Updates:
Summary: rewrite sign(x) (and possibly abs()) using Piecewise
Comment #6 on issue 1113 by asmeurer: rewrite sign(x) (and possibly abs())
using Piecewise
http://code.google.com/p/sympy/issues/detail?id=1113
Actually, you teach integrate how to integrate new functions by showing it
the derivative, not the integral,
ironically enough.
So if you do this:
diff --git a/sympy/functions/elementary/complexes.py
b/sympy/functions/elementary/complexes.py
index 6ff99e3..c942749 100644
--- a/sympy/functions/elementary/complexes.py
+++ b/sympy/functions/elementary/complexes.py
@@ -185,6 +185,9 @@ def eval(cls, arg):
is_bounded = True
+ def _eval_derivative(self, x):
+ return S.Zero
+
def _eval_conjugate(self):
return self
You get this:
In [2]: integrate(sign(x), x)
Out[2]: x⋅sign(x)
which is the same as abs(x).
Of course, the real fix here is to replace sign() with a Piecewise, which
already handles it:
In [14]: s = Piecewise((1, x>0), (0, Eq(x, 0)), (-1, x<0))
In [15]: s
Out[15]:
⎧1 for 0 < x
⎪
⎨0 for x = 0
⎪
⎩-1 for x < 0
In [16]: t = integrate(s, x)
In [17]: t # Look familiar? This is the same as abs(x)
Out[17]:
⎧x for 0 < x
⎪
⎨0 for x = 0
⎪
⎩-x for x < 0
In [18]: integrate(s, (x, -1, 1)) # OOPS! This is wrong!
Out[18]: 1
In [1]: integrate(sign(x), (x, -1, 1)) # The fixed sign() gets it right.
Out[1]: 0
In [12]: t.subs(x, 1) # and so does subs, oddly...
Out[12]: 1
In [13]: t.subs(x, -1)
Out[13]: 1
I will create another issue for the Piecewise._eval_interval problem
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