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I'm reading an article where an integral of the form:

[tex] \int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\infty}_{-\infty}\,\mathrm{d}\tau''...[/tex]

The author then splits this into the region whereby [tex]\tau' >\tau'' [/tex], and the region [tex] \tau''>\tau' [/tex].

[tex] \int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\infty}_{\tau'}\,\mathrm{d}\tau''...+\int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\tau'}_{-\infty}\,\mathrm{d}\tau''...[/tex]

For the first integral the change of variables [tex] u=\tau'',s=\tau''-\tau' [/tex] is made, (and for the second part the change [tex] u=\tau',s=\tau'-\tau'' [/tex])

Focusing on just say the first integral for clarity, is this change of variables really well defined? I mean we could write [tex] \tau' =u-s, \tau''=u [/tex] and then we arrive at [tex] \mathrm{d}\tau'=du-ds, \mathrm{d}\tau''=du [/tex]. How does one uniquely change the s integral measure? My instincts say I should just use [tex] \mathrm{d}\tau'' \rightarrow du, \mathrm{d}\tau'\rightarrow -ds [/tex] but I don't really know how to justify this.

Secondly the limits, starting with the [tex] \tau'' [/tex] integral, [tex] \tau'' \in [\tau', +\infty]\rightarrow u \in [u-s ,+\infty] [/tex]....but the variable u can't be in it's own lower limit??

I can see by eye, if you like, that the max and min values of [tex] \tau' [/tex] are going to be [tex] u \in [-\infty,+\infty], s \in [0,\infty] [/tex], and indeed this is what the author has written, but how to show this analytically?

I tried to use Maple, just to see if it could do it using the change of variables commands, but it said it was "unable to solve the change of variable equations"