sympy/series/tests/test_nseries.py

   1 from sympy import Symbol, Rational, ln, exp, log, sqrt, E, O, pi, I, sinh, \
   2         sin, cosh, cos, tanh, coth, asinh, acosh, atanh, acoth, tan, Integer, \
   3         PoleError, floor, ceiling
   4 from sympy.abc import x, y, z
   5 from sympy.utilities.pytest import XFAIL
   6 import py
   7
   8 def test_simple_1():
   9     assert x.nseries(x, 0, 5) == x
  10     assert y.nseries(x, 0, 5) == y
  11     assert (1/(x*y)).nseries(y, 0, 5) == 1/(x*y)
  12     assert Rational(3,4).nseries(x, 0, 5) == Rational(3,4)
  13
  14 def test_mul_0():
  15     assert (x*ln(x)).nseries(x, 0, 5) == x*ln(x)
  16
  17 def test_mul_1():
  18     assert (x*ln(2+x)).nseries(x, 0, 4) == x*log(2)+x**2/2-x**3/8+x**4/24+ \
  19             O(x**5)
  20     assert (x*ln(1+x)).nseries(x, 0, 4) == x**2- x**3/2 + x**4/3 + O(x**5)
  21
  22 def test_pow_0():
  23     assert (x**2).nseries(x, 0, 5) == x**2
  24     assert (1/x).nseries(x, 0, 5) == 1/x
  25     assert (1/x**2).nseries(x, 0, 5) == 1/x**2
  26     assert (x**(Rational(2,3))).nseries(x, 0, 5) == (x**(Rational(2,3)))
  27     assert (x**(Rational(3,2))).nseries(x, 0, 5) == (x**(Rational(3,2)))
  28
  29 def test_pow_1():
  30     assert ((1+x)**2).nseries(x, 0, 5) == 1+2*x+x**2
  31
  32 def test_geometric_1():
  33     assert (1/(1-x)).nseries(x, 0, 5) == 1+x+x**2+x**3+x**4+O(x**5)
  34     assert (x/(1-x)).nseries(x, 0, 5) == x+x**2+x**3+x**4+x**5+O(x**6)
  35     assert (x**3/(1-x)).nseries(x, 0, 5) == x**3+x**4+x**5+x**6+x**7+O(x**8)
  36
  37 def test_sqrt_1():
  38     assert sqrt(1+x).nseries(x, 0, 5) == 1+x/2-x**2/8+x**3/16-5*x**4/128+O(x**5)
  39
  40 def test_exp_1():
  41     assert exp(x).nseries(x, 0, 5) == 1+x+x**2/2+x**3/6+x**4/24 + O(x**5)
  42     assert exp(x).nseries(x, 0, 12) == 1+x+x**2/2+x**3/6+x**4/24+x**5/120+  \
  43             x**6/720+x**7/5040+x**8/40320+x**9/362880+x**10/3628800+  \
  44             x**11/39916800 + O(x**12)
  45     assert exp(1/x).nseries(x, 0, 5) == exp(1/x)
  46     assert exp(1/(1+x)).nseries(x, 0, 4) ==  \
  47             (E*(1-x-13*x**3/6+3*x**2/2)).expand() + O(x**4)
  48     assert exp(2+x).nseries(x, 0, 5) ==  \
  49             (exp(2)*(1+x+x**2/2+x**3/6+x**4/24)).expand() + O(x**5)
  50
  51 def test_exp_sqrt_1():
  52     assert exp(1+sqrt(x)).nseries(x, 0, 3) ==  \
  53         (exp(1)*(1+sqrt(x)+x/2+sqrt(x)*x/6)).expand() + O(sqrt(x)**3)
  54
  55 def test_power_x_x1():
  56     assert (exp(x*ln(x))).nseries(x, 0, 4) == \
  57             1+x*log(x)+x**2*log(x)**2/2+x**3*log(x)**3/6 + O(x**4*log(x)**4)
  58
  59 def test_power_x_x2():
  60     assert (x**x).nseries(x, 0, 4) == \
