Problems with identity in complex calc

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SUMMARY

The discussion centers on the evaluation of the integral \(\int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}\), where the user is struggling to derive the correct sign between the integrals \(I_+\) and \(I_-\). The user consistently obtains a positive sign instead of the expected negative sign. The integrals are defined as \(I_+ = \int_1^{\infty}(\ldots)\) and \(I_- = \int_{-\infty}^{-1}(\ldots)\), with the integrand specified as \(\sqrt{x^2-1}\). The discussion highlights the importance of clarifying the integrand to resolve the sign discrepancy.

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  • Knowledge of the behavior of functions involving imaginary components, such as \(\epsilon\).
  • Experience with the manipulation of square root functions in integrals.
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  • Review the properties of contour integration in complex analysis.
  • Study the implications of the small parameter \(\epsilon\) in integrals involving complex variables.
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betel
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Hello,

in a paper I have the identity

[tex]\int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}= I_+ - I_- + i \int_{-1}^{1}(\ldots)[/tex]


where [tex]I_+ = \int_1^{\infty}(\ldots), I_=\int_{-\infty}^{-1}(\ldots)[/tex] and [tex]\epsilon[/tex] is a small positive number that will be taken to zero at the end.

My Problem is to get the minus sign between [tex]I_+[/tex] and [tex]I_-[/tex]. In all my calculations I get +. The integrand for both should be [tex]\sqrt{x^2-1}[/tex].

Can anybody tell me what I am missing here to get the correct sign.
Thanks in advance.
betel
 
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I think it's essential what you (or the author) means by "(...)" for the integrand.
 
In further calculations the author uses [tex]\sqrt{x^2-1}[/tex]. The integrand is definitely real.
 

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