# Critical point

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For an analytic function , a critical point of order is a point of the complex plane at which is regular but its derivative has a zero of order , where is a natural number. In other words, a critical point is defined by the conditions A critical point at infinity, , of order , for a function which is regular at infinity, is defined by the conditions Under the analytic mapping , the angle between two curves emanating from a critical point of order is increased by a factor . If is regarded as the complex potential of some planar flow of an incompressible liquid, a critical point is characterized by the property that through it pass not one but stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function (i.e. the function for which ), a critical point is an algebraic branch point of order .

A point of a complex -dimensional irreducible analytic set the latter being defined in a neighbourhood of in the complex space by the conditions , where are holomorphic functions on in complex variables, , is called a critical point if the rank of the Jacobian matrix , , , is less than . The other points of are called regular. There are relatively few critical points on : They form an analytic set of complex dimension at most . In particular, when , i.e. if , and the dimension of is , the dimension of the set of critical points is at most .

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How to Cite This Entry:
Critical point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_point&oldid=16181
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article