integrate can return a meijerg object that evaluates to nan

[ethankward]

>>> from sympy import *
>>> x =  Symbol('x')
>>> r = integrate(log(1 + x**2) / (1 + x**2), [x, 0, oo])
>>> r
meijerg(((1/2, 1, 1), ()), ((1/2, 1), (0,)), 1)/2
>>> N(r)
nan

The meijerg result might be correct and just be evaluating incorrectly (I don't know), but the actual value of the integral is pi log(2).

[asmeurer]

It seems the value is correct. There are a lot of issues here.

If you replace 1 with z, you get

>>> pprint(hyperexpand(meijerg(((S(1)/2, 1, 1), ()), ((S(1)/2, 1), (0,)), z)/2))
     ⎛⎧            ⅈ⋅π             ⎞
     ⎜⎪acoth(√z) + ───             ⎟     ⎛⎧log(z - 1) - ⅈ⋅π  for │z│ > 1⎞
     ⎜⎪             2              ⎟   π⋅⎜⎨                             ⎟
     ⎜⎪───────────────  for │z│ > 1⎟     ⎝⎩  log(-z + 1)      otherwise ⎠
π⋅√z⋅⎜⎨       √z                   ⎟ + ──────────────────────────────────
     ⎜⎪                            ⎟                   2
     ⎜⎪   atanh(√z)                ⎟
     ⎜⎪   ─────────      otherwise ⎟
     ⎝⎩       √z                   ⎠

Now limit can't handle this directly (issue 1):

>>> limit(hyperexpand(meijerg(((S(1)/2, 1, 1), ()), ((S(1)/2, 1), (0,)), z)/2), z, 1)
Traceback (most recent call last):
  File "./sympy/series/limits.py", line 209, in doit
    r = gruntz(e, z, z0, dir)
  File "./sympy/series/gruntz.py", line 658, in gruntz
    r = limitinf(e0, z)
  File "./sympy/series/gruntz.py", line 428, in limitinf
    c0, e0 = mrv_leadterm(e, x)
  File "./sympy/series/gruntz.py", line 513, in mrv_leadterm
    series = calculate_series(f, w, logx=logw)
  File "./sympy/series/gruntz.py", line 466, in calculate_series
    for t in e.lseries(x, logx=logx):
  File "./sympy/core/expr.py", line 2696, in yield_lseries
    for si in s:
  File "./sympy/core/expr.py", line 2761, in _eval_lseries
    series = self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/add.py", line 386, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/add.py", line 386, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/mul.py", line 1606, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/mul.py", line 1606, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/add.py", line 386, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/add.py", line 386, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/function.py", line 669, in _eval_nseries
    raise PoleError("Cannot expand %s around 0" % (self))
sympy.core.function.PoleError: Cannot expand acoth(sqrt(_w + 1)) around 0

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "./sympy/series/limits.py", line 209, in doit
    r = gruntz(e, z, z0, dir)
  File "./sympy/series/gruntz.py", line 658, in gruntz
    r = limitinf(e0, z)
  File "./sympy/series/gruntz.py", line 428, in limitinf
    c0, e0 = mrv_leadterm(e, x)
  File "./sympy/series/gruntz.py", line 513, in mrv_leadterm
    series = calculate_series(f, w, logx=logw)
  File "./sympy/series/gruntz.py", line 466, in calculate_series
    for t in e.lseries(x, logx=logx):
  File "./sympy/core/expr.py", line 2696, in yield_lseries
    for si in s:
  File "./sympy/core/expr.py", line 2761, in _eval_lseries
    series = self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/mul.py", line 1606, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/mul.py", line 1606, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/add.py", line 386, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/add.py", line 386, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/function.py", line 669, in _eval_nseries
    raise PoleError("Cannot expand %s around 0" % (self))
sympy.core.function.PoleError: Cannot expand acoth(sqrt(_w + 1)) around 0

