mrewrite(mrv(e, x), x, m) == (1/(m*exp(m)) - 1/m, -x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_rewrite2():
|
||
|
|
e = exp(x)*log(log(exp(x)))
|
||
|
|
assert mmrv(e, x) == {exp(x)}
|
||
|
|
assert mrewrite(mrv(e, x), x, m) == (1/m*log(x), -x)
|
||
|
|
|
||
|
|
#sometimes infinite recursion due to log(exp(x**2)) not simplifying
|
||
|
|
|
||
|
|
|
||
|
|
def test_rewrite3():
|
||
|
|
e = exp(-x + 1/x**2) - exp(x + 1/x)
|
||
|
|
#both of these are correct and should be equivalent:
|
||
|
|
assert mrewrite(mrv(e, x), x, m) in [(-1/m + m*exp(
|
||
|
|
1/x + 1/x**2), -x - 1/x), (m - 1/m*exp(1/x + x**(-2)), x**(-2) - x)]
|
||
|
|
|
||
|
|
|
||
|
|
def test_mrv_leadterm1():
|
||
|
|
assert mrv_leadterm(-exp(1/x), x) == (-1, 0)
|
||
|
|
assert mrv_leadterm(1/exp(-x + exp(-x)) - exp(x), x) == (-1, 0)
|
||
|
|
assert mrv_leadterm(
|
||
|
|
(exp(1/x - exp(-x)) - exp(1/x))*exp(x), x) == (-exp(1/x), 0)
|
||
|
|
|
||
|
|
|
||
|
|
def test_mrv_leadterm2():
|
||
|
|
#Gruntz: p51, 3.25
|
||
|
|
assert mrv_leadterm((log(exp(x) + x) - x)/log(exp(x) + log(x))*exp(x), x) == \
|
||
|
|
(1, 0)
|
||
|
|
|
||
|
|
|
||
|
|
def test_mrv_leadterm3():
|
||
|
|
#Gruntz: p56, 3.27
|
||
|
|
assert mmrv(exp(-x + exp(-x)*exp(-x*log(x))), x) == {exp(-x - x*log(x))}
|
||
|
|
assert mrv_leadterm(exp(-x + exp(-x)*exp(-x*log(x))), x) == (exp(-x), 0)
|
||
|
|
|
||
|
|
|
||
|
|
def test_limit1():
|
||
|
|
assert gruntz(x, x, oo) is oo
|
||
|
|
assert gruntz(x, x, -oo) is -oo
|
||
|
|
assert gruntz(-x, x, oo) is -oo
|
||
|
|
assert gruntz(x**2, x, -oo) is oo
|
||
|
|
assert gruntz(-x**2, x, oo) is -oo
|
||
|
|
assert gruntz(x*log(x), x, 0, dir="+") == 0
|
||
|
|
assert gruntz(1/x, x, oo) == 0
|
||
|
|
assert gruntz(exp(x), x, oo) is oo
|
||
|
|
assert gruntz(-exp(x), x, oo) is -oo
|
||
|
|
assert gruntz(exp(x)/x, x, oo) is oo
|
||
|
|
assert gruntz(1/x - exp(-x), x, oo) == 0
|
||
|
|
assert gruntz(x + 1/x, x, oo) is oo
|
||
|
|
|
||
|
|
|
||
|
|
def test_limit2():
|
||
|
|
assert gruntz(x**x, x, 0, dir="+") == 1
|
||
|
|
assert gruntz((exp(x) - 1)/x, x, 0) == 1
|
||
|
|
assert gruntz(1 + 1/x, x, oo) == 1
|
||
|
|
assert gruntz(-exp(1/x), x, oo) == -1
|
||
|
|
assert gruntz(x + exp(-x), x, oo) is oo
|
||
|
|
assert gruntz(x + exp(-x**2), x, oo) is oo
|
||
|
|
assert gruntz(x + exp(-exp(x)), x, oo) is oo
|
||
|
|
assert gruntz(13 + 1/x - exp(-x), x, oo) == 13
|
||
|
|
|
||
|
|
|
||
|
|
def test_limit3():
|
||
|
|
a = Symbol('a')
|
||
|
|
assert gruntz(x - log(1 + exp(x)), x, oo) == 0
|
||
|
|
assert gruntz(x - log(a + exp(x)), x, oo) == 0
|
||
|
|
assert gruntz(exp(x)/(1 + exp(x)), x, oo) == 1
|
||
|
|
assert gruntz(exp(x)/(a + exp(x)), x, oo) == 1
|
||
|
|
|
||
|
|
|
||
|
|
def test_limit4():
|
||
|
|
#issue 3463
|
||
|
|
assert gruntz((3**x + 5**x)**(1/x), x, oo) == 5
|
||
|
|
#issue 3463
|
||
|
|
assert gruntz((3**(1/x) + 5**(1/x))**x, x, 0) == 5
|
||
|
|
|
||
|
|
|
||
|
|
@XFAIL
|
||
|
|
def test_MrvTestCase_page47_ex3_21():