  61             1+x*log(x)+x**2*log(x)**2/2+x**3*log(x)**3/6 + O(x**4*log(x)**4)
  62
  63 def test_log_singular1():
  64     assert log(1+1/x).nseries(x, 0, 5) == x - log(x) - x**2/2 + x**3/3 - \
  65             x**4/4 + O(x**5)
  66
  67 def test_log_power1():
  68     e = 1 / (1/x + x ** (log(3)/log(2)))
  69     assert e.nseries(x, 0, 5) == x - x**(2 + log(3)/log(2)) + O(x**5)
  70
  71 def test_log_series():
  72     e = 1/(1-log(x))
  73     assert e.nseries(x, 0, 5) == -1/log(x) - log(x)**(-2) - log(x)**(-3) - \
  74             log(x)**(-4) + O(log(x)**(-5))
  75
  76 def test_log2():
  77     e = log(-1/x)
  78     assert e.nseries(x, 0, 5) == -log(x) + log(-1)
  79
  80 def test_log3():
  81     e = 1/log(-1/x)
  82     assert e.nseries(x, 0, 4) == -1/log(x) - pi*I*log(x)**(-2) + \
  83         pi**2*log(x)**(-3) + O(log(x)**(-4))
  84
  85 def testseries1():
  86     x = Symbol("x")
  87     e = sin(x)
  88     assert e.nseries(x,0,0) != 0
  89     assert e.nseries(x,0,0) == O(1, x)
  90     assert e.nseries(x,0,1) == O(x, x)
  91     assert e.nseries(x,0,2) == x + O(x**2, x)
  92     assert e.nseries(x,0,3) == x + O(x**3, x)
  93     assert e.nseries(x,0,4) == x-x**3/6 + O(x**4, x)
  94
  95     e = (exp(x)-1)/x
  96     assert e.nseries(x,0,3) == 1+x/2+O(x**2, x)
  97
  98     #assert x.nseries(x,0,0) == O(1, x)
  99     #assert x.nseries(x,0,1) == O(x, x)
 100     assert x.nseries(x,0,2) == x
 101
 102 def testseriesbug1():
 103     x = Symbol("x")
 104     assert (1/x).nseries(x,0,3) == 1/x
 105     assert (x+1/x).nseries(x,0,3) == x+1/x
 106
 107 def testseries2():
 108     x = Symbol("x")
 109     assert ((x+1)**(-2)).nseries(x,0,4) == 1-2*x+3*x**2-4*x**3+O(x**4, x)
 110     assert ((x+1)**(-1)).nseries(x,0,4) == 1-x+x**2-x**3+O(x**4, x)
 111     assert ((x+1)**0).nseries(x,0,3) == 1
 112     assert ((x+1)**1).nseries(x,0,3) == 1+x
 113     assert ((x+1)**2).nseries(x,0,3) == 1+2*x+x**2
 114     assert ((x+1)**3).nseries(x,0,3) == 1+3*x+3*x**2+x**3
 115
 116     assert (1/(1+x)).nseries(x,0,4) == 1-x+x**2-x**3+O(x**4, x)
 117     assert (x+3/(1+2*x)).nseries(x,0,4) == 3-5*x+12*x**2-24*x**3+O(x**4, x)
 118
 119     assert ((1/x+1)**3).nseries(x,0,3)== 1+x**(-3)+3*x**(-2)+3/x
 120     assert (1/(1+1/x)).nseries(x,0,4) == x-x**2+x**3-O(x**4, x)
 121     assert (1/(1+1/x**2)).nseries(x,0,6) == x**2-x**4+O(x**6, x)
 122
 123 def test_bug2(): ### 1/log(0) * log(0) problem
 124     w = Symbol("w")
 125     e = (w**(-1)+w**(-log(3)*log(2)**(-1)))**(-1)*(3*w**(-log(3)*log(2)**(-1))+2*w**(-1))
 126     e = e.expand()
 127     #should be 3, but is 2
 128     assert e.nseries(w,0,4).subs(w,0)==3
