View source for Hessian of a function

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471601.png" />''

The quadratic form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471602.png" /></td> </tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471603.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471604.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471605.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471606.png" /> is given on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471607.png" />-dimensional real space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471608.png" /> (or on the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h0471609.png" />) with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h04716010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h04716011.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h04716012.png" /> (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|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|Pseudo-convex and pseudo-concave]]) and of a [[Plurisubharmonic function|plurisubharmonic function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Postnikov,   "Introduction to Morse theory" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>



====Comments====
One usually calls the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h04716013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047160/h04716014.png" /> the complex Hessian.

If the Hessian of a real-valued function is a positive (semi-) definite form, then the function is convex; similarly, if the complex Hessian of a function is a positive (semi-) definite form, then the function is plurisubharmonic.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,   "Function theory of several complex variables" , Wiley  (1982)  pp. Chapt. 3</TD></TR></table>