Hi,
With Sage 6.3, I am getting:
sage: abs(x).diff(x)
x/abs(x)
sage: abs(I*x).diff(x)
-x/abs(I*x)
But abs(I*x) == abs(x). So also abs(x).diff(x) and abs(I*x).diff(x)
must be the same. But in the first case we get x/abs(x), and in the
second we got -x/abs(x).
In SymPy, the answer is:
In [1]: abs(x).diff(x)
Out[1]:
d d
re(x)⋅──(re(x)) + im(x)⋅──(im(x))
dx dx
─────────────────────────────────
│x│
In [2]: x = Symbol("x", real=True)
In [3]: abs(x).diff(x)
Out[3]: sign(x)
In [4]: abs(I*x).diff(x)
Out[4]: sign(x)
In [26]: var("x")
Out[26]: x
Which seems all correct --- in the complex case [1] we get a little
messy expression, but a correct one. For a real case, we get the
correct answer.
In Wolfram Alpha, the answer of abs(I*x).diff(x) is x/abs(x):
http://www.wolframalpha.com/input/?i=Diff%5BAbs%5Bi*x%5D%2C+x%5D
Which is only correct for real "x", but at least it is correct for
this special case.
The Sage result seems wrong for any "x".
Ondrej
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