 129
 130 def test_exp():
 131     x = Symbol("x")
 132     e = (1+x)**(1/x)
 133     assert e.nseries(x, 0, 3) == exp(1) - x*exp(1)/2 + O(x**2, x)
 134
 135 def test_exp2():
 136     x = Symbol("x")
 137     w = Symbol("w")
 138     e = w**(1-log(x)/(log(2) + log(x)))
 139     assert e.nseries(w,0,1) == e
 140
 141 def test_bug3():
 142     x = Symbol("x")
 143     e = (2/x+3/x**2)/(1/x+1/x**2)
 144     assert e.nseries(x, 0, 3) == 3 + O(x)
 145
 146 def test_generalexponent():
 147     x = Symbol("x")
 148     p = 2
 149     e = (2/x+3/x**p)/(1/x+1/x**p)
 150     assert e.nseries(x,0,3) == 3 + O(x)
 151     p = Rational(1,2)
 152     e = (2/x+3/x**p)/(1/x+1/x**p)
 153     assert e.nseries(x,0,2) == 2 + sqrt(x) + O(x)
 154
 155     e=1+x**Rational(1,2)
 156     assert e.nseries(x,0,4) == 1+x**Rational(1,2)
 157
 158 # more complicated example
 159 def test_genexp_x():
 160     x = Symbol("x")
 161     e=1/(1+x**Rational(1,2))
 162     assert e.nseries(x,0,2) == \
 163                 1+x-x**Rational(1,2)-x**Rational(3,2)+O(x**2, x)
 164
 165 # more complicated example
 166 def test_genexp_x2():
 167     x = Symbol("x")
 168     p = Rational(3,2)
 169     e = (2/x+3/x**p)/(1/x+1/x**p)
 170     assert e.nseries(x,0,3) == 3 - sqrt(x) + x + O(sqrt(x)**3)
 171
 172 def test_seriesbug2():
 173     w = Symbol("w")
 174     #simple case (1):
 175     e = ((2*w)/w)**(1+w)
 176     assert e.nseries(w,0,1) == 2 + O(w, w)
 177     assert e.nseries(w,0,1).subs(w,0) == 2
 178
 179 def test_seriesbug2b():
 180     w = Symbol("w")
 181     #test sin
 182     e = sin(2*w)/w
 183     assert e.nseries(w,0,3) == 2 + O(w**2, w)
 184
 185 def test_seriesbug2d():
 186     w = Symbol("w", real=True)
 187     e = log(sin(2*w)/w)
 188     assert e.nseries(w, 0, 5) == log(2) - 2*w**2/3 - 4*w**4/45 + O(w**5)
 189
 190 def test_seriesbug2c():
 191     w = Symbol("w", real=True)
 192     #more complicated case, but sin(x)~x, so the result is the same as in (1)
 193     e=(sin(2*w)/w)**(1+w)
 194     assert e.nseries(w,0,1) == 2 + O(w)
 195     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)
 196     assert e.nseries(w,0,2).subs(w,0) == 2
 197
 198 def test_expbug4():
 199     x = Symbol("x", real=True)
 200     assert (log(sin(2*x)/x)*(1+x)).nseries(x,0,2) == log(2) + x*log(2) + O(x**2, x)
 201     #assert exp(log(2)+O(x)).nseries(x,0,2) == 2 +O(x**2, x)
 202     assert exp(log(sin(2*x)/x)*(1+x)).nseries(x,0,2) == 2 + 2*x*log(2) + O(x**2)
 203     #assert ((2+O(x))**(1+x)).nseries(x,0,2) == 2 + O(x**2, x)
 204
 205 def test_logbug4():
 206     x = Symbol("x")
 207     assert log(2+O(x)).nseries(x,0,2) == log(2) + O(x, x)
 208
