# 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?