I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. For example the normal MGF is $ \exp{(\mu t + \frac{\sigma^2 t^2}{2})} $ but if we substitute $it$ into that we get $ \exp{(\mu it - \frac{\sigma^2 t^2}{2})} $ which is the characteristic function. Can we always do this if the MGF exists in a neighborhood of 0?
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You are correct about the link between these two functions, provided the MGF exists. This was discussed in another question in this site: Link between moment-generating function and characteristic function