 209 def test_expbug5():
 210     x = Symbol("x")
 211     #assert exp(O(x)).nseries(x,0,2) == 1 + O(x**2, x)
 212     assert exp(log(1+x)/x).nseries(x, 0, 3) == exp(1) + -exp(1)*x/2 + O(x**2)
 213
 214 def test_sinsinbug():
 215     x = Symbol("x")
 216     assert sin(sin(x)).nseries(x,0,8) == x-x**3/3+x**5/10-8*x**7/315+O(x**8)
 217
 218 def test_issue159():
 219     x = Symbol("x")
 220     a=x/(exp(x)-1)
 221     assert a.nseries(x,0,6) == 1 - x/2 - x**4/720 + x**2/12 + O(x**5)
 222
 223 def test_issue105():
 224     x = Symbol("x")
 225     f = sin(x**3)**Rational(1,3)
 226     assert f.nseries(x,0,17) == x - x**7/18 - x**13/3240 + O(x**17)
 227
 228 def test_issue125():
 229     y = Symbol("y")
 230     f=(1-y**(Rational(1)/2))**(Rational(1)/2)
 231     assert f.nseries(y,0,2) == 1 - sqrt(y)/2-y/8-y**Rational(3,2)/16+O(y**2)
 232
 233 def test_issue364():
 234     w = Symbol("w")
 235     x = Symbol("x")
 236     e = 1/x*(-log(w**(1 + 1/log(3)*log(5))) + log(w + w**(1/log(3)*log(5))))
 237     e_ser = -log(5)*log(w)/(x*log(3)) + w**(log(5)/log(3) - 1)/x - \
 238             w**(2*log(5)/log(3) - 2)/(2*x) + O(w)
 239     assert e.nseries(w, 0, 1) == e_ser
 240
 241 def test_sin():
 242     x = Symbol("x")
 243     y = Symbol("y")
 244     assert sin(8*x).nseries(x, 0, 4) == 8*x - 256*x**3/3 + O(x**4)
 245     assert sin(x+y).nseries(x, 0, 1) == sin(y) + O(x)
 246     assert sin(x+y).nseries(x, 0, 2) == sin(y) + cos(y)*x + O(x**2)
 247     assert sin(x+y).nseries(x, 0, 5) == sin(y) + cos(y)*x - sin(y)*x**2/2 - \
 248         cos(y)*x**3/6 + sin(y)*x**4/24 + O(x**5)
 249
 250 def test_issue416():
 251     x = Symbol("x")
 252     e = sin(8*x)/x
 253     assert e.nseries(x, 0, 6) == 8 - 256*x**2/3 + 4096*x**4/15 + O(x**5)
 254
 255 def test_issue406():
 256     x = Symbol("x")
 257     e = sin(x)**(-4)*(cos(x)**Rational(1,2)*sin(x)**2 - \
 258             cos(x)**Rational(1,3)*sin(x)**2)
 259     assert e.nseries(x, 0, 8) == -Rational(1)/12 - 7*x**2/288 - \
 260             43*x**4/10368 + O(x**5)
 261
 262 def test_issue402():
 263     x = Symbol("x")
 264     a = Symbol("a")
 265     e = x**(-2)*(x*sin(a + x) - x*sin(a))
 266     assert e.nseries(x, 0, 5) == cos(a) - sin(a)*x/2 - cos(a)*x**2/6 + \
 267             sin(a)*x**3/24 + O(x**4)
 268     e = x**(-2)*(x*cos(a + x) - x*cos(a))
 269     assert e.nseries(x, 0, 5) == -sin(a) - cos(a)*x/2 + sin(a)*x**2/6 + \
 270             cos(a)*x**3/24 + O(x**4)
 271
 272 def test_issue403():
 273     x = Symbol("x")
 274     e = sin(5*x)/sin(2*x)
 275     assert e.nseries(x, 0, 2) == Rational(5,2) + O(x)
 276     assert e.nseries(x, 0, 6) == Rational(5,2) - 35*x**2/4 + 329*x**4/48 + O(x**5)
