text: from sympy import (Symbol, Rational, ln, exp, log, sqrt, E, O, pi, I, sinh,
      sin, cosh, cos, tanh, coth, asinh, acosh, atanh, acoth, tan, cot, Integer,
      PoleError, floor, ceiling, asin, symbols, limit, Piecewise, Eq, sign,
      Derivative)
  from sympy.abc import x, y, z
  
  from sympy.utilities.pytest import raises
  
  def test_simple_1():
      assert x.nseries(x, n=5) == x
      assert y.nseries(x, n=5) == y
      assert (1/(x*y)).nseries(y, n=5) == 1/(x*y)
      assert Rational(3,4).nseries(x, n=5) == Rational(3,4)
      assert x.nseries() == x
  
  def test_mul_0():
      assert (x*ln(x)).nseries(x, n=5) == x*ln(x)
  
  def test_mul_1():
      assert (x*ln(2+x)).nseries(x, n=4) == x*log(2) + x**2/2 - x**3/8 + x**4/24 + O(x**5) # x*log(2)+x**2/2-x**3/8+O(x**4)
      assert (x*ln(1+x)).nseries(x, n=4) == x**2 - x**3/2 + x**4/3 + O(x**5) # x**2- x**3/2 + O(x**4)
  
  def test_pow_0():
      assert (x**2).nseries(x, n=5) == x**2
      assert (1/x).nseries(x, n=5) == 1/x
      assert (1/x**2).nseries(x, n=5) == 1/x**2
      assert (x**(Rational(2,3))).nseries(x, n=5) == (x**(Rational(2,3)))
      assert (x**(Rational(3,2))).nseries(x, n=5) == (x**(Rational(3,2)))
  
  def test_pow_1():
      assert ((1+x)**2).nseries(x, n=5) == 1+2*x+x**2
  
  def test_geometric_1():
      assert (1/(1-x)).nseries(x, n=5) == 1+x+x**2+x**3+x**4+O(x**5)
      assert (x/(1-x)).nseries(x, n=5) == x + x**2 + x**3 + x**4 + x**5 + O(x**6) # x+x**2+x**3+x**4+O(x**5)
      assert (x**3/(1-x)).nseries(x, n=5) == x**3 + x**4 + x**5 + x**6 + x**7 + O(x**8) # x**3+x**4+O(x**5)
  
  def test_sqrt_1():
      assert sqrt(1+x).nseries(x, n=5) == 1+x/2-x**2/8+x**3/16-5*x**4/128+O(x**5)
  
  def test_exp_1():
      assert exp(x).nseries(x, n=5) == 1+x+x**2/2+x**3/6+x**4/24 + O(x**5)
      assert exp(x).nseries(x, n=12) == 1+x+x**2/2+x**3/6+x**4/24+x**5/120+  \
              x**6/720+x**7/5040+x**8/40320+x**9/362880+x**10/3628800+  \
              x**11/39916800 + O(x**12)
      assert exp(1/x).nseries(x, n=5) == exp(1/x)
      assert exp(1/(1+x)).nseries(x, n=4) ==  \
              (E*(1-x-13*x**3/6+3*x**2/2)).expand() + O(x**4)
      assert exp(2+x).nseries(x, n=5) ==  \
              (exp(2)*(1+x+x**2/2+x**3/6+x**4/24)).expand() + O(x**5)
  
  def test_exp_sqrt_1():
      assert exp(1+sqrt(x)).nseries(x, n=3) ==  \
          (exp(1)*(1+sqrt(x)+x/2+sqrt(x)*x/6)).expand() + O(sqrt(x)**3)
  
  def test_power_x_x1():
      assert (exp(x*ln(x))).nseries(x, n=4) == \
              1+x*log(x)+x**2*log(x)**2/2+x**3*log(x)**3/6 + O(x**4*log(x)**4)
  
  def test_power_x_x2():
      assert (x**x).nseries(x, n=4) == \
              1+x*log(x)+x**2*log(x)**2/2+x**3*log(x)**3/6 + O(x**4*log(x)**4)
  
  def test_log_singular1():
      assert log(1+1/x).nseries(x, n=5) == x + log(1/x) - x**2/2 + x**3/3 - \
              x**4/4 + O(x**5)
  
  def test_log_power1():
      e = 1 / (1/x + x ** (log(3)/log(2)))
      assert e.nseries(x, n=5) == x - x**(2 + log(3)/log(2)) + O(x**5)
  
  def test_log_series():
      e = 1/(1-log(x))
      assert e.nseries(x, n=5) == -1/log(x) - log(x)**(-2) - log(x)**(-3) - \
              log(x)**(-4) + O(log(x)**(-5))
  
