Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 01 Introduction;
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Name: Complex plane
Definition: A 2 dimensional plane, with the point \((x,y)\) represented as \(x+iy\)
Page: 2
Chapter: 1
Section: I1
Notation: \(\mathbb C\)
Name2: Gauss plane
Name3: Argand plane

Question: In the complex plane, addition represents
Answer: parallelogram rule of ordinary vector addition
Page: 2
Chapter: 1
Section: I1

Question: In the complex plane, mulplication represents
Answer: sum of angles and product of distance
Page: 2
Chapter: 1
Section: I1

Question: Solution of \(x^3=3px+pq\)
Answer: is \(\sum_{\circ\in\{+,-\}}\sqrt[3]{q\circ \sqrt{q^2-p^3}}\)
Page: 5
Chapter: 1
Section: I2

Name: The modulus of \(z\)
Definition: \(\sqrt{x^2+y^2}\)
Page: 6
Chapter: 1
Section: I3
Variables: \(z=x+iy\in\mathbb C\)
Notation: [$]\abs{z}[/$]
Intuition: Length of \(z\)

Name: The argument of \(z\)
Definition: \(\theta\)
Page: 6
Chapter: 1
Section: I3
Variables: \(z=re^{i\theta}\in\mathbb C\)
Abbreviation: \(\arg{z}\)
Intuition: Angle of \(z\) with the real axis

Name: The real part of \(z\)
Definition: \(x\)
Page: 6
Chapter: 1
Section: I3
Variables: \(z=x+iy\in\mathbb C\)
Abbreviation: \(\text{Re}{z}\)
Intuition: \(x\) coordinate of \(z\)

Name: The imaginary part of \(z\)
Definition: \(y\)
Page: 6
Chapter: 1
Section: I3
Variables: \(z=x+iy\in\mathbb C\)
Abbreviation: \(\text{Im}{z}\)
Intuition: \(y\) coordinate of \(z\)

Name: An imaginary number
Definition: A real multiple of \(i\)
Page: 6
Chapter: 1
Section: I3
Intuition: An element of the complex axis

Name: Real axis
Definition: The set of the real numbers
Page: 6
Chapter: 1
Section: I3
Context: complex geometry

Name: Complex axis
Definition: Set of complex number
Page: 6
Chapter: 1
Section: I3

Name: The complex conjugate of \(z\)
Definition: \(x-iy\)
Page: 6
Chapter: 1
Section: I3
Variables: \(z=x+iy\in\mathbb C\)
Notation: \(\bar z\)
Intuition: Reflection of \(z\) in the real axis

Definition: \(re^{i\theta}\)
Page: 7
Chapter: 1
Section: I3
Variables: \(r,\theta\in\mathbb R\)
Notation: \(r\angle\theta\)
Intuition: The point with length \(r\) and angle \(\theta\)

Smaller0: \(\tan(\arg z)\)
Greater0: \(\frac{\text{Im}(z)}{\text{Re}(z)}\)
Page: 7
Chapter: 1I
Section: 4
Variables: \(z\in\mathbb C\)
Smaller to greater: =

Question: \(\frac{1}{r\angle\theta}\) in \(\angle\) form
Answer: \(\frac{1}{r}\angle-\theta\)
Page: 7
Chapter: 1I
Section: 4
Variables: \(z\in\mathbb C\)
\(r>0\)
\(\theta\in\mathbb R\)

Question: \(\frac{1}{x+iy}\) in normal form
Answer: [$]\frac{\overline{z}}{\abs{z}^2}[/$] with \(z=x+iy\)
Page: 7
Chapter: 1I
Section: 4
Variables: \(r>0\)
\(x,y\in\mathbb R\)

Question: geometricaly, multiplication by \(R\angle\theta\) is
Answer: rotation of the plane thourgh angle \(\theta\) and expansion of the plane by factor \(R\)
Page: 8
Chapter: 1I
Section: 5

Smaller0: \(e^{i\theta}\)
Greater0: \(\cos\theta+i\sin\theta\)
Page: 8
Chapter: 1I
Section: 5
Variables: \(\theta\in\mathbb R\)
Smaller to greater: =
Name: Euler's formula
Construction: If you think that \(e^{it}\) is a function of time, at any moment, the derivative at position \(e^{it}\) (for \(t\) fixed) is orthogonal to the vector from \(0\) to \(e^{it}\). Its modulus remain constant. Thus [$]\defFun{\mathbb R}{\mathbb C}{t}{e^{it}}[/$] is a circle around \(0\) of radius \(1\)

