Complex Analysis/Curves

Introduction

In the Mathematics a curve (of lat. 'curvus 'bent, curved') is a eindimensionales object in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space15105.

Parameter representations

Multidimensional analysis: A continuous mapping ${\textstyle f:[a,b]\to \mathbb {R} ^{n}}$ is a curve in the ${\textstyle \mathbb {R} ^{n}}$ .
Complex Analysis: Continuous mapping ${\textstyle f:[a,b]\to \mathbb {C} }$ is a path in ${\textstyle \mathbb {C} }$ (see also path for integration).

Explanatory notes

A curve/a way is a mapping. It is necessary to distinguish the track of the path or the image of a path from the mapping graph. A path is a steady mapping of a interval in the space considered (e.g. ${\textstyle \mathbb {R} ^{n}}$ or ${\textstyle \mathbb {C} }$ ).

Example 1 - Plot

${\textstyle \gamma _{1}\colon \mathbb {R} \to \mathbb {R} ^{2},}$ ${\textstyle t\mapsto \gamma _{1}(t)={\big (}t^{2}-1,t(t^{2}-1){\big )}}$

EXAMPLE 1 Curve as a solution of an algebraic equation

$\gamma _{1}\colon \mathbb {R} \to \mathbb {R} ^{2},$ $t\mapsto \gamma _{1}(t)={\big (}t^{2}-1,t(t^{2}-1){\big )}$ resp. $y^{2}=x^{2}(x+1)$ .

Determine for the curve all ${\textstyle (t_{1},t_{2})\in \mathbb {R} ^{2}}$ with ${\textstyle \gamma _{1}(t_{1})=\gamma _{1}(t_{2})\in \mathbb {R} ^{2}}$

Examples 2

The mapping

${\textstyle {\widetilde {\gamma _{2}}}\colon [0,2\pi )\to \mathbb {R} ^{2},\quad t\mapsto {\widetilde {\gamma _{2}}}(t)=(\cos t,\sin t)}$

describes the Unit circle in the plane ${\textstyle \mathbb {R} ^{2}}$ .

${\textstyle \gamma _{2}\colon [0,2\pi )\to \mathbb {C} ,\quad t\mapsto \gamma _{2}(t)=\cos(t)+i\sin(t)}$

describes the Unit circle in the Gaussian number level ${\textstyle \mathbb {C} }$ .

Examples 3

The mapping

{\textstyle \gamma _{3}\colon \mathbb {R} \to \mathbb {R} ^{2},\quad t\mapsto \gamma _{3}(t)={\big (}t^{2}-1,t(t^{2}-1){\big )}}

describes a curve with a simple double point at ${\textstyle (0,0)}$ , corresponding to the parameter values ${\textstyle t=1}$ and ${\textstyle t=-1}$ .

Direction

As a result of the parameter representation, the curve receives a directional direction in the direction of increasing parameter.^[1]^[2]

Curve as Image of Path

Let ${\textstyle \gamma :[a,b]\to \mathbb {C} }$ or ${\textstyle \gamma :[a,b]\to \mathbb {R} ^{n}}$ be a path. is the image of a path

{\textstyle Spur(\gamma ):=\left\{(t,\gamma (t))\ |\ a\leq t\leq b\right\}}

.

Animation of the track

Curves in Geogebra

First create a slider for the variable ${\textstyle t\in [0,2\pi ]}$ and two points ${\textstyle K_{1}=(2\cos(t),2\sin(t))\in \mathbb {R} ^{2}}$ or ${\textstyle K_{2}=(\cos(3t),\sin(3t))\in \mathbb {R} ^{2}}$ and generate with ${\textstyle K:=K_{1}+K_{2}}$ the sum of both location vectors of ${\textstyle K_{1}}$ and ${\textstyle K_{2}}$ . Analyze the parameterization of the curves.

