# Hessian of a function

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or

where (or ) and is given on the -dimensional real space (or on the complex space ) with coordinates (or ). Introduced in 1844 by O. Hesse. With the aid of a local coordinate system this definition is transferred to functions defined on a real manifold of class (or on a complex space), at critical points of the functions. In both cases the Hessian is a quadratic form given on the tangent space and is independent of the choice of coordinates. In Morse theory the Hessian is used to define the concepts of a (non-)degenerate critical point, the Morse form and the Bott form. In complex analysis the Hessian is used in the definition of a pseudo-convex space (cf. Pseudo-convex and pseudo-concave) and of a plurisubharmonic function.

#### References

 [1] M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) [2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)