# Analytic curve

analytic arc

A curve $K$ in an $n$- dimensional Euclidean space $\mathbf R ^ {n}$, $n \geq 2$, which has an analytic parametrization. This means that the coordinates of its points $x = ( x _ {1} \dots x _ {n} )$ can be expressed as analytic functions of a real parameter $x _ {i} = x _ {i} (t)$, $i = 1 \dots n$, $\alpha \leq t \leq \beta$, i.e. in a certain neighbourhood of each point $t _ {0}$, $\alpha \leq t _ {0} \leq \beta$, the functions $x _ {i} (t)$ can be represented as convergent power series in $t - t _ {0}$, and the derivatives $x _ {i} ^ \prime (t _ {0} )$, $i = 1 \dots n$, do not simultaneously vanish at any point of the segment $[ \alpha , \beta ]$. This last condition is sometimes treated separately, and an analytic curve which satisfies it is called a regular analytic curve. An analytic curve is called closed if $x _ {i} ( \alpha ) = x _ {i} ( \beta )$, $i = 1 \dots n$.

An analytic curve in the plane $\mathbf C = \mathbf C ^ {1}$ of the complex variable $z = x _ {1} + ix _ {2}$ can be represented as a complex-analytic function of a real parameter $z = f(t)$, $\alpha \leq t \leq \beta$, $f ^ {\ \prime } (t) \neq 0$ on $[ \alpha , \beta ]$. If the analytic curve is located in a domain $D \subset \mathbf C$, then a conformal mapping of $D$ into any domain will also yield an analytic curve. If the set of intersection points of two analytic curves is infinite, these analytic curves coincide.

In general, in a complex space $\mathbf C ^ {n}$, $n \geq 1$, the complex coordinates $z _ {i}$ of the points of an analytic curve can be represented as analytic functions of a real parameter $z _ {i} = z _ {i} (t)$, $\alpha \leq t \leq \beta$, $i = 1 \dots n$. It should be noted, however, that if $n > 1$, the term "analytic curve" may sometimes denote an analytic surface of complex dimension one.

On a Riemann surface $S$ an analytic curve $K$ can be represented as $f(t) = \psi ( \phi (t))$, where $z = \psi (P)$ is a local uniformizing parameter of the points $P$ on $S$ and $f(t)$ is an analytic function of a real parameter in a neighbourhood of any point $t _ {0} \in [ \alpha , \beta ]$.

#### References

 [1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) [2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1969) pp. Chapt. 3 (In Russian)