How Does Integral Rewriting Transform Two Integrals into One?

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Homework Help Overview

The discussion revolves around the transformation of a double integral into a single integral through variable substitution, specifically involving the variables τ and μ. Participants are exploring the implications of this rewriting in the context of integral calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to understand the mechanics of the integral transformation and questioning the validity of certain assumptions, such as the role of beta and the presence of a Green's function in the integrand. Some are suggesting specific substitutions to clarify the transformation process.

Discussion Status

The discussion is active, with participants raising questions about the steps involved in the transformation and expressing uncertainty about the outcomes of their proposed substitutions. There is no clear consensus, but various interpretations and approaches are being explored.

Contextual Notes

Some participants express concern over potential typos in the original formulation, indicating that the clarity of the problem setup may be a factor in their understanding.

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On the attached picture the double integral in the first line is rewritten in the second line by introducting the variable τ=τ1-τ2
But how exactly does this happen? I simply can't see how two integrals can turn into one.
 

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The first integral is just beta. It eventually cancels out.
 
I don't understand why it is just beta. There is a Green's function in the integrand. Do you say this because there is no beta in the second line? Because for all I know this could be a typo.
 
Can you do the full substitution where you introduce new variables, say
[tex]\tau = \tau_1 - \tau_2 \text{ and } \mu = \tau_1 + \tau_2[/tex]
to turn a two dimensional integral into a two dimensional integral? You will see the integration over [itex]\mu[/itex] become trivial.
 
hmm I'd would say it would give something like the attached, but I don't see any trivial integral. What did I do wrong?
 

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Last edited:

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