In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. One traditional technique is to modify the integral by adding a ghost. Therefore, the above integral can be modified to

$$\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}-\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-\Lambda^{2})^{2}}\sim i \log\frac{m^2}{\Lambda^2}$$

This defines a curve in the complex plane $\Lambda^{2}\rightarrow i \log\frac{m^2}{\Lambda^2}$

So we don't associate a finite complex number to the integral, but we associate a curve to the integral. My question is: Do mathematicians study divergent integrals in this way? By generalizing the notion of integration, such that it is possible to be valued in the space of curves in the complex plane?