Complex Analysis of a Function (Arg)

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.)

Re: Complex Analysis of a Function (Arg)

Let's see, I don't understand the exact meaning of your question. It seems we have the function $\displaystyle f:A\subset \mathbb{R}\to \mathbb{R}$ defined by $\displaystyle f(x)=\arg (1-e^{-i\pi x})$ where $\displaystyle \arg$ means the principal argument. Right?. Now, suppose $\displaystyle A=[a,b]$ is a closed interval such that $\displaystyle f(x)=\ldots=\arctan\dfrac {\sin \pi x}{1-\cos \pi x}$ is continuous . Are you asking about $\displaystyle \int_a^bf(x)\;dx$ ?

Re: Complex Analysis of a Function (Arg)

Ah! While trying to respond I realized that I had written down the wrong function! Answered my own question, thanks!