Variation of steepest descent/Laplace methods for non-exponential integrands

I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type

$ $ \int_C f(z) M(\lambda g(z)) dz$ $

for $ \lambda >>0$ functions $ f(z), g(z): \mathbb C \rightarrow \mathbb C$ and where $ M(z)$ is a “stretched exponential” of some sort (e.g. Mittag Leffler function, Hypergeometric functions).

As far as I can see, for the case when $ f,g,M$ are real functions it is easy to mimick the formal derivation of the Laplace method, if, say, we can explicitly compute the integral of $ M(-x^2)$ and $ M(-\lambda x^2)$ looks like a delta around the origin. Can all this be made precise?