  Analytic continuations of integer-valued parameters can lead to profound
insights, such as angular momentum in Regge theory, the number of replicas in
spin glasses, the number of internal degrees of freedom, the spacetime
dimension in dimensional regularization and Wilson's renormalization group. In
this work, we consider a new kind of analytic continuation of correlation
functions, inspired by two recent approaches to underdetermined Dyson-Schwinger
equations in $D$-dimensional spacetime. If the Green's functions
$G_n=\langle\phi^n\rangle$ admit analytic continuation to complex values of
$n$, the two different approaches are unified by a novel principle for
self-consistent problems: Singularities in the complex plane should be minimal.
This principle manifests as the merging of different branches of Green's
functions in the quartic theories. For $D=0$, we obtain the closed-form
solutions of the general $g\phi^m$ theories, including the cases with complex
coupling constant $g$ or non-integer power $m$. For $D=1$, we derive rapidly
convergent results for the Hermitian quartic and non-Hermitian cubic theories
by minimizing the complexity of the singularity at $n=\infty$.