 277
 278 def test_issue404():
 279     x = Symbol("x")
 280     e = sin(2 + x)/(2 + x)
 281     assert e.nseries(x, 0, 2) == sin(2)/2 + x*cos(2)/2 - x*sin(2)/4 + O(x**2)
 282
 283 def test_issue407():
 284     x = Symbol("x")
 285     e = (x + sin(3*x))**(-2)*(x*(x + sin(3*x)) - (x + sin(3*x))*sin(2*x))
 286     assert e.nseries(x, 0, 6) == -Rational(1,4) + 5*x**2/96 + 91*x**4/768 + O(x**5)
 287
 288 def test_issue409():
 289     x = Symbol("x", real=True)
 290     assert log(sin(x)).nseries(x, 0, 5) == log(x) - x**2/6 - x**4/180 + O(x**5)
 291     e = -log(x) + x*(-log(x) + log(sin(2*x))) + log(sin(2*x))
 292     assert e.nseries(x, 0, 5) == log(2)+log(2)*x-2*x**2/3-2*x**3/3-4*x**4/45+O(x**5)
 293
 294 def test_issue408():
 295     x = Symbol("x")
 296     e = x**(-4)*(x**2 - x**2*cos(x)**Rational(1,2))
 297     assert e.nseries(x, 0, 7) == Rational(1,4) + x**2/96 + 19*x**4/5760 + O(x**5)
 298
 299 def test_issue540():
 300     x = Symbol("x")
 301     assert sin(cos(x)).nseries(x, 0, 5) == sin(1) -x**2*cos(1)/2 - x**4*sin(1)/8 + x**4*cos(1)/24 + O(x**5)
 302
 303 def test_hyperbolic():
 304     x = Symbol("x")
 305     assert sinh(x).nseries(x, 0, 6) == x + x**3/6 + x**5/120 + O(x**6)
 306     assert cosh(x).nseries(x, 0, 5) == 1 + x**2/2 + x**4/24 + O(x**5)
 307     assert tanh(x).nseries(x, 0, 6) == x - x**3/3 + 2*x**5/15 + O(x**6)
 308     assert coth(x).nseries(x, 0, 6) == 1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
 309     assert asinh(x).nseries(x, 0, 6) == x - x**3/6 + 3*x**5/40 + O(x**6)
 310     assert acosh(x).nseries(x, 0, 6) == pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
 311     assert atanh(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + O(x**6)
 312     assert acoth(x).nseries(x, 0, 6) == x + x**3/3 + x**5/5 + pi*I/2 + O(x**6)
 313
 314 def test_series2():
 315     w = Symbol("w", real=True)
 316     x = Symbol("x", real=True)
 317     e =  w**(-2)*(w*exp(1/x - w) - w*exp(1/x))
 318     assert e.nseries(w, 0, 3) == -exp(1/x) + w * exp(1/x) / 2  + O(w**2)
 319
 320 def test_series3():
 321     w = Symbol("w", real=True)
 322     x = Symbol("x", real=True)
 323     e = w**(-6)*(w**3*tan(w) - w**3*sin(w))
 324     assert e.nseries(w, 0, 5) == Integer(1)/2 + O(w**2)
 325
 326 def test_bug4():
 327     w = Symbol("w")
 328     x = Symbol("x")
 329     e = x/(w**4 + x**2*w**4 + 2*x*w**4)*w**4
 330     assert e.nseries(w, 0, 2) in [x/(1 + 2*x + x**2), 1/(1+x/2+1/x/2)/2, 1/x/(1 + 2/x + x**(-2))]
 331
 332 def test_bug5():
 333     w = Symbol("w")
 334     x = Symbol("x")
 335     e = (-log(w) + log(1 + w*log(x)))**(-2)*w**(-2)*((-log(w) + log(1 + \
