I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question:
It sounds maybe a bit strange but I don't really see the conceptual difference between
WKB approximation ( https://en.wikipedia.org/wiki/WKB_approximation ),
the semi-classical approximation ( or is there a varied range of semi-classical approximations? My definition is $h \rightarrow 0$ )
the stationary phase approximation ( https://en.wikipedia.org/wiki/Stationary_phase_approximation )
and the methods of steepest descent ( https://en.wikipedia.org/wiki/Method_of_steepest_descent) ( the last is a bit more complex when integrating in the complex plane but rely also on a majoration of the function by the taylor approximation from what I have seen ).
Following me, the 4 methods are always approximating the integral thanks to a Taylor development to the second order in order to analytically solve the integral. Is there a difference of validity for these methods ( by that I mean one case where one method is a good approximation and not the others ).
I don't think I have a really good comprehension of these methods so excuse me if it is way off the mark.