What's the right way to make a change of variables under another integral?

Solution 1:

Subtracting one divergent integral from another is not, without additional information, going to yield a meaningful result (even if the singularities are in some sense "the same"). To take a silly example, suppose you're asked to calculate $$ \int_{0}^{1}\frac{dx}{x}-\int_{0}^{1}\frac{dt}{t}. $$ Obviously this should be zero. Right? But by making the change of variable $t=ax$ in the second integral, it is seen to also equal $$ \int_{0}^{1}\frac{dx}{x}-\int_{0}^{1/a}\frac{dx}{x}=\int_{1/a}^{1}\frac{dx}{x}=\ln x\Big\vert_{1/a}^{1}=\ln a, $$ where $a$ is completely arbitrary. Hopefully the relevance to your problem is clear: changes of variables can change the overall result if the individual integrals are improper. In order to nail down a particular result, you need to specify the regularization procedure. Here, you might make the individual integrals meaningful by calculating $$ \int_{\varepsilon}^{1}\frac{dx}{x}-\int_{\varepsilon}^{1}\frac{dt}{t} $$ and then letting $\varepsilon\rightarrow 0$. (The result would then be zero no matter what changes of variable were carried out midstream.) If you adopt the same procedure in your problem (i.e., restrict the first integral to $|t_\perp|>\varepsilon$ and the second to $|r'_{\perp}|>\varepsilon$, then make your change of variable and subtract the results, and finally let $\varepsilon$ go to $0$), you will get an additional contribution from the annular region between radii $\varepsilon$ and $\xi' \varepsilon$. Specifically, the new term is $$ \int_{\varepsilon}^{\xi'\varepsilon}\frac{dt_\perp}{2\pi t_\perp} J_0\left(\xi'k_\perp t_\perp\right). $$ In the limit as $\varepsilon\rightarrow 0$, the Bessel function is irrelevant (it just goes to $1$), and the rest integrates to $$ \frac{1}{2\pi}\ln \xi' = \frac{1}{4\pi}\ln\xi'^2, $$ which is the term you were missing.