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "./sympy/series/limits.py", line 209, in doit
    r = gruntz(e, z, z0, dir)
  File "./sympy/series/gruntz.py", line 658, in gruntz
    r = limitinf(e0, z)
  File "./sympy/series/gruntz.py", line 428, in limitinf
    c0, e0 = mrv_leadterm(e, x)
  File "./sympy/series/gruntz.py", line 513, in mrv_leadterm
    series = calculate_series(f, w, logx=logw)
  File "./sympy/series/gruntz.py", line 466, in calculate_series
    for t in e.lseries(x, logx=logx):
  File "./sympy/core/expr.py", line 2696, in yield_lseries
    for si in s:
  File "./sympy/core/expr.py", line 2761, in _eval_lseries
    series = self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/mul.py", line 1606, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/mul.py", line 1606, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/add.py", line 386, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/add.py", line 386, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "./sympy/core/expr.py", line 2848, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "./sympy/core/function.py", line 669, in _eval_nseries
    raise PoleError("Cannot expand %s around 0" % (self))
sympy.core.function.PoleError: Cannot expand acoth(sqrt(_w + 1)) around 0

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "./sympy/series/limits.py", line 209, in doit
    r = gruntz(e, z, z0, dir)
  File "./sympy/series/gruntz.py", line 658, in gruntz
    r = limitinf(e0, z)
  File "./sympy/series/gruntz.py", line 428, in limitinf
    c0, e0 = mrv_leadterm(e, x)
  File "./sympy/series/gruntz.py", line 487, in mrv_leadterm
    Omega, exps = mrv(e, x)
  File "./sympy/series/gruntz.py", line 309, in mrv
    "Don't know how to calculate the mrv of '%s'" % e)
NotImplementedError: Don't know how to calculate the mrv of '((acoth(sqrt(1 + 1/_p)) + I*pi/2)/sqrt(1 + 1/_p), True)'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "./sympy/series/limits.py", line 68, in limit
    return Limit(e, z, z0, dir).doit(deep=False)
  File "./sympy/series/limits.py", line 213, in doit
    r = heuristics(e, z, z0, dir)
  File "./sympy/series/limits.py", line 80, in heuristics
    l = limit(a, z, z0, dir)
  File "./sympy/series/limits.py", line 68, in limit
    return Limit(e, z, z0, dir).doit(deep=False)
  File "./sympy/series/limits.py", line 213, in doit
    r = heuristics(e, z, z0, dir)
  File "./sympy/series/limits.py", line 80, in heuristics
    l = limit(a, z, z0, dir)
  File "./sympy/series/limits.py", line 68, in limit
    return Limit(e, z, z0, dir).doit(deep=False)
  File "./sympy/series/limits.py", line 213, in doit
    r = heuristics(e, z, z0, dir)
  File "./sympy/series/limits.py", line 80, in heuristics
    l = limit(a, z, z0, dir)
  File "./sympy/series/limits.py", line 68, in limit
    return Limit(e, z, z0, dir).doit(deep=False)
  File "./sympy/series/limits.py", line 224, in doit
    raise NotImplementedError()
NotImplementedError

I'm guessing limits of Piecewise not working is already a known issue.

If you extract the z = 1 part, you get

>>> pprint(pi*atanh(sqrt(z)) + pi*log(1 - z)/2)
π⋅log(-z + 1)
───────────── + π⋅atanh(√z)
      2

But that still doesn't work with limit (issue 2):

>>> limit(pi*atanh(sqrt(z)) + pi*log(1 - z)/2, z, 1)
Limit(pi*log(-z + 1)/2 + pi*atanh(sqrt(z)), z, 1)

If you rewrite the atanh as log, though, it works, and gives the answer you said:

>>> limit(pi*atanh(sqrt(z)).rewrite(log) + pi*log(1 - z)/2, z, 1)
pi*log(2)

So there are a couple of issues with limit. And there are also two issues with meijerg:

• hyperexpand doesn't work on the expression unless it is generalized (1 -> z)
• evalf gives the wrong answer (nan)

[asmeurer]

Also want to say that in an ideal world even if SymPy can't compute the limit, hyperexpand would return Limit(pi*log(-z + 1)/2 + pi*atanh(sqrt(z)), z, 1).

[skirpichev]

evalf issue sounds like mpmath's bug

[skirpichev]

mpmath/mpmath#378