Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 07 Winding numbers=Critical point
Tags: examplesSwitchExample1 riemanIntegralOfF récupéré

Name: The winding number of \(L\) around \(p\)
Definition: Take any half-line from \(p\) to infiny, not crossing an intersection of \(L\). Then take the number of times it crosses \(L\) going from left to right, minus the number of times it crosses \(L\) going from right to left
Page: 339, 340, 343
Chapter: 7I
Section: 1, 3
Variables: \(L:[0,1]\to\mathbb C\) a closed loop
\(p\in\mathbb C\)
Notation: \(v(L,p)\)
Name2: The degree of \(\mathcal L\) (around \(p\))
Intuition: The net number of revolutions of the direction of \(z\) as it traces out \(L\) once in its given sense

Name: A simple loop
Definition: A loop which does not intersect itself
Page: 339
Chapter: 7I
Section: 2

Smaller0: \(q\) and \(q'\) belong to the same area defined by \(L\)
Greater0: \(v(L,q)=v(L,q')\)
Page: 339
Chapter: 7I
Section: 3
Variables: \(L:[0,1]\to\mathbb C\) a closed loop,
\(q,q'\in\mathbb C\)

Name: The inside of \(L\)
Definition: The set of \(q\) such that \(v(L,q)\ne 0\)
Page: 339
Chapter: 7I
Section: 3
Variables: \(L:[0,1]\to\mathbb C\) a closed loop,

Name: The midpoint Riemann integral of \(f\) between \(a\) and \(b\)
Definition: \(\frac {b-a}n\sum_{i=1}^n f(a+(b-a)(\frac{i-\frac12}{n}))\)
Page: 378
Chapter: 8II
Section: 2
Variables: \(f:\mathbb {R\to R}\),
\(a<b\)
\(n\in\mathbb N\)
Notation: \(R_n\)
Intuition: Riemann sum, with each interval of equal length, \(x_i\) in the middle of intervals

Name: Trapeizodal Rule
Definition: Approximating \(\int_a^b f\) by \(\frac {b-a}n\left(\frac{f(a)}2+\sum_{i=1}^{n-1} f(a+(b-a)(\frac{i}{n}))+\frac{f(b)}2\right)\)
Page: 380
Chapter: 8II
Section: 2
Variables: \(f:\mathbb {R\to R}\),
\(a<b\)
\(n\in\mathbb N\)

Question: [$]\abs{\left(\int_a^bf(x)\dd x\right)-R_n}[/$]
Answer: is at most \(\frac{1}{n^2}\)
Page: 381
Chapter: 8II
Section: 3
Variables: \(f:\mathbb {R\to R}\),
\(a<b\)
\(n\in\mathbb N\)
\(R_n\) the midpoint Riemann Sum

Question: What can be said from this figure ?
Answer: \(PQ\cdot QR=AQ\cdot QB\)
Page: 381
Chapter: 8II
Section: 3
Variables: \(C\) a circle,
\(A,P,B,Q\) on the circle in this order
\(Q\) the intersection of \(AB\) and \(PR\)

Smaller0: the error of \(R_m\)
Greater0: the error from the trapeizodal formula
Page: 383
Chapter: 8II
Section: 3
Variables: \(f:\mathbb {R\to R}\),
\(a<b\)
\(m\in\mathbb N\)
Smaller to greater: is at most half of
Greater to smaller: is at least the double of

Smaller0: \(v(L,p)=v(K,p)\)
Greater0: \(K\) and \(L\) can be deformed without crossing \(p\)
Page: 341-343
Chapter: 7
Section: II1
Variables: \(p\in\mathbb C\),
\(K,L\) closed loop not containing \(p\)
Smaller to greater: iff
Name: Hopf's degree theorem
Construction: The problem is reduced to the fact that they are similar to the unit circle around \(z\) with the same number of turn.
For \(\mathcal L:[0,1]\to\mathbb C\) the path of \(L\), transform it into [$]\hat L:z\mapsto \frac{z}{\abs{z}}[/$]. And then deform in the unit cercle, which around \(z\)

Name: The degree of \(\mathcal L\) (around \(p\))
Definition: The winding number of the loop defined by \(\mathcal L\)
Page: 343
Chapter: 7
Section: II2
Variables: \(p\in\mathbb C\),
\(\mathcal L:[0,1]\to\mathbb L\) a closed loop
Example: as degree \(k\)
Applied to: \(z\mapsto z^k\) around \(0\)

Smaller0: the number of roots of \(P(z)\) around which \(\Gamma\) winds
Greater0: \(\nu[P(\Gamma),0]\)
Page: 344
Chapter: 7II
Variables: \(\Gamma\) a simple loop
\(P\in\mathbb C[z]\)
Smaller to greater: =

Name: a \(p\)-point of the mapping
Definition: some element of \(f^{-1}(p)\)
Page: 344
Chapter: 7II
Variables: \(f:\mathbb C\to\mathbb C\) a map
\(p\in\mathbb C\)
Context: Complex map

Smaller0: \(f\) analytic on \(\Gamma\) and its inside
Greater0: \(\nu[f(\Gamma),p]\)
Page: 344
Chapter: 7II
Variables: \(f:\mathbb C\to\mathbb C\) a map
\(p\in\mathbb C\)
\(\Gamma\) a loop
Smaller1: \(\Gamma\) a simple loop
Greater1: the number of \(p\)-points counted with their multiplicities inside \(\Gamma\)
Connect Greater0: equals

Name: The multiplicity of \(a\) as a root
Definition: the greatest \(n\) such that \((z-a)^n\) divides \(f\)
Page: 346
Chapter: 7IV
Section: 1
Variables: \(f\in\mathbb C[z]\)
\(a,p\in\mathbb C\)