summation over poles of rational functions

[akritas]

summation( 1 / (k * (k + 1)), (k, -3, n) ) returns -(n + 4)/(3*(n + 1)) whereas it should return nothing as Wolframalpha and Maxima.

[gxyd]

I have modified the 'summation' method in the Concrete part. It seems to be working fine for me.. Can anyone tell me about it... ?

def summation(f, _symbols, *_kwargs): r""" Compute the summation of f with respect to symbols.

The notation for symbols is similar to the notation used in Integral.
summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
i.e.,

::

                                b
                              ____
                              \   `
    summation(f, (i, a, b)) =  )    f
                              /___,
                              i = a

If it cannot compute the sum, it returns an unevaluated Sum object.
Repeated sums can be computed by introducing additional symbols tuples::

>>> from sympy import summation, oo, symbols, log
>>> i, n, m = symbols('i n m', integer=True)

>>> summation(2*i - 1, (i, 1, n))
n**2
>>> summation(1/2**i, (i, 0, oo))
2
>>> summation(1/log(n)**n, (n, 2, oo))
Sum(log(n)**(-n), (n, 2, oo))
>>> summation(i, (i, 0, n), (n, 0, m))
m**3/6 + m**2/2 + m/3

>>> from sympy.abc import x
>>> from sympy import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
exp(x)

See Also
========

Sum
Product, product

"""

''' def L(expr):
e1 = simplify(expr)
denom = e1.as_numer_denom()[1]
r = solve(denom)
r_int = []
for i in r:
    if isinstance(i, Integer)==True:
        r_int.append(i)
return r_int'''

e1 = simplify(f)
denom = e1.as_numer_denom()[1]
r = solve(denom)
r_int = []
for i in r:
    if isinstance(i, Integer):
        r_int.append(i)

if r_int!=[]:
    if isinstance(symbols[0][1], int) and isinstance(symbols[0][2], Symbol):
        for val in r_int:
            if val>symbols[0][1]:
                return "Sum(%s, %s)" %(f, symbols[0])
        for val in r_int:
            if val==symbols[0][1]:
                return "Sum(%s, %s)" %(f, symbols[0])
    elif isinstance(symbols[0][1], Symbol) and isinstance(symbols[0][2], int):
        gr = 0
        lt = 0
        for val in r_int:
            if val>symbols[0][2]:
                gr=gr+1
            elif val<symbols[0][2]:
                lt=lt+1
            if gr!=0 and lt!=0:
                return "Sum(%s, %s)" %(f, symbols[0])
        if gr==0 or lt==0:
            for val in r_int:
                if val==symbols[0][2]:
                    return "Sum(%s, %s)" %(f, symbols[0])

return Sum(f, *symbols, **kwargs).doit(deep=False)

///

The tests are as:

print(summation(1/(k(k+3)), (k, -5, n))) Sum(1/(k(k + 3)), (k, -5, n))

print(summation(1/(k(k+3)), (k, -3, n))) Sum(1/(k(k + 3)), (k, -3, n))

print(summation(1/(k(k+3)), (k, -2, n))) Sum(1/(k(k + 3)), (k, -2, n))

print(summation(1/(k(k+3)), (k, 0, n))) Sum(1/(k(k + 3)), (k, 0, n))

print(summation(1/(k(k+3)), (k, 2, n))) (n - 1)(13_n__2 + 55n + 54)/(36(n + 1)_(n + 2)*(n + 3))

print(summation(1/(k(k+3)), (k, n, -5))) (n + 4)(13_n__2 + 23_n + 6)/(36n(n + 1)*(n + 2))

print(summation(1/(k(k+3)), (k, n, -2))) Sum(1/(k(k + 3)), (k, n, -2))

print(summation(1/(k(k+3)), (k, n, 0))) Sum(1/(k(k + 3)), (k, n, 0))

print(summation(1/(k(k+3)), (k, n, 2))) -(n - 3)(47_n__2 + 102_n + 40)/(180n(n + 1)*(n + 2))

[skirpichev]

@gxyd, please read this

[asmeurer]

@gxyd it would be easier to review the code if you open a pull request, even if the code is not ready to be merged yet.

[asmeurer]

Wouldn't a better answer be a Piecewise (c.f. what integrate() does in similar situations). In the example above, if n <= -2 then the summation does not go across any poles.

[gxyd]

@asmeurer the pull request is at #8839 . i think it will be difficult to represent the summation in that case. For ex. summation(1/(k*(k+1)), (k, 2, n)) . If n== pi, in this case the documentation says should include all the integer value from 2 to pi. for that 'greatest integer function should be used'.

[asmeurer]

I'm not sure what our policy is regarding summation where the bounds are noninteger, but note that summations can have reversed bounds (using the Karr convention, see the docstring of Sum).

[asmeurer]

@raoulb thoughts on this?

[gxyd]

@asmeurer According to the current definition of the summation in the docstring, the issue i am fixing does not impact the answer.