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Math Help - Complex Analysis of a Function (Arg)

  1. #1
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    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.)
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    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 f:A\subset \mathbb{R}\to \mathbb{R} defined by f(x)=\arg (1-e^{-i\pi x}) where \arg means the principal argument. Right?. Now, suppose A=[a,b] is a closed interval such that f(x)=\ldots=\arctan\dfrac {\sin \pi x}{1-\cos \pi x} is continuous . Are you asking about \int_a^bf(x)\;dx ?
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  3. #3
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    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!
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