558 lines
18 KiB
Python
558 lines
18 KiB
Python
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from sympy.calculus.util import AccumBounds
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from sympy.core.function import (Derivative, PoleError)
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from sympy.core.numbers import (E, I, Integer, Rational, pi)
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from sympy.core.singleton import S
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from sympy.core.symbol import (Symbol, symbols)
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from sympy.functions.elementary.complexes import sign
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from sympy.functions.elementary.exponential import (exp, log)
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from sympy.functions.elementary.hyperbolic import (acosh, acoth, asinh, atanh, cosh, coth, sinh, tanh)
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from sympy.functions.elementary.integers import (ceiling, floor, frac)
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from sympy.functions.elementary.miscellaneous import (cbrt, sqrt)
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from sympy.functions.elementary.trigonometric import (asin, cos, cot, sin, tan)
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from sympy.series.limits import limit
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from sympy.series.order import O
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from sympy.abc import x, y, z
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from sympy.testing.pytest import raises, XFAIL
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def test_simple_1():
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assert x.nseries(x, n=5) == x
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assert y.nseries(x, n=5) == y
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assert (1/(x*y)).nseries(y, n=5) == 1/(x*y)
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assert Rational(3, 4).nseries(x, n=5) == Rational(3, 4)
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assert x.nseries() == x
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def test_mul_0():
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assert (x*log(x)).nseries(x, n=5) == x*log(x)
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def test_mul_1():
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assert (x*log(2 + x)).nseries(x, n=5) == x*log(2) + x**2/2 - x**3/8 + \
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x**4/24 + O(x**5)
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assert (x*log(1 + x)).nseries(
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x, n=5) == x**2 - x**3/2 + x**4/3 + O(x**5)
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def test_pow_0():
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assert (x**2).nseries(x, n=5) == x**2
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assert (1/x).nseries(x, n=5) == 1/x
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assert (1/x**2).nseries(x, n=5) == 1/x**2
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assert (x**Rational(2, 3)).nseries(x, n=5) == (x**Rational(2, 3))
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assert (sqrt(x)**3).nseries(x, n=5) == (sqrt(x)**3)
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def test_pow_1():
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assert ((1 + x)**2).nseries(x, n=5) == x**2 + 2*x + 1
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# https://github.com/sympy/sympy/issues/21075
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assert ((sqrt(x) + 1)**2).nseries(x) == 2*sqrt(x) + x + 1
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assert ((sqrt(x) + cbrt(x))**2).nseries(x) == 2*x**Rational(5, 6)\
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+ x**Rational(2, 3) + x
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def test_geometric_1():
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assert (1/(1 - x)).nseries(x, n=5) == 1 + x + x**2 + x**3 + x**4 + O(x**5)
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assert (x/(1 - x)).nseries(x, n=6) == x + x**2 + x**3 + x**4 + x**5 + O(x**6)
