A multiple-valued function can be considered as a collection of single-valued functions, each member of which is called a branch of the function. In general, we consider on particular member as a principal branch of the multiple-valued function and the value of the function corresponding to this branch as the principal branch.
I think the last word should be "value". The sentence goes as "the value of the function corresponding to this branch as the principal value".
Third paragraph
I think the last word should be "value". The sentence goes as "the value of the function corresponding to this branch as the principal value".