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We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs $f:({ℂ}^2,0)⟶(ℂ,0)$ which take into account the inflection points of the fibres of $f$. We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.

We prove a $D=1$ analytic versal deformation theorem in the Heisenberg algebra. We define the spectrum of an element in the Heisenberg algebra. The quantised version of the Morse lemma already shows that the perturbation series arising in a perturbed harmonic oscillator become analytic after a formal Borel transform.