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assert (x**3/(1 - x)).nseries(x, n=8) == x**3 + x**4 + x**5 + x**6 + \
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x**7 + O(x**8)
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def test_sqrt_1():
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assert sqrt(1 + x).nseries(x, n=5) == 1 + x/2 - x**2/8 + x**3/16 - 5*x**4/128 + O(x**5)
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def test_exp_1():
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assert exp(x).nseries(x, n=5) == 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)
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assert exp(x).nseries(x, n=12) == 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + \
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x**6/720 + x**7/5040 + x**8/40320 + x**9/362880 + x**10/3628800 + \
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x**11/39916800 + O(x**12)
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assert exp(1/x).nseries(x, n=5) == exp(1/x)
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assert exp(1/(1 + x)).nseries(x, n=4) == \
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(E*(1 - x - 13*x**3/6 + 3*x**2/2)).expand() + O(x**4)
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assert exp(2 + x).nseries(x, n=5) == \
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(exp(2)*(1 + x + x**2/2 + x**3/6 + x**4/24)).expand() + O(x**5)
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def test_exp_sqrt_1():
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assert exp(1 + sqrt(x)).nseries(x, n=3) == \
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(exp(1)*(1 + sqrt(x) + x/2 + sqrt(x)*x/6)).expand() + O(sqrt(x)**3)
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def test_power_x_x1():
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assert (exp(x*log(x))).nseries(x, n=4) == \
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1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4)
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def test_power_x_x2():
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assert (x**x).nseries(x, n=4) == \
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1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4)
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def test_log_singular1():
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assert log(1 + 1/x).nseries(x, n=5) == x - log(x) - x**2/2 + x**3/3 - \
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x**4/4 + O(x**5)
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def test_log_power1():
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e = 1 / (1/x + x ** (log(3)/log(2)))
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assert e.nseries(x, n=5) == -x**(log(3)/log(2) + 2) + x + O(x**5)
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def test_log_series():
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l = Symbol('l')
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e = 1/(1 - log(x))
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assert e.nseries(x, n=5, logx=l) == 1/(1 - l)
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def test_log2():
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e = log(-1/x)
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assert e.nseries(x, n=5) == -log(x) + log(-1)
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def test_log3():
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l = Symbol('l')
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e = 1/log(-1/x)
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assert e.nseries(x, n=4, logx=l) == 1/(-l + log(-1))
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def test_series1():
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e = sin(x)
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assert e.nseries(x, 0, 0) != 0
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assert e.nseries(x, 0, 0) == O(1, x)
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assert e.nseries(x, 0, 1) == O(x, x)
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assert e.nseries(x, 0, 2) == x + O(x**2, x)
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assert e.nseries(x, 0, 3) == x + O(x**3, x)
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assert e.nseries(x, 0, 4) == x - x**3/6 + O(x**4, x)