  def test_log2():
      e = log(-1/x)
      assert e.nseries(x, n=5) == -log(x) + log(-1)
  
  def test_log3():
      e = 1/log(-1/x)
      assert e.nseries(x, n=4) == -1/log(x) - pi*I*log(x)**(-2) + \
          pi**2*log(x)**(-3) + O(log(x)**(-4))
  
  def test_series1():
      x = Symbol("x")
      e = sin(x)
      assert e.nseries(x,0,0) != 0
      assert e.nseries(x,0,0) == O(1, x)
      assert e.nseries(x,0,1) == O(x, x)
      assert e.nseries(x,0,2) == x + O(x**2, x)
      assert e.nseries(x,0,3) == x + O(x**3, x)
      assert e.nseries(x,0,4) == x-x**3/6 + O(x**4, x)
  
      e = (exp(x)-1)/x
      assert e.nseries(x,0,3) == 1+x/2+O(x**2, x)
  
      #assert x.nseries(x,0,0) == O(1, x)
      #assert x.nseries(x,0,1) == O(x, x)
      assert x.nseries(x,0,2) == x
  
  def test_seriesbug1():
      x = Symbol("x")
      assert (1/x).nseries(x,0,3) == 1/x
      assert (x+1/x).nseries(x,0,3) == x+1/x
  
  def test_series2x():
      x = Symbol("x")
      assert ((x+1)**(-2)).nseries(x,0,4) == 1-2*x+3*x**2-4*x**3+O(x**4, x)
      assert ((x+1)**(-1)).nseries(x,0,4) == 1-x+x**2-x**3+O(x**4, x)
      assert ((x+1)**0).nseries(x,0,3) == 1
      assert ((x+1)**1).nseries(x,0,3) == 1+x
      assert ((x+1)**2).nseries(x,0,3) == 1+2*x+x**2
      assert ((x+1)**3).nseries(x,0,3) == 1 + 3*x + 3*x**2 + x**3 # 1+3*x+3*x**2+O(x**3)
  
      assert (1/(1+x)).nseries(x,0,4) == 1-x+x**2-x**3+O(x**4, x)
      assert (x+3/(1+2*x)).nseries(x,0,4) == 3-5*x+12*x**2-24*x**3+O(x**4, x)
  
      assert ((1/x+1)**3).nseries(x,0,3)== 1+x**(-3)+3*x**(-2)+3/x
      assert (1/(1+1/x)).nseries(x,0,4) == x-x**2+x**3-O(x**4, x)
      assert (1/(1+1/x**2)).nseries(x,0,6) == x**2-x**4+O(x**6, x)
  
  def test_bug2(): ### 1/log(0) * log(0) problem
      w = Symbol("w")
      e = (w**(-1)+w**(-log(3)*log(2)**(-1)))**(-1)*(3*w**(-log(3)*log(2)**(-1))+2*w**(-1))
      e = e.expand()
      assert e.nseries(w, 0, 4).subs(w, 0) == 3
  
  def test_exp():
      x = Symbol("x")
      e = (1+x)**(1/x)
      assert e.nseries(x, n=3) == exp(1) - x*exp(1)/2 + O(x**2, x)
  
  def test_exp2():
      x = Symbol("x")
      w = Symbol("w")
      e = w**(1-log(x)/(log(2) + log(x)))
      assert e.nseries(w,0,1) == e
  
  def test_bug3():
      x = Symbol("x")
      e = (2/x+3/x**2)/(1/x+1/x**2)
      assert e.nseries(x, n=3) == 3 + O(x)
  
  def test_generalexponent():
      x = Symbol("x")
      p = 2
      e = (2/x+3/x**p)/(1/x+1/x**p)
      assert e.nseries(x,0,3) == 3 + O(x)
      p = Rational(1,2)
      e = (2/x+3/x**p)/(1/x+1/x**p)
      assert e.nseries(x,0,2) == 2 + sqrt(x) + O(x)
  
      e=1+x**Rational(1,2)
      assert e.nseries(x,0,4) == 1+x**Rational(1,2)
  
  # more complicated example
  def test_genexp_x():
      x = Symbol("x")
      e=1/(1+x**Rational(1,2))
      assert e.nseries(x,0,2) == \
                  1+x-x**Rational(1,2)-x**Rational(3,2)+O(x**2, x)
  