 336         x*w))*(-log(w) + log(1 + w*log(x)))*w - x*(-log(w) + log(1 + \
 337             w*log(x)))*w)
 338     assert e.nseries(w, 0, 1) == x/w/log(w) + 1/w + O(1/log(w))
 339     assert e.nseries(w, 0, 2) == x/w/log(w) + 1/w - x/log(w) + 1/log(w)*log(x)\
 340             + x*log(x)/log(w)**2 + O(w/log(w))
 341
 342
 343 def test_issue1016():
 344     x = Symbol("x")
 345     assert ( sin(x)/(1 - cos(x)) ).nseries(x, 0, 2) == O(1/x)
 346     assert ( sin(x)**2/(1 - cos(x)) ).nseries(x, 0, 2) == O(1, x)
 347
 348 def test_pole():
 349     x = Symbol("x")
 350     py.test.raises(PoleError, "sin(1/x).series(x, 0, 5)")
 351     py.test.raises(PoleError, "sin(1+1/x).series(x, 0, 5)")
 352     py.test.raises(PoleError, "(x*sin(1/x)).series(x, 0, 5)")
 353
 354 def test_expsinbug():
 355     x = Symbol("x")
 356     assert exp(sin(x)).series(x, 0, 0) == O(1, x)
 357     assert exp(sin(x)).series(x, 0, 1) == 1+O(x)
 358     assert exp(sin(x)).series(x, 0, 2) == 1+x+O(x**2)
 359     assert exp(sin(x)).series(x, 0, 3) == 1+x+x**2/2+O(x**3)
 360     assert exp(sin(x)).series(x, 0, 4) == 1+x+x**2/2+O(x**4)
 361     assert exp(sin(x)).series(x, 0, 5) == 1+x+x**2/2-x**4/8+O(x**5)
 362
 363 def test_floor():
 364     x = Symbol('x')
 365     assert floor(x).series(x) == 0
 366     assert floor(-x).series(x) == -1
 367     assert floor(sin(x)).series(x) == 0
 368     assert floor(sin(-x)).series(x) == -1
 369     assert floor(x**3).series(x) == 0
 370     assert floor(-x**3).series(x) == -1
 371     assert floor(cos(x)).series(x) == 0
 372     assert floor(cos(-x)).series(x) == 0
 373     assert floor(5+sin(x)).series(x) == 5
 374     assert floor(5+sin(-x)).series(x) == 4
 375
 376     assert floor(x).series(x, 2) == 2
 377     assert floor(-x).series(x, 2) == -3
 378
 379     x = Symbol('x', negative=True)
 380     assert floor(x+1.5).series(x) == 1
 381
 382 def test_ceiling():
 383     x = Symbol('x')
 384     assert ceiling(x).series(x) == 1
 385     assert ceiling(-x).series(x) == 0
 386     assert ceiling(sin(x)).series(x) == 1
 387     assert ceiling(sin(-x)).series(x) == 0
 388     assert ceiling(1-cos(x)).series(x) == 1
 389     assert ceiling(1-cos(-x)).series(x) == 1
 390     assert ceiling(x).series(x, 2) == 3
 391     assert ceiling(-x).series(x, 2) == -2
 392
 393 def test_abs():
 394     x = Symbol('x')
 395     assert abs(x).nseries(x, 0, 4) == x
 396     assert abs(-x).nseries(x, 0, 4) == x
 397     assert abs(x+1).nseries(x, 0, 4) == x+1
 398     assert abs(sin(x)).nseries(x, 0, 4) == x - Rational(1, 6)*x**3 + O(x**4)
 399     assert abs(sin(-x)).nseries(x, 0, 4) == x - Rational(1, 6)*x**3 + O(x**4)