Why the moment-generating function, rather than the characteristic function? I'm wondering why the moment-generating function is worth discussing (say, in basic probability courses, or in textbooks, rather than research), when the characteristic function appears to completely supersede it.
The "technology" involved is approximately the same, and a lot of the properties correspond, but the characteristic function has several huge advantages over the MGF, such as


*

*Existence for any probability distribution

*Lévy's continuity theorem

*Inversion formula using the Fourier inversion theorem.


The only (pretty trivial) disadvantage that I can see is that the characteristic function requires knowledge of complex numbers.
Does the MGF have some advantage I'm missing?
 A: I agree that the MGF's advantage that you've identified (its avoidance of complex numbers) seems trivial; but then again I was trained in Engineering, where the complex plane is taught sooner. 
The thinking seems quite different in Statistics. For example: 


*

*In DeGroot and Schervish's Probability and Statistics, are complex numbers ever even mentioned?

*Ditto for Wackerly et al's Mathematical Statistics with Applications, another popular textbook.

*In John Rice's Mathematical Statistics and Data Analysis, after MGFs are described there's a brief mention of the CF. However, this ends with a warning that "using the characteristic function requires some familiarity with the techniques of complex variables". 


So, perhaps you've answered your question better than you thought!
A: The MGF does not always exist, so for example, you cannot use MGF to prove the central limit theorem without imposing a very strong assumption.  Note MGF's existence implies the existence of all moments (but this is not sufficient), while the CLT requires only second moment to exist.
