Coverage for /usr/lib/python3/dist-packages/sympy/series/residues.py: 21%

1""" 

2This module implements the Residue function and related tools for working 

3with residues. 

4""" 

5 

6from sympy.core.mul import Mul 

7from sympy.core.singleton import S 

8from sympy.core.sympify import sympify 

9from sympy.utilities.timeutils import timethis 

10 

11 

12@timethis('residue') 

13def residue(expr, x, x0): 

14 """ 

15 Finds the residue of ``expr`` at the point x=x0. 

16 

17 The residue is defined as the coefficient of ``1/(x-x0)`` in the power series 

18 expansion about ``x=x0``. 

19 

20 Examples 

21 ======== 

22 

23 >>> from sympy import Symbol, residue, sin 

24 >>> x = Symbol("x") 

25 >>> residue(1/x, x, 0) 

26 1 

27 >>> residue(1/x**2, x, 0) 

28 0 

29 >>> residue(2/sin(x), x, 0) 

30 2 

31 

32 This function is essential for the Residue Theorem [1]. 

33 

34 References 

35 ========== 

36 

37 .. [1] https://en.wikipedia.org/wiki/Residue_theorem 

38 """ 

39 # The current implementation uses series expansion to 

40 # calculate it. A more general implementation is explained in 

41 # the section 5.6 of the Bronstein's book {M. Bronstein: 

42 # Symbolic Integration I, Springer Verlag (2005)}. For purely 

43 # rational functions, the algorithm is much easier. See 

44 # sections 2.4, 2.5, and 2.7 (this section actually gives an 

45 # algorithm for computing any Laurent series coefficient for 

46 # a rational function). The theory in section 2.4 will help to 

47 # understand why the resultant works in the general algorithm. 

48 # For the definition of a resultant, see section 1.4 (and any 

49 # previous sections for more review). 

50 

51 from sympy.series.order import Order 

52 from sympy.simplify.radsimp import collect 

53 expr = sympify(expr) 

54 if x0 != 0: 

55 expr = expr.subs(x, x + x0) 

56 for n in (0, 1, 2, 4, 8, 16, 32): 

57 s = expr.nseries(x, n=n) 

58 if not s.has(Order) or s.getn() >= 0: 

59 break 

60 s = collect(s.removeO(), x) 

61 if s.is_Add: 

62 args = s.args 

63 else: 

64 args = [s] 

65 res = S.Zero 

66 for arg in args: 

67 c, m = arg.as_coeff_mul(x) 

68 m = Mul(*m) 

69 if not (m in (S.One, x) or (m.is_Pow and m.exp.is_Integer)): 

70 raise NotImplementedError('term of unexpected form: %s' % m) 

71 if m == 1/x: 

72 res += c 

73 return res