Name: Translation by \(v\)
Definition: [$]\defFun{\mathbb C}{\mathbb C}{z}{z+v}[/$]
Page: 17
Chapter: 1III
Section: 3
Variables: \(v\in\mathbb C\)
Notation: \(\mathcal T_v\)

Name: Rotation of \(\theta\) around \(a\)
Definition: [$]\defFun{\mathbb C}{\mathbb C}{z}{(z-a)e^{i\theta}+a=a(1-e^{i\theta})+ze^{i\theta}}[/$]
Page: 17,38
Chapter: 1III,IV
Section: 3, 3
Variables: \(a\in\mathbb C\)
\(\theta\in\mathbb R\)
Notation: \(\mathcal R_a^\theta\)
Definition2: \(\mathfrak R_{L_1}\circ\mathfrak R_{L_2}\) with \(\angle L_1L_2=\theta\) and \(L_1\cap L_2=\{a\}\)

Name: Cross product
Definition: [$]\defFun{\mathbb C^2}{\mathbb R}{\vec a,\vec b}{\abs{a}\abs{b}\sin\theta}[/$] with \(\theta\) the angle between \(\vec a\) and \(\vec b\)
Page: 28
Chapter: 1III
Section: 6
Notation: \(\times\)
Intuition: Signed area of of the parallelogram spanned by \(\vec a\) and \(\vec b\)

Smaller0: \(\overline ab\)
Greater0: \(\mathbf{a\cdot b}+i(\mathbf{a\times b})\)
Page: 28
Chapter: 1III
Section: 6
Variables: \(\mathbf{a,b}\in\mathbb C\)
Smaller to greater: =

Smaller0: the area of the polygon \(a_1\dots a_n\)
Greater0: \(\frac12\sum_{i=1}^na_i\times a_{i+1\mod n}\)
Page: 30
Chapter: 1III
Section: 6
Variables: \(\mathbf{a_1,\dots, a_n}\in\mathbb C\)
Smaller to greater: =

Name: geometric figure
Definition: anything we might choose to draw on an infinitely large piece of flat paper with an infinitely fine pen
Page: 30
Chapter: 1
Section: IV1

Name: geometric properties
Definition: Property unaltered by all possible motions of the figure.
Page: 30, 32, 34
Chapter: 1
Section: IV1, 2
Property: Determined by its effect on an arbitrary triangle

Name: A (rigid) motion
Definition: \(f:\mathbb C\to\mathbb C\) such that \(d(f(x),f(y))=d(x,y)\)
Page: 31
Chapter: 1
Section: IV1
Context: Complex analysis
Definition2: composition of two or three reflections
Name2: An isometry
Extra: not a meseometry
Denoted by: \(M\)

Page: 32
Definition: on geometric figures such that there exists a motion sending one figure to the other ones
Chapter: 1
Section: IV1
Notation: \(\cong\)
Name: Congruent figures
Context: Complex analysis

Page: 32
Chapter: 1
Section: IV1
Notation: \(\sim\)
Name: Geometric equality
Context: Complex analysis
Transitive: Equivalence relation

Name: Geometry (according to Klein)
Definition: study of the invariants of a group
Page: 33
Chapter: 1
Section: IV1
Context: Complex analysis

Name: A direct motion
Definition: A motion preserving angles
Page: 36
Chapter: 1
Section: IV1
Context: Complex analysis
Definition2: Either rotation or translation
Definition3: [$]\defFun{\mathbb C}{\mathbb C}{z}{e^{i\theta}z+v}[/$] for some \(\)
Definition4: composition of two reflections

Name: An opposite motion
Definition: A motion reversing angles
Page: 36, 37
Chapter: 1
Section: IV1, 3
Context: Complex analysis
Definition2: Composition of three reflexions

Question: The set of motions sending \(A\) to \(A'\) and \(B\) to \(B'\)
Answer: contains exactly a direct and an opposite motion
Page: 36
Chapter: 1
Section: IV1
Variables: \(A,B\) distinct points
\(A',B'\) distinct points
Context: Complex analysis

Name: direct similarities
Page: 41
Section: IV4
Chapter: 1
Under: composition
Under2: inverse ?

