Let $X$ and $Y$ be real-valued non-independent random variables and suppose I have their joint moment generating function $$\phi(s,t)=\mathsf{E}\left[e^{sX+tY}\right],$$ which I assume is a "nice" function. (In particular, all the derivatives of $\phi$ exist.)
If I want to compute the expectation of $X$ or $Y$ or any other moment from $\phi$, there is a straightforward formula: $$\mathsf{E}\left[X\right] = \left.\frac{\partial}{\partial s} \phi(s,t) \right|_{s=t=0}, \qquad \mathsf{E}\left[Y\right] = \left.\frac{\partial}{\partial t} \phi(s,t) \right|_{s=t=0}, \qquad \mathsf{E}\left[X^mY^n\right] = \left.\left(\frac{\partial}{\partial s}\right)^m\left(\frac{\partial}{\partial t}\right)^n \phi(s,t) \right|_{s=t=0}.$$ I want to compute the conditional expectation $$h(x) = \mathsf{E}\left[Y\mid X=x\right].$$ One way to compute $h$ is to apply the inverse Laplace transform to $\phi$ to obtain the probability density function of $(X,Y)$ and then integrate that. That is very messy and is not very useful to me.
My question is: Is there a straightforward way to compute $h(x) = \mathsf{E}\left[Y\mid X=x\right]$ from $\phi(s,t)=\mathsf{E}\left[e^{sX+tY}\right]$?
In particular, I would like a formula that allows me to approximate $h$ given an approximation to $\phi$, which means I'd like to keep the number of integrals that appear to a minimum.