Trac #22209: Differentiation of conj/imag/real/abs functions

At the moment the result from differentiation of the functions `conj`,
`real`, `imag`,  `abs` does not take the argument domain into account.
{{{
sage: _ = var('x', domain='real')
sage: _ = var('z')

sage: (x^z).conjugate().diff(x)
x^(z - 1)*z*D[0](conjugate)(x^z)
sage: atan(x).imag_part().diff(x)
D[0](imag_part)(arctan(x))/(x^2 + 1)
sage: abs(log(z)).diff(z)
log(z)/(z*abs(log(z)))
}}}

GiNaC has functionality that is only now included in Pynac. The ticket
should doctest it when it is implemented. It all benefits from recent
domain logic improvements in Pynac as well.

pynac/pynac#45

URL: https://trac.sagemath.org/22209
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/calculus/wester.py

@@ -509,6 +509,8 @@
     sage: #    Verify(D(x) Abs(x), Sign(x));
     sage: diff(abs(x))
     1/2*(x + conjugate(x))/abs(x)
+    sage: _.simplify_full()
+    x/abs(x)
     sage: _ = var('x', domain='real')
     sage: diff(abs(x))
     x/abs(x)

src/sage/symbolic/expression.pyx

@@ -3822,7 +3822,28 @@ cdef class Expression(CommutativeRingElement):
             sage: f.derivative(2)
             x |--> 6*x - sin(x)
 
-        ::
+        Some expressions can't be cleanly differentiated by the
+        chain rule::
+
+            sage: _ = var('x', domain='real')
+            sage: _ = var('w z')
+            sage: (x^z).conjugate().diff(x)
+            conjugate(x^(z - 1))*conjugate(z)
+            sage: (w^z).conjugate().diff(w)
+            w^(z - 1)*z*D[0](conjugate)(w^z)
+            sage: atanh(x).real_part().diff(x)
+            -1/(x^2 - 1)
+            sage: atanh(x).imag_part().diff(x)
+            0
+            sage: atanh(w).real_part().diff(w)
+            -D[0](real_part)(arctanh(w))/(w^2 - 1)
+            sage: atanh(w).imag_part().diff(w)
+            -D[0](imag_part)(arctanh(w))/(w^2 - 1)
+            sage: abs(log(x)).diff(x)
+            1/2*(conjugate(log(x))/x + log(x)/x)/abs(log(x))
+            sage: abs(log(z)).diff(z)
+            1/2*(conjugate(log(z))/z + log(z)/conjugate(z))/abs(log(z))
+            sage: forget()
 
             sage: t = sin(x+y^2)*tan(x*y)
             sage: t.derivative(x)