  # more complicated example
  def test_genexp_x2():
      x = Symbol("x")
      p = Rational(3,2)
      e = (2/x+3/x**p)/(1/x+1/x**p)
      assert e.nseries(x,0,3) == 3 - sqrt(x) + x + O(sqrt(x)**3)
  
  def test_seriesbug2():
      w = Symbol("w")
      #simple case (1):
      e = ((2*w)/w)**(1+w)
      assert e.nseries(w,0,1) == 2 + O(w, w)
      assert e.nseries(w,0,1).subs(w,0) == 2
  
  def test_seriesbug2b():
      w = Symbol("w")
      #test sin
      e = sin(2*w)/w
      assert e.nseries(w,0,3) == 2 + O(w**2, w)
  
  def test_seriesbug2d():
      w = Symbol("w", real=True)
      e = log(sin(2*w)/w)
      assert e.nseries(w, n=5) == log(2) - 2*w**2/3 - 4*w**4/45 + O(w**5)
  
  def test_seriesbug2c():
      w = Symbol("w", real=True)
      #more complicated case, but sin(x)~x, so the result is the same as in (1)
      e=(sin(2*w)/w)**(1+w)
      assert e.nseries(w,0,1) == 2 + O(w)
      assert e.nseries(w,0,3) == 2-Rational(4,3)*w**2+w**2*log(2)**2+2*w*log(2)+O(w**3, w)
      assert e.nseries(w,0,2).subs(w,0) == 2
  
  def test_expbug4():
      x = Symbol("x", real=True)
      assert (log(sin(2*x)/x)*(1+x)).nseries(x,0,2) == log(2) + x*log(2) + O(x**2, x)
      #assert exp(log(2)+O(x)).nseries(x,0,2) == 2 +O(x**2, x)
      assert exp(log(sin(2*x)/x)*(1+x)).nseries(x,0,2) == 2 + 2*x*log(2) + O(x**2)
      #assert ((2+O(x))**(1+x)).nseries(x,0,2) == 2 + O(x**2, x)
  
  def test_logbug4():
      x = Symbol("x")
      assert log(2+O(x)).nseries(x,0,2) == log(2) + O(x, x)
  
  def test_expbug5():
      x = Symbol("x")
      #assert exp(O(x)).nseries(x,0,2) == 1 + O(x**2, x)
      assert exp(log(1+x)/x).nseries(x, n=3) == exp(1) + -exp(1)*x/2 + O(x**2)
  
  def test_sinsinbug():
      x = Symbol("x")
      assert sin(sin(x)).nseries(x,0,8) == x-x**3/3+x**5/10-8*x**7/315+O(x**8)
  
  def test_issue159():
      x = Symbol("x")
      a=x/(exp(x)-1)
      assert a.nseries(x,0,6) == 1 - x/2 - x**4/720 + x**2/12 + O(x**5)
  
  def test_issue105():
      x = Symbol("x", nonnegative=True)
      f = sin(x**3)**Rational(1,3)
      assert f.nseries(x,0,17) == x - x**7/18 - x**13/3240 + O(x**17)
  
  def test_issue125():
      y = Symbol("y")
      f=(1-y**(Rational(1)/2))**(Rational(1)/2)
      assert f.nseries(y,0,2) == 1 - sqrt(y)/2-y/8-y**Rational(3,2)/16+O(y**2)
  
  def test_issue364():
      w = Symbol("w")
      x = Symbol("x")
      e = 1/x*(-log(w**(1 + 1/log(3)*log(5))) + log(w + w**(1/log(3)*log(5))))
      e_ser = -log(5)*log(w)/(x*log(3)) + w**(log(5)/log(3) - 1)/x - \
              w**(2*log(5)/log(3) - 2)/(2*x) + O(w)
      assert e.nseries(w, n=1) == e_ser
  
  def test_sin():
      x = Symbol("x")
      y = Symbol("y")
      assert sin(8*x).nseries(x, n=4) == 8*x - 256*x**3/3 + O(x**4)
      assert sin(x+y).nseries(x, n=1) == sin(y) + O(x)
      assert sin(x+y).nseries(x, n=2) == sin(y) + cos(y)*x + O(x**2)
      assert sin(x+y).nseries(x, n=5) == sin(y) + cos(y)*x - sin(y)*x**2/2 - \
          cos(y)*x**3/6 + sin(y)*x**4/24 + O(x**5)
  
  def test_issue416():
      x = Symbol("x")
      e = sin(8*x)/x
      assert e.nseries(x, n=6) == 8 - 256*x**2/3 + 4096*x**4/15 + O(x**5)
  
  def test_issue406():
      x = Symbol("x")
      e = sin(x)**(-4)*(cos(x)**Rational(1,2)*sin(x)**2 - \
              cos(x)**Rational(1,3)*sin(x)**2)
      assert e.nseries(x, n=8) == -Rational(1)/12 - 7*x**2/288 - \
              43*x**4/10368 + O(x**5)
  
