Rotation number (mathematics)

The number of turns (also called number of turns or index ) is a topological invariant that plays a decisive role in function theory .

Preview

The number of revolutions of a curve in relation to a point represents the number of turns in counterclockwise direction when following the course of the curve. A turn around in clockwise direction results in the negative number of turns −1. ${\displaystyle \gamma }$ ${\displaystyle z_{0}}$ ${\displaystyle z_{0}}$

Number of turns

1	−1	0	1	2

definition

If a closed curve is in and if there is also a point that does not lie on, then the revolution number of is defined in relation to : ${\displaystyle \gamma }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z_{0}}$ ${\displaystyle \mathbb {C} ,}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$ ${\displaystyle z_{0}}$

${\displaystyle \operatorname {ind} _{\gamma }(z_{0})=n(\gamma ,z_{0}):={\frac {1}{2\pi \mathrm {i} }}\int _{\gamma }{\frac {\mathrm {d} \zeta }{\zeta -z_{0}}}}$

The circulation number (according to the English index ) is often referred to in the literature as or . The number of revolutions of a closed curve is always an integer regardless of the reference point. ${\displaystyle \operatorname {ind} }$ ${\displaystyle I}$ ${\displaystyle \chi }$

calculation

Number of turns = 2

Number of turns = 0

The number of turns can be determined intuitively using

${\displaystyle \operatorname {ind} _{\gamma }(z_{0})=}$Number of revolutions from um counterclockwise - number of revolutions from um clockwise ${\displaystyle \gamma }$ ${\displaystyle z_{0}}$ ${\displaystyle \gamma }$ ${\displaystyle z_{0}}$

to calculate. The calculation via the definition is often not easily possible. As an example we choose the unit circle

${\displaystyle \gamma \colon [0,2\pi ]\to \mathbb {C} ,\,t\mapsto e^{\mathrm {i} t}}$

as a curve. According to the intuitive rule is for all points in its interior and for all points outside of the closed circular disk . The latter follows immediately from Cauchy's integral theorem and the definition. Be now ${\displaystyle \operatorname {ind} _{\gamma }(z)=1}$ ${\displaystyle z\in \mathbb {E} }$ ${\displaystyle \mathbb {E} }$ ${\displaystyle \operatorname {ind} _{\gamma }(z)=0}$ ${\displaystyle z\in \mathbb {C} \setminus {\bar {\mathbb {E} }}}$ ${\displaystyle {\bar {\mathbb {E} }}}$

${\displaystyle f\colon \mathbb {E} \to \mathbb {C} ,\,z\mapsto \operatorname {ind} _{\gamma }(z).}$

It applies

${\displaystyle \operatorname {ind} _{\gamma }(0)=f(0)={\frac {1}{2\pi \mathrm {i} }}\int _{\gamma }{\frac {\mathrm {d} \zeta }{\zeta }}={\frac {1}{2\pi \mathrm {i} }}\int \limits _{0}^{2\pi }{\frac {\mathrm {i} e^{\mathrm {i} t}}{e^{\mathrm {i} t}}}\mathrm {d} t=1.}$

Interchanging differentiation and integration results in

${\displaystyle f'(z)={\frac {1}{2\pi \mathrm {i} }}\int _{\gamma }{\frac {\mathrm {d} \zeta }{\left(\zeta -z\right)^{2}}}}$

and because a primitive of the integrand is, is because coherently is, so for all ${\displaystyle \zeta \mapsto -{\frac {1}{\zeta -z}}}$ ${\displaystyle f'\equiv 0.}$ ${\displaystyle \mathbb {E} }$ ${\displaystyle f(z)=\operatorname {ind} _{\gamma }(z)=1}$ ${\displaystyle z\in \mathbb {E} .}$

Application in function theory

The number of revolutions is mainly used in the calculation of curve integrals in the complex number plane. Be

${\displaystyle f\colon \mathbb {C} \setminus \left\{a_{1},\dotsc ,a_{n}\right\}\to \mathbb {C} }$

a meromorphic function with singularities then one can after the residue theorem , the integral of a (by any of the singularities extending) curve by ${\displaystyle a_{1},\dotsc ,a_{n},}$ ${\displaystyle f}$ ${\displaystyle \gamma }$

${\displaystyle \int _{\gamma }f\mathrm {d} z=2\pi \mathrm {i} \sum _{k=1}^{n}\operatorname {ind} _{\gamma }(a_{k})\operatorname {Res} _{a_{k}}f}$

to calculate.

algorithm

Number of turns of the surfaces of a nontrivial polygon: The number of turns for the surface in which the point is located is −1, i.e. This means that it lies within the polygon (the gray area). Each surface has a fixed number of turns.

In algorithmic geometry , the revolution number is used to determine whether a point is outside or inside a non-simple polygon (a polygon whose edges intersect). For simple polygons the algorithm for the even-odd rule is simplified.

For polygons (closed edges) the following algorithm is used to calculate the number of revolutions:

1. Find a half-line (starting at the point to be examined outwards ) that does not contain any corner points of the polygon.
2. Set ${\displaystyle w=0.}$
3. For all intersections of the half-line with the polygon:
   • If the half-line intersects a polygon edge that is oriented "from right to left" (if the point is on the left side of the edge), increase by 1. ${\displaystyle w}$
   • If the half-line intersects a polygon edge that is oriented "from left to right" (if the point is on the right side of the edge), reduce by 1. ${\displaystyle w}$
4. ${\displaystyle w}$ is now the number of revolutions of the point.

If the rotation number is 0, the point lies outside the polygon, otherwise inside.

In the example opposite, the half-line that starts with is the vertical arrow. It cuts three edges of the polygon. With regard to the red edge, the point is to the right. With regard to the next edge, the point is also to the right and, with regard to the last edge, the point is to the left. The point lies within the polygon. The polygon area has a gray background. ${\displaystyle \left(w=-1\right).}$ ${\displaystyle \left(w=-2\right)}$ ${\displaystyle \left(w=-1\right).}$

An analogous algorithm also gives the number of revolutions around one point for non-straight (closed) curves, but checking the intersection points is not so easy to implement there.

Generalization for n -dimensional manifolds

A generalization for -dimensional manifolds comes from Nikolai Nikolajewitsch Bogolyubov : Using the general Stokes' theorem for one can ${\displaystyle n}$ ${\displaystyle z_{0}=0}$

${\displaystyle \mathrm {ind} _{\gamma }(0)={\frac {1}{n\mathrm {Vol(B)} }}\oint _{\gamma }{\frac {{\vec {x}}\cdot \mathrm {d} {\vec {S}}}{\|x\|^{n}}}}$

write. is the unit sphere im is the considered -dimensional closed manifold on which to integrate. ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{n},}$ ${\displaystyle \gamma }$ ${\displaystyle (n-1)}$

See also

• Cauchy's integral formula
• Residual theorem
• Curve integral

literature

• Eberhard Freitag, Rolf Busam: Theory of functions 1. Springer-Verlag, Berlin, ISBN 3-540-67641-4 .