Integrating and deriving absolute functions sympy

Question

Sympy is generally a great tool for calculating both the integral and derivative of a function. When the function happens to contain an absolute component though (|x|), for some reason it doesn't seem to be able to figure that out.

when for example you write something like this:

diff(abs(x+1))

you'll get the following output:

sign⁡(x+1)

The answer shoud be (x+1)/|x+1|, so is there something wrong with sympy and is there a way around it?

Answer 1

In SymPy sign(z) is defined as z/|z| for complex non-zero z. In fact another definition of sign(z) is precisely as the derivative of abs(z): https://en.wikipedia.org/wiki/Sign_function#Definition

It should ideally be possible to use rewrite(Abs) with sign but is not currently: https://github.com/sympy/sympy/issues/19277

Note that rewrite wouldn't work in your case without any way to know that x+1 is nonzero (e.g. if x is declared as positive`).

You can force the rewrite manually using replace though:

In [4]: s
Out[4]: sign(x + 1)

In [5]: s.replace(sign, lambda arg: arg/Abs(arg))
Out[5]: 
 x + 1 
───────
│x + 1│