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)

References

[a1]	R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)