Bug in integration solver

[stopslavery404]

For calculating the integration of pdf of cauchy distribution I tried sympy, but it produced wrong result.

integrate(1/(pi*(1+(x-a)**2)),(x,-oo,oo))
the expected output was 1 but it produced output 0 which is wrong.

[oscarbenjamin]

If a is declared as real then the correct answer is given:

In [8]: a = Symbol('a', real=True)

In [9]: integrate(1/(pi*(1+(x-a)**2)),(x,-oo,oo))
Out[9]: 1

[stopslavery404]

Thank you so much :)

[oscarbenjamin]

Reopening because the answer should not be given as zero when a has not been declared as real

[oscarbenjamin]

Seems to be a bug in the limit code:

In [1]: a = Symbol('a')

In [2]: I = integrate(1/(pi*(1+(x-a)**2)),x)

In [3]: I
Out[3]: 
  ⅈ⋅log(-a + x - ⅈ)   ⅈ⋅log(-a + x + ⅈ)
- ───────────────── + ─────────────────
          2                   2        
───────────────────────────────────────
                   π                   

In [4]: I.limit(x, oo)
Out[4]: 0

In [5]: I.limit(x, -oo)   # <-- should be -1
Out[5]: 0

Seems to be also a bug for this limit (even though the integral works in this case):

In [9]: a = Symbol('a', real=True)

In [10]: I = integrate(1/(pi*(1+(x-a)**2)),x)

In [11]: I
Out[11]: 
-atan(a - x) 
─────────────
      π      

In [12]: I.limit(x, oo)  # <-- should be +1/2
Out[12]: -1/2

In [13]: I.limit(x, -oo)
Out[13]: -1/2

[oscarbenjamin]

In [12]: I.limit(x, oo)  # <-- should be +1/2
Out[12]: -1/2

Actually this is a regression on master since sympy 1.8. Bisected to 2b6b233 from #21589
CC @0sidharth @jksuom

[jksuom]

In [4]: I.limit(x, oo)
Out[4]: 0

In [5]: I.limit(x, -oo)   # <-- should be -1
Out[5]: 0

I get these results also on sympy-1.8. It looks like that would be correct if im(a) > 1 so that the arguments of the two log terms are on the same side of the real axis (on the same branch).

[oscarbenjamin]

I get these results also on sympy-1.8.

I wonder if it depends on the Python version because I do not get the same in sympy 1.8 with Python 3.6 on Linux...

EDIT: I see that actually you were referring to the example without real=True.

[0sidharth]

The limit regression has been fixed in #21731, although the answer is still returned as 0 when a is not declared real.

[oscarbenjamin]

Some of the problems mentioned above are fixed but some remain. The part that was identified as a regression has been fixed:

In [28]: a = Symbol('a', real=True)

In [29]: I = integrate(1/(pi*(1+(x-a)**2)),x)

In [30]: I.limit(x, oo)  # <-- should be +1/2
Out[30]: 1/2

I'm removing the 1.9 milestone.