I've been messing around with a problem and converted it from an arc-cotangent function, to an arctangent function, to a complex logarithm, to an inverse hyperbolic tangent function, and finally argument form.

The function is:-

arg(1-e^(-i pi x)) : This is not the whole function, but its a section of it.

How is the above function calculated? I've seen references to it, but no actual explanations. (Note: The conversion from its previous form to the current form was achieved with the help of software I just purchased, mathematica! Too bad it cannot explain WHY ;(

When it comes to integration, what do I do? What is the definition for the integration of the arg(z) function, where z = 1-e^(-i pi x)?

Thanks for your help!

(Note: I put this question in the Analysis section because the problem is considered a Complex Analysis topic.)