complex - ComplexExpand and Arg function

Why does the command ComplexExpand[Arg[a]] gives Arg[a] instead of 0? Isn't it that ComplexExpand supposed to assume all unspecified variable to be real, and the argument of real variable is 0?

Answer

The function ComplexExpand itself does not know what is a, complex, real and so on. One needs to instruct Mma about it. For example,

    Simplify[ComplexExpand[Arg[a]], a > 0]

(*  0  *)

Here is another example:

    Simplify[ComplexExpand[Arg[a]], a < 0]

(*  \[Pi]  *)

On the other hand, if one does not fix the sign of a, but only that it is Real, Mma assumes that it can have the both signs and leaves the both possibilities:

Simplify[ComplexExpand[Arg[a]], a \[Element] Reals]

(*  Arg[a]  *)

Even more this is valid, if a may be complex:

Simplify[ComplexExpand[Arg[a]]]

(*  Arg[a] *)

Have fun!