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e = (exp(x) - 1)/x
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assert e.nseries(x, 0, 3) == 1 + x/2 + x**2/6 + O(x**3)
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assert x.nseries(x, 0, 2) == x
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@XFAIL
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def test_series1_failing():
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assert x.nseries(x, 0, 0) == O(1, x)
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assert x.nseries(x, 0, 1) == O(x, x)
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def test_seriesbug1():
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assert (1/x).nseries(x, 0, 3) == 1/x
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assert (x + 1/x).nseries(x, 0, 3) == x + 1/x
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def test_series2x():
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assert ((x + 1)**(-2)).nseries(x, 0, 4) == 1 - 2*x + 3*x**2 - 4*x**3 + O(x**4, x)
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assert ((x + 1)**(-1)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x)
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assert ((x + 1)**0).nseries(x, 0, 3) == 1
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assert ((x + 1)**1).nseries(x, 0, 3) == 1 + x
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assert ((x + 1)**2).nseries(x, 0, 3) == x**2 + 2*x + 1
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assert ((x + 1)**3).nseries(x, 0, 3) == 1 + 3*x + 3*x**2 + O(x**3)
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assert (1/(1 + x)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x)
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assert (x + 3/(1 + 2*x)).nseries(x, 0, 4) == 3 - 5*x + 12*x**2 - 24*x**3 + O(x**4, x)
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assert ((1/x + 1)**3).nseries(x, 0, 3) == 1 + 3/x + 3/x**2 + x**(-3)
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assert (1/(1 + 1/x)).nseries(x, 0, 4) == x - x**2 + x**3 - O(x**4, x)
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assert (1/(1 + 1/x**2)).nseries(x, 0, 6) == x**2 - x**4 + O(x**6, x)
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def test_bug2(): # 1/log(0)*log(0) problem
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w = Symbol("w")
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e = (w**(-1) + w**(
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-log(3)*log(2)**(-1)))**(-1)*(3*w**(-log(3)*log(2)**(-1)) + 2*w**(-1))
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e = e.expand()
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assert e.nseries(w, 0, 4).subs(w, 0) == 3
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def test_exp():
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e = (1 + x)**(1/x)
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assert e.nseries(x, n=3) == exp(1) - x*exp(1)/2 + 11*exp(1)*x**2/24 + O(x**3)
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def test_exp2():
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w = Symbol("w")
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e = w**(1 - log(x)/(log(2) + log(x)))
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logw = Symbol("logw")
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assert e.nseries(
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w, 0, 1, logx=logw) == exp(logw*log(2)/(log(x) + log(2)))
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def test_bug3():
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e = (2/x + 3/x**2)/(1/x + 1/x**2)
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assert e.nseries(x, n=3) == 3 - x + x**2 + O(x**3)
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def test_generalexponent():
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p = 2
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e = (2/x + 3/x**p)/(1/x + 1/x**p)
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assert e.nseries(x, 0, 3) == 3 - x + x**2 + O(x**3)
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p = S.Half
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e = (2/x + 3/x**p)/(1/x + 1/x**p)
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assert e.nseries(x, 0, 2) == 2 - x + sqrt(x) + x**(S(3)/2) + O(x**2)
