Corless & Davenport provide a whole bestiarium of rules. This is a small part of the most simple cases, which I sampled more or less for 'all day use' as reference. They are based on the 'unwinding number' (which is a sheet number of according Riemannian surfaces). It turns out, that Maple can 'proof' the identities, if one does not use the definition, but uses the version given in the help pages (= version 2 in the following).

  restart; interface(version);
    Classic Worksheet Interface, Maple 12.02, Windows, Dec 10 2008 Build ID 377066

  # intended to be used for Reals
  p:= y -> PIECEWISE([-1, y < 0],[1, 0 <= y]);

                               { -1        y <...

Peter Stone's Lectures about Math & using Maple: I always liked them, missed that
for a longer time (ok, had them filed to my disk) and now find them back on the web.

Does Maple 13 still have strange behaviours with empty sums like M12?

  interface(version);
  Classic Worksheet Interface, Maple 12.02, Windows, Dec 10 2008 Build ID 377066

  #Sum(t^j*(z-a)^(-2*n+j)/j!,j = 0 .. -1+n); eval(%, n=1);
  #S:=Sum(t^j*(z-a)^(-2+j)/j!,j = 0 .. 0);
  S:=Sum(1,j = 0 .. 0);
  value(S);
  #subs(Sum = add, S); value(%);
  #eval(S, Sum = add);
  #eval(S, Sum = sum);

r := abs(z)^(Re(a)) * exp(-Im(a) * argument(z));
w:= r * abs(z)^(Im(a)*I) * (z/abs(z))^Re(a);

I want to see, that z^a = w.

But simplify(w) gives a wrong result, it differs from w:

tstData:= [z=1+3*I, a=-3+I];
z^a; eval(%, tstData):  evalf(%);
'w'; eval(%, tstData): evalf(%);
'simplify(w)'; eval(%, tstData): evalf(%);

tstData:= [z=-2*I, a=+I];
z^a; eval(%, tstData):  evalf(%);
'w'; eval(%, tstData): evalf(%);
'simplify(w)'; eval(%, tstData): evalf(%);

In the last case simplify(w) results in a purely real value,
while w has a nonvanishing imaginary part.