For just a bit of background... I'm familiar with how to compute real integrals using complex integrals. But I'm a bit concerned with the logical legitimacy of it.
For example, to compute the real integral (real as in the Riemann integral defined over the set $\mathbb{R}$)
$$\int_{-\infty}^\infty {\frac{1}{(x^2 + 1)^2} } dx$$ We eventually end up computing instead the complex integral $$\int_{-\infty}^\infty {\frac{1}{(z^2 + 1)^2} }dz $$ along the "real line" embedded in $\mathbb{C}$.
This may sound a bit nitpicky... what exactly are the steps involved in going from saying that the complex integral evaluates to $\pi/2$ and that the real integral evaluates to $\pi/2$?
I know model theory deals with the embedding of structures within other structures. But it too seems restricted in saying only stuff regarding particular models which we consider to "be" $\mathbb{R}$ and $\mathbb{C}$.
[Posted as an answer at the OP's request]
Since the reals are a subfield of the complex numbers, any computation with real numbers gives the same result as the same computation with those numbers considered as complex numbers. Since the absolute value of a real number is its absolute value as a complex number, a limit of real numbers is the same whether you consider these as real or complex numbers. An integral is expressed in terms of limits of expressions...