- Analytization trick
The analytization trick is a
heuristic often applied byphysicist s.Suppose we have a function "f" of a
complex variable "z" which is not analytic, but happens to bedifferentiable with respect to its real andimaginary components separately. Differentiating "f" with respect to "z" is out of the question, but it turns out if:
for some
analytic function "g" oftwo complex variables , we can pretend "f" is "g" (physicists do this sort of thing all the time) and work with:
and
:
instead. Physicists write these as
:
and
:
and give some
handwaving explanation as to why and "z" may be treated as if they are "independent" when they really are not.Note that if "g" exists, it is unique (due to the theorem about the uniqueness of
analytic continuation s), at least if we ignore complications likebranch cut s and so on.Conceptually, whenever this trick is used, it probably means on a physical level that the variable z that they are working with "really" has a real structure and physicists are merely pigeonholing it into a complex variable.
Actually, it's not even necessary for there to be an analytic "g". It's enough for "f" to be
differentiable with respect to its real andimaginary components (or n times differentiable, as the case may be). In that case,:
has to be treated purely formally.
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