|
||
|
|
h = exp(-x/(1 + exp(-x)))
|
||
|
|
expr = exp(h)*exp(-x/(1 + h))*exp(exp(-x + h))/h**2 - exp(x) + x
|
||
|
|
assert mmrv(expr, x) == {1/h, exp(-x), exp(x), exp(x - h), exp(x/(1 + h))}
|
||
|
|
|
||
|
|
|
||
|
|
def test_gruntz_I():
|
||
|
|
y = Symbol("y")
|
||
|
|
assert gruntz(I*x, x, oo) == I*oo
|
||
|
|
assert gruntz(y*I*x, x, oo) == y*I*oo
|
||
|
|
assert gruntz(y*3*I*x, x, oo) == y*I*oo
|
||
|
|
assert gruntz(y*3*sin(I)*x, x, oo).simplify().rewrite(_sign) == _sign(y)*I*oo
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4814():
|
||
|
|
assert gruntz((x + 1)**(1/log(x + 1)), x, oo) == E
|
||
|
|
|
||
|
|
|
||
|
|
def test_intractable():
|
||
|
|
assert gruntz(1/gamma(x), x, oo) == 0
|
||
|
|
assert gruntz(1/loggamma(x), x, oo) == 0
|
||
|
|
assert gruntz(gamma(x)/loggamma(x), x, oo) is oo
|
||
|
|
assert gruntz(exp(gamma(x))/gamma(x), x, oo) is oo
|
||
|
|
assert gruntz(gamma(x), x, 3) == 2
|
||
|
|
assert gruntz(gamma(Rational(1, 7) + 1/x), x, oo) == gamma(Rational(1, 7))
|
||
|
|
assert gruntz(log(x**x)/log(gamma(x)), x, oo) == 1
|
||
|
|
assert gruntz(log(gamma(gamma(x)))/exp(x), x, oo) is oo
|
||
|
|
|
||
|
|
|
||
|
|
def test_aseries_trig():
|
||
|
|
assert cancel(gruntz(1/log(atan(x)), x, oo)
|
||
|
|
- 1/(log(pi) + log(S.Half))) == 0
|
||
|
|
assert gruntz(1/acot(x), x, -oo) is -oo
|
||
|
|
|
||
|
|
|
||
|
|
def test_exp_log_series():
|
||
|
|
assert gruntz(x/log(log(x*exp(x))), x, oo) is oo
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3644():
|
||
|
|
assert gruntz(((x**7 + x + 1)/(2**x + x**2))**(-1/x), x, oo) == 2
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_6843():
|
||
|
|
n = Symbol('n', integer=True, positive=True)
|
||
|
|
r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
|
||
|
|
assert gruntz(r, x, 1).simplify() == n/2
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4190():
|
||
|
|
assert gruntz(x - gamma(1/x), x, oo) == S.EulerGamma
|
||
|
|
|
||
|
|
|
||
|
|
@XFAIL
|
||
|
|
def test_issue_5172():
|
||
|
|
n = Symbol('n')
|
||
|
|
r = Symbol('r', positive=True)
|
||
|
|
c = Symbol('c')
|
||
|
|
p = Symbol('p', positive=True)
|
||
|
|
m = Symbol('m', negative=True)
|
||
|
|
expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c + \
|
||
|
|
(r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n)
|
||
|
|
expr = expr.subs(c, c + 1)
|
||
|
|
assert gruntz(expr.subs(c, m), n, oo) == 1
|
||
|
|
# fail:
|
||
|
|
assert gruntz(expr.subs(c, p), n, oo).simplify() == \
|
||
|
|
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4109():
|
||
|
|
assert gruntz(1/gamma(x), x, 0) == 0
|
||
|
|
assert gruntz(x*gamma(x), x, 0) == 1
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_6682():
|
||
|
|
assert gruntz(exp(2*Ei(-x))/x**2, x, 0) == exp(2*EulerGamma)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_7096():
|
||
|
|
from sympy.functions import sign
|
||
|
|
assert gruntz(x**-pi, x, 0, dir='-') == oo*sign((-1)**(-pi))
|