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e = 1 + sqrt(x)
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assert e.nseries(x, 0, 4) == 1 + sqrt(x)
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# more complicated example
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def test_genexp_x():
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e = 1/(1 + sqrt(x))
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assert e.nseries(x, 0, 2) == \
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1 + x - sqrt(x) - sqrt(x)**3 + O(x**2, x)
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# more complicated example
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def test_genexp_x2():
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p = Rational(3, 2)
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e = (2/x + 3/x**p)/(1/x + 1/x**p)
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assert e.nseries(x, 0, 3) == 3 + x + x**2 - sqrt(x) - x**(S(3)/2) - x**(S(5)/2) + O(x**3)
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def test_seriesbug2():
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w = Symbol("w")
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#simple case (1):
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e = ((2*w)/w)**(1 + w)
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assert e.nseries(w, 0, 1) == 2 + O(w, w)
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assert e.nseries(w, 0, 1).subs(w, 0) == 2
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def test_seriesbug2b():
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w = Symbol("w")
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#test sin
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e = sin(2*w)/w
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assert e.nseries(w, 0, 3) == 2 - 4*w**2/3 + O(w**3)
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def test_seriesbug2d():
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w = Symbol("w", real=True)
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e = log(sin(2*w)/w)
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assert e.series(w, n=5) == log(2) - 2*w**2/3 - 4*w**4/45 + O(w**5)
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def test_seriesbug2c():
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w = Symbol("w", real=True)
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#more complicated case, but sin(x)~x, so the result is the same as in (1)
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e = (sin(2*w)/w)**(1 + w)
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assert e.series(w, 0, 1) == 2 + O(w)
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assert e.series(w, 0, 3) == 2 + 2*w*log(2) + \
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w**2*(Rational(-4, 3) + log(2)**2) + O(w**3)
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assert e.series(w, 0, 2).subs(w, 0) == 2
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def test_expbug4():
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x = Symbol("x", real=True)
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assert (log(
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sin(2*x)/x)*(1 + x)).series(x, 0, 2) == log(2) + x*log(2) + O(x**2, x)
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assert exp(
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log(sin(2*x)/x)*(1 + x)).series(x, 0, 2) == 2 + 2*x*log(2) + O(x**2)
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assert exp(log(2) + O(x)).nseries(x, 0, 2) == 2 + O(x)
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assert ((2 + O(x))**(1 + x)).nseries(x, 0, 2) == 2 + O(x)
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def test_logbug4():
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assert log(2 + O(x)).nseries(x, 0, 2) == log(2) + O(x, x)
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def test_expbug5():
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assert exp(log(1 + x)/x).nseries(x, n=3) == exp(1) + -exp(1)*x/2 + 11*exp(1)*x**2/24 + O(x**3)
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assert exp(O(x)).nseries(x, 0, 2) == 1 + O(x)
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def test_sinsinbug():
