Connection between the Beta Function and Residue Theorem? When we define the bare coupling in Minimal Subtraction we write it as a Laurent series where the analytic part is identified with the finite, renormalized coupling and the nonanalytic part is precisely the poles which cancel those in the physical amplitudes (from Dimensional Regularization).
It can then be shown that the corresponding Beta Function depends only on the coefficient of the simple pole (residue) in the bare coupling (see for instance Weinberg Vol. II pg. 150, Ramond 2nd ed. pg. 138, Srednicki pg. 171). Weinberg and Srednicki do not use the language "Laurent series" for the bare coupling or "residue" for the coefficient of the simple pole, whereas Ramond uses both but does not mention any connection with the residue theorem.
Motivated by this, is there any deeper way to think of the Beta Function as related to a contour integral of the bare coupling in the complex $d$-plane (picking up the pole at $d$=4 and evaluating with the residue theorem)?
 A: TL;DR: The fact that the beta function
$$\beta(g)~=~ \lim_{\epsilon\to 0}\frac{\partial g}{\partial \ln \mu} ~=~ g^2c_{-1}^{\prime}(g)\tag{1}$$
only depends on the residue (=the coefficient $c_{-1}(g)$ of the simple pole in the Laurent series for $\epsilon$) seems to be an interplay of upper and lower truncated real Laurent series rather than complex function theory and the residue theorem.
Sketched proof for a prototype theory: Specifically, the following 3 facts are used:

*

*In the ${\rm MS}$ & $\overline{\rm MS}$ schemes, the  formal Laurent series for $Z-1$, or equivalently $$\ln Z~=~\sum_{n=1}^{\infty} \frac{c_{-n}(g)}{\epsilon^n},\tag{2}$$
contains only singular terms in $\epsilon$.


*The bare coupling constant
$$g_0~=~Zg \tilde{\mu}^{\epsilon}\quad\Leftrightarrow\quad\ln g_0~=~\ln Z +\ln g +\epsilon(\ln\mu+{\rm const}) \tag{3}$$
does not depend on the renormalization scale $\tilde{\mu} \sim \mu$:
$$ 0~=~\frac{\partial \ln g_0}{\partial \ln \mu}
~\stackrel{(3)}{=}~\frac{\partial \ln g}{\partial \ln \mu}\left(1+\frac{\partial \ln Z}{\partial \ln g} \right) +\epsilon. \tag{4}$$


*The physical/renormalized coupling constant $g$ should not contain singular terms in $\epsilon$:
$$\sum_{n=0}^{\infty} c_n(g)\epsilon^n~=~
\frac{\partial \ln g}{\partial \ln \mu}
~\stackrel{(4)}{=}~
-\frac{\epsilon}{1+\frac{\partial \ln Z}{\partial \ln g} }
~\stackrel{(2)}{=}~-\epsilon + \frac{\partial c_{-1}(g)}{\partial \ln g}+{\cal O}(\epsilon^{-1}).\tag{5}$$
In particular, the zeroth-order coefficient ${\cal O}(\epsilon^0)$ in eq. (5) reads
$$ c_0(g)~=~g^{-1}\beta(g)~=~ gc_{-1}^{\prime}(g),\tag{6} $$
which leads to eq. (1). $\Box$
