Integral Help: \int arg(x^{ix} + x^{-ix}) dx

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Homework Help Overview

The discussion revolves around the integral of the argument of a complex expression, specifically \(\int arg(x^{ix} + x^{-ix}) dx\). The subject area includes complex analysis and integration techniques.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of the expression when \(x\) is real, questioning whether the argument can be simplified or if it remains complex. There are inquiries about the conditions under which the argument is defined and the implications of the integral.

Discussion Status

The discussion is ongoing, with participants raising questions about the assumptions made regarding the nature of \(x\) and the expression itself. Some guidance is offered regarding the complexities of defining the integral in the context of complex numbers, but no consensus has been reached.

Contextual Notes

There is mention of the potential complexity in defining the integral, particularly regarding the contour of integration and the nature of the complex powers involved. Additionally, one participant notes that this is not a homework problem, which may influence the approach taken in the discussion.

epkid08
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I really have no idea how to do this, please show some steps.

[tex]\int arg(x^{ix} + x^{-ix}) dx[/tex]
 
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If x is real the exp(ix)+exp(-ix) is real. Why? What is arg of a real number? Are you sure that's the whole problem?
 
Dick said:
If x is real the exp(ix)+exp(-ix) is real. Why? What is arg of a real number? Are you sure that's the whole problem?

I'm pretty sure that when x is real, the inside does not necessarily equal a real number. I'm looking at the graph right now and y = arg(...) is not zero, and also this isn't homework.
 
Hmm. Ok, what is x? Do they give you a contour to integrate along? It's pretty tricky to define the general complex power of a complex number.
 

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