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assert sin(sin(x)).nseries(x, 0, 8) == x - x**3/3 + x**5/10 - 8*x**7/315 + O(x**8)
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def test_issue_3258():
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a = x/(exp(x) - 1)
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assert a.nseries(x, 0, 5) == 1 - x/2 - x**4/720 + x**2/12 + O(x**5)
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def test_issue_3204():
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x = Symbol("x", nonnegative=True)
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f = sin(x**3)**Rational(1, 3)
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assert f.nseries(x, 0, 17) == x - x**7/18 - x**13/3240 + O(x**17)
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def test_issue_3224():
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f = sqrt(1 - sqrt(y))
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assert f.nseries(y, 0, 2) == 1 - sqrt(y)/2 - y/8 - sqrt(y)**3/16 + O(y**2)
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def test_issue_3463():
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w, i = symbols('w,i')
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r = log(5)/log(3)
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p = w**(-1 + r)
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e = 1/x*(-log(w**(1 + r)) + log(w + w**r))
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e_ser = -r*log(w)/x + p/x - p**2/(2*x) + O(w)
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assert e.nseries(w, n=1) == e_ser
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def test_sin():
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assert sin(8*x).nseries(x, n=4) == 8*x - 256*x**3/3 + O(x**4)
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assert sin(x + y).nseries(x, n=1) == sin(y) + O(x)
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assert sin(x + y).nseries(x, n=2) == sin(y) + cos(y)*x + O(x**2)
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assert sin(x + y).nseries(x, n=5) == sin(y) + cos(y)*x - sin(y)*x**2/2 - \
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cos(y)*x**3/6 + sin(y)*x**4/24 + O(x**5)
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def test_issue_3515():
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e = sin(8*x)/x
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||
|
|
assert e.nseries(x, n=6) == 8 - 256*x**2/3 + 4096*x**4/15 + O(x**6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3505():
|
||
|
|
e = sin(x)**(-4)*(sqrt(cos(x))*sin(x)**2 -
|
||
|
|
cos(x)**Rational(1, 3)*sin(x)**2)
|
||
|
|
assert e.nseries(x, n=9) == Rational(-1, 12) - 7*x**2/288 - \
|
||
|
|
43*x**4/10368 - 1123*x**6/2488320 + 377*x**8/29859840 + O(x**9)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3501():
|
||
|
|
a = Symbol("a")
|
||
|
|
e = x**(-2)*(x*sin(a + x) - x*sin(a))
|
||
|
|
assert e.nseries(x, n=6) == cos(a) - sin(a)*x/2 - cos(a)*x**2/6 + \
|
||
|
|
x**3*sin(a)/24 + x**4*cos(a)/120 - x**5*sin(a)/720 + O(x**6)
|
||
|
|
e = x**(-2)*(x*cos(a + x) - x*cos(a))
|
||
|
|
assert e.nseries(x, n=6) == -sin(a) - cos(a)*x/2 + sin(a)*x**2/6 + \
|
||
|
|
cos(a)*x**3/24 - x**4*sin(a)/120 - x**5*cos(a)/720 + O(x**6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3502():
|
||
|
|
e = sin(5*x)/sin(2*x)
|
||
|
|
assert e.nseries(x, n=2) == Rational(5, 2) + O(x**2)
|
||
|
|
assert e.nseries(x, n=6) == \
|
||
|
|
Rational(5, 2) - 35*x**2/4 + 329*x**4/48 + O(x**6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3503():
|
||
|
|
e = sin(2 + x)/(2 + x)
|
||
|
|
assert e.nseries(x, n=2) == sin(2)/2 + x*cos(2)/2 - x*sin(2)/4 + O(x**2)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3506():
|
||
|
|
e = (x + sin(3*x))**(-2)*(x*(x + sin(3*x)) - (x + sin(3*x))*sin(2*x))
|
||
|
|
assert e.nseries(x, n=7) == \
|
||
|
|
Rational(-1, 4) + 5*x**2/96 + 91*x**4/768 + 11117*x**6/129024 + O(x**7)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3508():
|
||
|
|
x = Symbol("x", real=True)
|
||
|
|
assert log(sin(x)).series(x, n=5) == log(x) - x**2/6 - x**4/180 + O(x**5)
|
||
|
|
e = -log(x) + x*(-log(x) + log(sin(2*x))) + log(sin(2*x))
|
||
|
|
assert e.series(x, n=5) == \
|
||
|
|
log(2) + log(2)*x - 2*x**2/3 - 2*x**3/3 - 4*x**4/45 + O(x**5)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3507():