  def test_issue402():
      x = Symbol("x")
      a = Symbol("a")
      e = x**(-2)*(x*sin(a + x) - x*sin(a))
      assert e.nseries(x, n=5) == cos(a) - sin(a)*x/2 - cos(a)*x**2/6 + \
              sin(a)*x**3/24 + O(x**4)
      e = x**(-2)*(x*cos(a + x) - x*cos(a))
      assert e.nseries(x, n=5) == -sin(a) - cos(a)*x/2 + sin(a)*x**2/6 + \
              cos(a)*x**3/24 + O(x**4)
  
  def test_issue403():
      x = Symbol("x")
      e = sin(5*x)/sin(2*x)
      assert e.nseries(x, n=2) == Rational(5,2) + O(x)
      assert e.nseries(x, n=6) == Rational(5,2) - 35*x**2/4 + 329*x**4/48 + O(x**5)
  
  def test_issue404():
      x = Symbol("x")
      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_issue407():
      x = Symbol("x")
      e = (x + sin(3*x))**(-2)*(x*(x + sin(3*x)) - (x + sin(3*x))*sin(2*x))
      assert e.nseries(x, n=6) == -Rational(1,4) + 5*x**2/96 + 91*x**4/768 + O(x**5)
  
  def test_issue409():
      x = Symbol("x", real=True)
      assert log(sin(x)).nseries(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.nseries(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_issue408():
      x = Symbol("x")
      e = x**(-4)*(x**2 - x**2*cos(x)**Rational(1,2))
      assert e.nseries(x, n=7) == Rational(1,4) + x**2/96 + 19*x**4/5760 + O(x**5)
  
  def test_issue540():
      x = Symbol("x")
      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():
      x = Symbol("x")
      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) == x + x**3/3 + x**5/5 + pi*I/2 + 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=3) == -exp(1/x) + w * exp(1/x) / 2  + O(w**2)
  
  def test_series3():
      w = Symbol("w", real=True)
      x = Symbol("x", real=True)
      e = w**(-6)*(w**3*tan(w) - w**3*sin(w))
      assert e.nseries(w, n=5) == Integer(1)/2 + O(w**2)
  
  def test_bug4():
      w = Symbol("w")
      x = Symbol("x")
      e = x/(w**4 + x**2*w**4 + 2*x*w**4)*w**4
      assert e.nseries(w, n=2) 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")
      x = Symbol("x")
      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=1) == x/w/log(w) + 1/w + O(1/log(w))
      assert e.nseries(w, n=2) == x/w/log(w) + 1/w - x/log(w) + 1/log(w)*log(x)\
              + x*log(x)/log(w)**2 + O(w/log(w))
  
  
  def test_issue1016():
      x = Symbol("x")
      assert ( sin(x)/(1 - cos(x)) ).nseries(x, n=2) == O(1/x)
      assert ( sin(x)**2/(1 - cos(x)) ).nseries(x, n=2) == O(1, x)
  
  def test_pole():
      x = Symbol("x")
      raises(PoleError, "sin(1/x).series(x, 0, 5)")
      raises(PoleError, "sin(1+1/x).series(x, 0, 5)")
      raises(PoleError, "(x*sin(1/x)).series(x, 0, 5)")
  
  def test_expsinbug():
      x = Symbol("x")
      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_ceiling():
      x = Symbol('x')
      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():
      x = Symbol('x')
      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) == Piecewise((x - 1, Eq(1 - a, 0)),
                                                  ((x - a)*sign(1 - a), True))
  
  def test_dir():
      x = Symbol('x')
      y = Symbol('y')
      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_issue405():
      a = Symbol("a")
      e = asin(a*x)/x
      assert e.series(x, 4, n=2).removeO().subs(x, x - 4) == (
             asin(4*a)/4 +
             (4 - x)*asin(4*a)/16 -
             a*(4 - x)/(4*sqrt(1 - 16*a**2)))
  
  def test_issue1342():
      x, a, b = symbols('x 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 - a*x - b*x + a**2*x**2 + b**2*x**2 + \
              2*a*b*x**2 + O(x**3)
  
  def test_issue1230():
      assert tan(x).series(x, pi/2, n=3).removeO().subs(x, x - pi/2) == \
             -pi/6 + x/3 - 1/(x - pi/2)
      assert cot(x).series(x, pi, n=3).removeO().subs(x, x - pi) == \
             -x/3 + pi/3 + 1/(x - pi)
      assert limit(tan(x)**tan(2*x), x, pi/4) == exp(-1)
  
  def test_issue2084():
      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) == 1 + 2*x + x**2
      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)
  