Geogebra:

 
  K_1:(2*cos(t),2 * sin(t)) 
  K_2:  (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2

\gamma _{4}(t):=(2\cos(t),2\sin(t))+(\cos(3t),\sin(3t))\in \mathbb {R} ^{2}

Equation representations

A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve:

The equation ${\textstyle x^{2}+y^{2}=1}$ describes the unit circle in the plane.
The equation ${\textstyle y^{2}=x^{2}(x+1)}$ describes the curve indicated above in parameter representation with double point.

If the equation is given by a w:en:w:en:Polynomialialialial, the curve is called algebraisch'.

Graph of a function

graph of a functionen are a special case of the two forms indicated above: The graph of a function

{\textstyle f\colon D\to \mathbb {R} ,\quad x\mapsto f(x)}

can be either as a parameter representation ${\textstyle \gamma \colon D\to \mathbb {R} ^{2},\quad t\mapsto (t,f(t))\}}$ or as equation ${\textstyle y=f(x)}$ , wherein the solution quantity of the equation represents the curve by ${\textstyle \{(x,y)\in \mathbb {R} ^{2}\mid y=f(x)\}}$ . If the Mathematics Education of w:en:Curve sketching is spoken, this special case is usually only said.

Closed curves

Closed curves ${\textstyle \gamma \colon [a,b]\to \mathbb {C} }$ are continuous mappings with ${\textstyle \gamma (a)=\gamma (b)}$ . In the function theory, we need curves ${\textstyle \gamma \colon [a,b]\to \mathbb {C} }$ in ${\textstyle \mathbb {C} }$ , which can be continuously differentiated. These are called integration paths.

Number of circulations in the complex numbers

Smooth closed curves can be assigned a further number, the Umlaufzahl, which curve is parameterized according to the arc curve ${\textstyle \gamma \colon [a,b]\to \mathbb {C} }$ by

\mu (\gamma ,z):={\frac {1}{2\pi i}}\int _{\gamma }{\frac {1}{\xi -z}}\,d\xi :=\int _{a}^{b}{\frac {1}{\gamma (t)-z}}\cdot \gamma '(t)\,dt

is given. The Umlaufsatz analogously to a curve in ${\textstyle R^{2}}$ , states that a simple closed curve has the number of revolutions ${\textstyle 1}$ or ${\textstyle -1}$ .

Curves as independent object

Curves without surrounding space are relatively uninteresting in the Differential geometry because each one-dimensional Manifold diffeomorph for real straight lines ${\textstyle \mathbb {R} }$ or for the unit circle line ${\textstyle S^{1}}$ . Also properties such as the Curvature of a curve cannot be determined intrinsically. In the algebraischen Geometrie and associated in the komplexen Analysis, “curves” are generally understood as one-dimensional w:en:complex manifolden, often also referred to as Riemann surface. These curves are independent study objects, the most prominent example being the elliptischen Kurven. See' Curve (algebraic geometry)

Historical

The first book of Elemente of Euclid began with the definition

A point is what has no parts. A curve is a length without width.

This definition can no longer be maintained today, as there are, for example, Peano-curven, i.e., continuous surjectivee mappings ${\textstyle f\colon \mathbb {R} \to \mathbb {R} ^{2}}$ , which fill the entire plane ${\textstyle \mathbb {R} ^{2}}$ . On the other hand, it follows from Sard's theorem that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'.

Interactive display of curves in geogebra

Tangent vector in ${\textstyle \mathbb {R} ^{2}}$ for a path ${\textstyle \gamma :[a,b]\to \mathbb {R} ^{2}}$ with a tangent vector ${\textstyle \gamma '(t)\in \mathbb {R} ^{2}}$ defined by the derivation ${\textstyle \gamma ':[a,b]\to \mathbb {R} ^{2}}$
Curves created by reflector on wheels of a bicycle as an example of curves - Cycloid
Example of Roulette curves - see also Roulette (curve)

Literature

Ethan D. Bloch: A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997.
Wilhelm Klingenberg: A Course in Differential Geometry'. Springer, New York 1978.

Individual evidence

↑ H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5
↑ H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9

Web links

{Commonscat|Curves|Kurven} {Wiktionary|Kurve}

Page Information

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

[CITE1-1] H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5

[cite2-2] H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9

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