|
||
|
|
e = x**(-4)*(x**2 - x**2*sqrt(cos(x)))
|
||
|
|
assert e.nseries(x, n=9) == \
|
||
|
|
Rational(1, 4) + x**2/96 + 19*x**4/5760 + 559*x**6/645120 + 29161*x**8/116121600 + O(x**9)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3639():
|
||
|
|
assert sin(cos(x)).nseries(x, n=5) == \
|
||
|
|
sin(1) - x**2*cos(1)/2 - x**4*sin(1)/8 + x**4*cos(1)/24 + O(x**5)
|
||
|
|
|
||
|
|
|
||
|
|
def test_hyperbolic():
|
||
|
|
assert sinh(x).nseries(x, n=6) == x + x**3/6 + x**5/120 + O(x**6)
|
||
|
|
assert cosh(x).nseries(x, n=5) == 1 + x**2/2 + x**4/24 + O(x**5)
|
||
|
|
assert tanh(x).nseries(x, n=6) == x - x**3/3 + 2*x**5/15 + O(x**6)
|
||
|
|
assert coth(x).nseries(x, n=6) == \
|
||
|
|
1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
|
||
|
|
assert asinh(x).nseries(x, n=6) == x - x**3/6 + 3*x**5/40 + O(x**6)
|
||
|
|
assert acosh(x).nseries(x, n=6) == \
|
||
|
|
pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
|
||
|
|
assert atanh(x).nseries(x, n=6) == x + x**3/3 + x**5/5 + O(x**6)
|
||
|
|
assert acoth(x).nseries(x, n=6) == -I*pi/2 + x + x**3/3 + x**5/5 + O(x**6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_series2():
|
||
|
|
w = Symbol("w", real=True)
|
||
|
|
x = Symbol("x", real=True)
|
||
|
|
e = w**(-2)*(w*exp(1/x - w) - w*exp(1/x))
|
||
|
|
assert e.nseries(w, n=4) == -exp(1/x) + w*exp(1/x)/2 - w**2*exp(1/x)/6 + w**3*exp(1/x)/24 + O(w**4)
|
||
|
|
|
||
|
|
|
||
|
|
def test_series3():
|
||
|
|
w = Symbol("w", real=True)
|
||
|
|
e = w**(-6)*(w**3*tan(w) - w**3*sin(w))
|
||
|
|
assert e.nseries(w, n=8) == Integer(1)/2 + w**2/8 + 13*w**4/240 + 529*w**6/24192 + O(w**8)
|
||
|
|
|
||
|
|
|
||
|
|
def test_bug4():
|
||
|
|
w = Symbol("w")
|
||
|
|
e = x/(w**4 + x**2*w**4 + 2*x*w**4)*w**4
|
||
|
|
assert e.nseries(w, n=2).removeO().expand() in [x/(1 + 2*x + x**2),
|
||
|
|
1/(1 + x/2 + 1/x/2)/2, 1/x/(1 + 2/x + x**(-2))]
|
||
|
|
|
||
|
|
|
||
|
|
def test_bug5():
|
||
|
|
w = Symbol("w")
|
||
|
|
l = Symbol('l')
|
||
|
|
e = (-log(w) + log(1 + w*log(x)))**(-2)*w**(-2)*((-log(w) +
|
||
|
|
log(1 + x*w))*(-log(w) + log(1 + w*log(x)))*w - x*(-log(w) +
|
||
|
|
log(1 + w*log(x)))*w)
|
||
|
|
assert e.nseries(w, n=0, logx=l) == x/w/l + 1/w + O(1, w)
|
||
|
|
assert e.nseries(w, n=1, logx=l) == x/w/l + 1/w - x/l + 1/l*log(x) \
|
||
|
|
+ x*log(x)/l**2 + O(w)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4115():
|
||
|
|
assert (sin(x)/(1 - cos(x))).nseries(x, n=1) == 2/x + O(x)
|
||
|
|
assert (sin(x)**2/(1 - cos(x))).nseries(x, n=1) == 2 + O(x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_pole():
|
||
|
|
raises(PoleError, lambda: sin(1/x).series(x, 0, 5))
|
||
|
|
raises(PoleError, lambda: sin(1 + 1/x).series(x, 0, 5))
|
||
|
|
raises(PoleError, lambda: (x*sin(1/x)).series(x, 0, 5))
|
||
|
|
|
||
|
|
|
||
|
|
def test_expsinbug():
|
||
|
|
assert exp(sin(x)).series(x, 0, 0) == O(1, x)
|
||
|
|
assert exp(sin(x)).series(x, 0, 1) == 1 + O(x)
|
||
|
|
assert exp(sin(x)).series(x, 0, 2) == 1 + x + O(x**2)
|
||
|
|
assert exp(sin(x)).series(x, 0, 3) == 1 + x + x**2/2 + O(x**3)
|
||
|
|
assert exp(sin(x)).series(x, 0, 4) == 1 + x + x**2/2 + O(x**4)
|
||
|
|
assert exp(sin(x)).series(x, 0, 5) == 1 + x + x**2/2 - x**4/8 + O(x**5)
|
||
|
|
|
||
|
|
|
||
|
|
def test_floor():
|
||
|
|
x = Symbol('x')
|
||
|
|
assert floor(x).series(x) == 0
|
||
|
|
assert floor(-x).series(x) == -1
|
||
|
|
assert floor(sin(x)).series(x) == 0
|
||
|
|
assert floor(sin(-x)).series(x) == -1
|
||
|
|
assert floor(x**3).series(x) == 0
|
||
|
|
assert floor(-x**3).series(x) == -1
|
||
|
|
assert floor(cos(x)).series(x) == 0
|
||
|
|
assert floor(cos(-x)).series(x) == 0
|
||
|
|
assert floor(5 + sin(x)).series(x) == 5
|
||
|
|
assert floor(5 + sin(-x)).series(x) == 4
|
||
|
|
|
||
|
|
assert floor(x).series(x, 2) == 2
|
||
|
|
assert floor(-x).series(x, 2) == -3
|
||
|
|
|
||
|
|
x = Symbol('x', negative=True)
|
||
|
|
assert floor(x + 1.5).series(x) == 1
|
||
|
|
|
||
|
|
|
||
|
|
def test_frac():
|
||
|
|
assert frac(x).series(x, cdir=1) == x
|
||
|
|
assert frac(x).series(x, cdir=-1) == 1 + x
|
||
|
|
assert frac(2*x + 1).series(x, cdir=1) == 2*x
|
||
|
|
assert frac(2*x + 1).series(x, cdir=-1) == 1 + 2*x
|
||
|
|
assert frac(x**2).series(x, cdir=1) == x**2
|
||
|
|
assert frac(x**2).series(x, cdir=-1) == x**2
|
||
|
|