Name: direct motions
Page: 36
Section: IV1
Chapter: 1
Under: composition

Name: motion
Page: 36
Section: IV1
Chapter: 1
Under: composition

Name: Three Reflections Theorem
Definition: Each motion is the composition of two or three reflections
Page: 37
Chapter: 1
Section: IV3

Question: \(\mathfrak R_{L_1}\circ\mathfrak R_{L_1}\) is
Answer: a rotation of \(2\angle L_1L_2\), with center \(L_1\cap L_2\) if they are not parallel
Page: 37
Chapter: 1
Section: IV3
Variables: \(L_1,L_2\) two lines
Answer2: a translation of \(2(L_1-L_2)\)

Name: An invariant curve of \(M\)
Definition: A geometric figure \(G\) such that \(M(G)=G\)
Page: 38
Chapter: 1
Section: IV3
Variables: \(M\) a mapping
Example: lines perpendicular to the vector of translation
Example2: circles whose center is the center of rotation
Applied to: translation
Applied to2: rotation

Question: \(\mathcal R_a^\theta\circ \mathcal R_b^\phi\) is
Answer: a rotation of angle \(\frac{\theta+\phi}2\), whose center is determined as in the figure
Page: 38
Chapter: 1
Section: IV3
Variables: \(a,c\in\mathbb C\)
\(\theta,\phi\in\mathbb R\)

Name: Similarity
Definition: Mapping of the plane to itself that preserves ratios of distances
Page: 40
Chapter: 1IV
Section: 4
Context: complex geometry
Definition2: \(\mathcal M\circ\mathcal D^r_p\) for any fixed \(p\)
Definition3: determined by the image of any triangle
Denoted by: \(\mathcal S\)

Name: The expansion of \(\mathcal S\)
Definition: The ratio by which segments are expanded by \(\mathcal S\)
Page: 40
Chapter: 1IV
Section: 4
Variables: \(\mathcal S\) a similarity
Context: complex geometry

Name: Euclidean geometry (according to Klein)
Definition: study of those properties of geometric figures that are invariant under the group of similarities
Page: 40
Chapter: 1IV
Section: 4
Variables: \(\mathcal S\) a similarity
Context: complex geometry

Name: A central dilatation of center \(o\) and expansion \(r\)
Definition: The similarity fixing \(o\) and expanding by \(r\)
Page: 40
Chapter: 1IV
Section: 4
Variables: \(o\in\mathbb C\)
\(r>0\)
Context: complex geometry
Notation: \(\mathcal D_o^r\)

Question: \(\left(\mathcal D_o^r\right)^{-1}\)
Answer: \(\mathcal D_o^{1/r}\)
Page: 40
Chapter: 1IV
Section: 4
Variables: \(o\in\mathbb C\)
\(r>0\)
Context: complex geometry

Name: The dilative rotation of center \(o\), expansion \(r\) and angle \(\theta\)
Definition: \(\mathcal D_o^{r}\circ\mathcal R_o^\theta=\mathcal R_o^\theta\circ\mathcal D_o^{r}\)
Page: 40
Chapter: 1IV
Section: 4
Variables: \(o\in\mathbb C\)
\(r>0\)
\(\theta\in\mathbb R\)
Context: complex geometry
Notation: \(\mathcal D_o^{r,\theta}\)
Definition2: \((re^{i\theta})z\) for \(o\) the center of the plane

Name: A direct similarity
Definition: A similarity preserving angle
Page: 41
Chapter: 1IV
Section: 4
Context: complex geometry
Definition2: dilative rotation or translation

Name: An opposite similarity
Definition: A similarity reversing angle
Page: 41
Chapter: 1IV
Section: 4
Context: complex geometry

Question: What is the relation between \(\mathcal D_{q}^{r,\theta}([A,B])\) and \(\mathcal D_{o}^{r,\theta}([A,B])\)
Answer: They are parallel segments of equal length
Page: 43
Chapter: 1IV
Section: 4
Variables: \(A,B\in\mathbb C\)
\(r>0\)
\(\theta\in\mathbb R\)
\(q,o\in\mathbb C\)
Context: complex geometry