Coverage for /usr/lib/python3/dist-packages/sympy/integrals/singularityfunctions.py: 19%

1from sympy.functions import SingularityFunction, DiracDelta 

2from sympy.integrals import integrate 

3 

4 

5def singularityintegrate(f, x): 

6 """ 

7 This function handles the indefinite integrations of Singularity functions. 

8 The ``integrate`` function calls this function internally whenever an 

9 instance of SingularityFunction is passed as argument. 

10 

11 Explanation 

12 =========== 

13 

14 The idea for integration is the following: 

15 

16 - If we are dealing with a SingularityFunction expression, 

17 i.e. ``SingularityFunction(x, a, n)``, we just return 

18 ``SingularityFunction(x, a, n + 1)/(n + 1)`` if ``n >= 0`` and 

19 ``SingularityFunction(x, a, n + 1)`` if ``n < 0``. 

20 

21 - If the node is a multiplication or power node having a 

22 SingularityFunction term we rewrite the whole expression in terms of 

23 Heaviside and DiracDelta and then integrate the output. Lastly, we 

24 rewrite the output of integration back in terms of SingularityFunction. 

25 

26 - If none of the above case arises, we return None. 

27 

28 Examples 

29 ======== 

30 

31 >>> from sympy.integrals.singularityfunctions import singularityintegrate 

32 >>> from sympy import SingularityFunction, symbols, Function 

33 >>> x, a, n, y = symbols('x a n y') 

34 >>> f = Function('f') 

35 >>> singularityintegrate(SingularityFunction(x, a, 3), x) 

36 SingularityFunction(x, a, 4)/4 

37 >>> singularityintegrate(5*SingularityFunction(x, 5, -2), x) 

38 5*SingularityFunction(x, 5, -1) 

39 >>> singularityintegrate(6*SingularityFunction(x, 5, -1), x) 

40 6*SingularityFunction(x, 5, 0) 

41 >>> singularityintegrate(x*SingularityFunction(x, 0, -1), x) 

42 0 

43 >>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x) 

44 f(1)*SingularityFunction(x, 1, 0) 

45 

46 """ 

47 

48 if not f.has(SingularityFunction): 

49 return None 

50 

51 if isinstance(f, SingularityFunction): 

52 x, a, n = f.args 

53 if n.is_positive or n.is_zero: 

54 return SingularityFunction(x, a, n + 1)/(n + 1) 

55 elif n in (-1, -2): 

56 return SingularityFunction(x, a, n + 1) 

57 

58 if f.is_Mul or f.is_Pow: 

59 

60 expr = f.rewrite(DiracDelta) 

61 expr = integrate(expr, x) 

62 return expr.rewrite(SingularityFunction) 

63 return None