assert frac(sin(x) + 5).series(x, cdir=1) == x - x**3/6 + x**5/120 + O(x**6)
|
||
|
|
assert frac(sin(x) + 5).series(x, cdir=-1) == 1 + x - x**3/6 + x**5/120 + O(x**6)
|
||
|
|
assert frac(sin(x) + S.Half).series(x) == S.Half + x - x**3/6 + x**5/120 + O(x**6)
|
||
|
|
assert frac(x**8).series(x, cdir=1) == O(x**6)
|
||
|
|
assert frac(1/x).series(x) == AccumBounds(0, 1) + O(x**6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_ceiling():
|
||
|
|
assert ceiling(x).series(x) == 1
|
||
|
|
assert ceiling(-x).series(x) == 0
|
||
|
|
assert ceiling(sin(x)).series(x) == 1
|
||
|
|
assert ceiling(sin(-x)).series(x) == 0
|
||
|
|
assert ceiling(1 - cos(x)).series(x) == 1
|
||
|
|
assert ceiling(1 - cos(-x)).series(x) == 1
|
||
|
|
assert ceiling(x).series(x, 2) == 3
|
||
|
|
assert ceiling(-x).series(x, 2) == -2
|
||
|
|
|
||
|
|
|
||
|
|
def test_abs():
|
||
|
|
a = Symbol('a')
|
||
|
|
assert abs(x).nseries(x, n=4) == x
|
||
|
|
assert abs(-x).nseries(x, n=4) == x
|
||
|
|
assert abs(x + 1).nseries(x, n=4) == x + 1
|
||
|
|
assert abs(sin(x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4)
|
||
|
|
assert abs(sin(-x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4)
|
||
|
|
assert abs(x - a).nseries(x, 1) == -a*sign(1 - a) + (x - 1)*sign(1 - a) + sign(1 - a)
|
||
|
|
|
||
|
|
|
||
|
|
def test_dir():
|
||
|
|
assert abs(x).series(x, 0, dir="+") == x
|
||
|
|
assert abs(x).series(x, 0, dir="-") == -x
|
||
|
|
assert floor(x + 2).series(x, 0, dir='+') == 2
|
||
|
|
assert floor(x + 2).series(x, 0, dir='-') == 1
|
||
|
|
assert floor(x + 2.2).series(x, 0, dir='-') == 2
|
||
|
|
assert ceiling(x + 2.2).series(x, 0, dir='-') == 3
|
||
|
|
assert sin(x + y).series(x, 0, dir='-') == sin(x + y).series(x, 0, dir='+')
|
||
|
|
|
||
|
|
|
||
|
|
def test_cdir():
|
||
|
|
assert abs(x).series(x, 0, cdir=1) == x
|
||
|
|
assert abs(x).series(x, 0, cdir=-1) == -x
|
||
|
|
assert floor(x + 2).series(x, 0, cdir=1) == 2
|
||
|
|
assert floor(x + 2).series(x, 0, cdir=-1) == 1
|
||
|
|
assert floor(x + 2.2).series(x, 0, cdir=1) == 2
|
||
|
|
assert ceiling(x + 2.2).series(x, 0, cdir=-1) == 3
|
||
|
|
assert sin(x + y).series(x, 0, cdir=-1) == sin(x + y).series(x, 0, cdir=1)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_3504():
|
||
|
|
a = Symbol("a")
|
||
|
|
e = asin(a*x)/x
|
||
|
|
assert e.series(x, 4, n=2).removeO() == \
|
||
|
|
(x - 4)*(a/(4*sqrt(-16*a**2 + 1)) - asin(4*a)/16) + asin(4*a)/4
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4441():
|
||
|
|
a, b = symbols('a,b')
|
||
|
|
f = 1/(1 + a*x)
|
||
|
|
assert f.series(x, 0, 5) == 1 - a*x + a**2*x**2 - a**3*x**3 + \
|
||
|
|
a**4*x**4 + O(x**5)
|
||
|
|
f = 1/(1 + (a + b)*x)
|
||
|
|
assert f.series(x, 0, 3) == 1 + x*(-a - b)\
|
||
|
|
+ x**2*(a + b)**2 + O(x**3)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_4329():
|
||
|
|
assert tan(x).series(x, pi/2, n=3).removeO() == \
|
||
|
|
-pi/6 + x/3 - 1/(x - pi/2)
|
||
|
|
assert cot(x).series(x, pi, n=3).removeO() == \
|
||
|
|
-x/3 + pi/3 + 1/(x - pi)
|
||
|
|
assert limit(tan(x)**tan(2*x), x, pi/4) == exp(-1)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5183():
|
||
|
|
assert abs(x + x**2).series(n=1) == O(x)
|
||
|
|
assert abs(x + x**2).series(n=2) == x + O(x**2)
|
||
|
|
assert ((1 + x)**2).series(x, n=6) == x**2 + 2*x + 1
|
||
|
|
assert (1 + 1/x).series() == 1 + 1/x
|
||
|
|
assert Derivative(exp(x).series(), x).doit() == \
|
||
|
|
1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5654():
|
||
|
|
a = Symbol('a')
|
||
|
|
assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=0) == \
|
||
|
|
-I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(1, (x, I*a))
|
||
|
|
assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=1) == 3/(16*a**4) \
|
||
|
|
-I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(-I*a + x, (x, I*a))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5925():
|
||
|
|
sx = sqrt(x + z).series(z, 0, 1)
|
||
|
|
sxy = sqrt(x + y + z).series(z, 0, 1)
|
||
|
|
s1, s2 = sx.subs(x, x + y), sxy
|
||
|
|
assert (s1 - s2).expand().removeO().simplify() == 0
|
||
|
|
|
||
|
|
sx = sqrt(x + z).series(z, 0, 1)
|
||
|
|
sxy = sqrt(x + y + z).series(z, 0, 1)
|
||
|
|
assert sxy.subs({x:1, y:2}) == sx.subs(x, 3)
|
||
|
|
|
||
|
|
|
||
|
|
def test_exp_2():
|
||
|
|
assert exp(x**3).nseries(x, 0, 14) == 1 + x**3 + x**6/2 + x**9/6 + x**12/24 + O(x**14)
|