Critical point

For an analytic function $f ( z)$, a critical point of order $m$ is a point $a$ of the complex plane at which $f ( z)$ is regular but its derivative $f ^ { \prime } ( z)$ has a zero of order $m$, where $m$ is a natural number. In other words, a critical point is defined by the conditions

$$\lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m} } = 0,\ \ \lim\limits _ {z \rightarrow a } \frac{f ( z) - f ( a) }{( z - a) ^ {m+} 1 } \neq 0.$$

A critical point at infinity, $a = \infty$, of order $m$, for a function $f ( z)$ which is regular at infinity, is defined by the conditions

$$\lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m} = 0,\ \ \lim\limits _ {z \rightarrow \infty } [ f ( z) - f ( \infty )] z ^ {m + 1 } \neq 0.$$

Under the analytic mapping $w = f ( z)$, the angle between two curves emanating from a critical point of order $m$ is increased by a factor $m + 1$. If $f ( z)$ 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 $m + 1$ stream lines, and the velocity of the flow at a critical point vanishes. In terms of the inverse function $z = \psi ( w)$( i.e. the function for which $f [ \psi ( w)] \equiv w$), a critical point is an algebraic branch point of order $m + 1$.

A point $a$ of a complex $( n - m)$- dimensional irreducible analytic set

$$M = \ \{ {z \in V } : { f _ {1} ( z) = \dots = f _ {m} ( z) = 0 } \} ,$$

the latter being defined in a neighbourhood $V$ of $a$ in the complex space $\mathbf C ^ {n}$ by the conditions $f _ {1} ( z) = \dots = f _ {m} ( z) = 0$, where $f _ {1} \dots f _ {m}$ are holomorphic functions on $V$ in $n$ complex variables, $z = ( z _ {1} \dots z _ {n} )$, is called a critical point if the rank of the Jacobian matrix $\| \partial f _ {j} / \partial z _ {k} \|$, $j = 1 \dots m$, $k = 1 \dots n$, is less than $m$. The other points of $M$ are called regular. There are relatively few critical points on $M$: They form an analytic set of complex dimension at most $n - m - 1$. In particular, when $m = 1$, i.e. if $M = \{ f _ {1} ( z) = 0 \}$, and the dimension of $M$ is $n - 1$, the dimension of the set of critical points is at most $n - 2$.

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How to Cite This Entry:
Critical point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_point&oldid=46555
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article