Is it true that $Arg (\prod _{n=1 } ^N p _n )= \sum _{n=1 } ^N Arg (p _n )$? Is it true that $Arg (\prod _{n=1 } ^N p _n )= \sum _{n=1 } ^N  Arg (p _n )$?
I know that $Arg(z _1 +z _2)=Arg(z _1)+Arg(z _2) $ and I wonder how this result generalizes.
(I think I have seen this given as $Arg (\prod _{n=1 } ^N p _n )= \sum _{n=1 } ^N  Arg (p _n )+ 2k \pi$, but I cannot figure why I should need the last term $2k \pi $.) 
here $p _n $ i a complex number.
Thanks in advance!
 A: When $z$ is a nonzero complex number then ${\rm arg}\, z$ denotes the set of all $\phi\in{\mathbb R}$ satisfying $z=|z|\>e^{i\phi}$. This set is an equivalence class with respect to the relation
$$\phi\sim\phi'\quad:\Leftrightarrow\quad e^{i\phi}=e^{i\phi'}\ ,$$ and is a one-dimensional lattice on ${\mathbb R}$ with distance $2\pi$ between successive lattice points.
When the given number $z$ is not negative real then the set (equivalence class) ${\rm arg}\, z$ contains exactly one representant in the interval $\ ]{-\pi},\pi[\ $. This representant is called the principal value of the argument and is denoted by ${\rm Arg}\,z$.
The "class-valued" function ${\rm arg}$ satisfies the functional equation of the logarithm:
$${\rm arg}(z_1\>z_2)={\rm arg}\,z_1+{\rm arg}\, z_2\qquad(z_1,\> z_2\in\dot{\mathbb C})\ ,\tag{1}$$
where on the right hand side the (well defined) sum of classes is understood. The principal value ${\rm Arg}$ on the other hand does not satisfy $(1)$, as is easily seen in suitable examples. One may, however, write
$${\rm Arg}(z_1\>z_2)={\rm Arg}\,z_1+{\rm Arg}\, z_2\>+2k\pi\ ,$$
the intended interpretation being that the terms on the two sides of the equation differ by an integer multiple of $2\pi$. But even this can lead to difficulties, e.g., if $z_1=z_2=i$.
