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I just ran across this: http://mathoverflow.net/questions/163377/what-is-the-complex-third-derivative. The answer points out that the real Hessian is not invariant under coordinate changes, which is exactly the problem of “Cauchy’s invariant formula” that’s solved by including d2x terms in d2f.
We can consider complex-valued cogerm differential forms, in particular on ℂn or a more general complex manifold. (Although should the domains of the germs still be intervals in ℝ, or should they be regions in ℂ?) Are there “Dolbeault” versions of the commutative cogerm differential for which the “complex Hessian” can be written as i∂ˉ∂f? If so, this might partially answer the second question asked at the link.
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