Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 02 Complex functions
Tags: hasSeenConjDef typWTF duplicateToOtherBook examplesSwitchExample1 fromMath trigoRelationToQuestion duplicateFromOtherBook functionVisualization markedtrigoRelationToQuestion

Name: A power series
Definition: A series of the form \(\sum_{i=0}^\infty c_i(z-z_0)^i\).
Page: 67, 196, 266
Chapter: 2I Abstract Algebra, Summit, Foote, 3, 5 Zorich
Section: 2, 3, 5.2
Index: ,, 4
Kind: ,,Ex
Name2: A formal power series in the indeterminate \(x\) with coefficient from \(R\)
Extra: Visual complex analysis ch 2I. 2. P. 67
French: Une série entière

Smaller0: \[\sin(a+b)\]
Greater0: \[\cos a\sin b+\sin a\cos b\]
Page: 044
Chapter: 2
Section: 3.2
Variables: \(a,b\in\mathbb C\).
Smaller to greater: =
Hide middle: yes
Extra: Complex, Ahlfors Ch 2, 3.2, p 44
Visual complex analysis, ch 2V, 1. p86

Name: \(\sinh\)
Definition: [$]\defFun{\mathbb C}{\mathbb C}{x}{\frac{e^x-e^{-x} }{2}}[/$]
Page: 86
Chapter: 2V
Section: 2
Index: 14
Kind: Ex
Definition2: [$]\defFun{\mathbb C}{\mathbb C}{x}{-i\sin ix}[/$]
Extra: Zorich, Analysis Ch 5. Sec. 2.3. P 199, 274
Visual complex analysis. Ch 2V. 2. P 86
Intuition: mean between \(e^x\) and \(-e^{-x}\)

Name: \(\sinh\)
Definition: mean between \(e^x\) and \(-e^{-x}\): [$]\defFun{\mathbb C}{\mathbb C}{x}{\frac{e^x-e^{-x} }{2}}[/$]
Page: 199, 274
Chapter: 5
Section: 2.3
Index: 14
Kind: Ex
Definition2: [$]\defFun{\mathbb C}{\mathbb C}{x}{-i\sin ix}[/$]
Extra: Zorich, Analysis Ch 5. Sec. 2.3. P 199, 274
Visual complex analysis. Ch 2V. 2. P 86

Name: An element of the image of \(f\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(w\)

Name: An element of the domain of \(f\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(z\)

Name: The image plane
Definition: \(\mathbb C\) seen as the range of \(f\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Name2: The \(w\)-plane

Name: The \(z\)-plane
Definition: \(\mathbb C\) seen as the domain of \(f\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)

Name: The real part of \(z\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(x\)

Name: The imaginary part of \(z\)
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(y\)

Name: The imaginary part of \(w\)
Definition: [$]\defFun{\mathbb C}{\mathbb R}{u+iv}{v}[/$]
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(v\)

Name: The real part of \(w\)
Definition: [$]\defFun{\mathbb C}{\mathbb R}{u+iv}{u}[/$]
Page: 55
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(u\)

Name: The modulus of \(w\)
Definition: [$]\defFun{\mathbb C}{\mathbb R^{\ge0}}{Re^{i\phi}}{R}[/$]
Page: 56
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(R\)

Name: The argument of \(w\)
Definition: [$]\defFun{\mathbb C}{\mathbb R^{[0,2\pi)}}{Re^{i\phi}}{\phi}[/$]
Page: 56
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(\phi\)

Name: The argument of \(z\)
Page: 56
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(\theta\)

Name: The modulus of \(z\)
Page: 56
Chapter: 2I
Variables: \(f:\mathbb{C\to C}\)
Denoted by: \(r\)

Name: The graph of \(f\)
Definition: The set of pair \((x,f(x))\) for \(x\in X\)
Page: 56
Chapter: 2I
Variables: \(f:\mathbb{X\to Y}\)

Name: A Riemann surface
Definition: The graph of a complex function
Page: 56
Chapter: 2I

Name: Modular surface of \(f\)
Definition: The function [$]\defFun{\mathbb C}{\mathbb R^{\ge0}}{z}{\abs{f(z)}}[/$]
Page: 56
Chapter: 2
Section: I
Variables: \(f:\mathbb C\to\mathbb C\)

Name: Power \(n\)
Page: 58
Chapter: 2
Section: II.1
Variables: \(n\in\mathbb N\)
Intuition: Angle multiplied by \(n\), distance from \(0\) power \(n\)

Question: The solutions of \(x^n=R\) are
Answer: \(\sqrt[n]Re^{i(\phi+2m\pi)/n}\)
Page: 58
Chapter: 2
Section: II.1
Variables: \(n\in\mathbb N\), \(Re^{i\phi}\)

Name: A foci of \(G\)
Definition: some \(a_1\) such that there exists \(a_2\) such that [$]\abs{g-a_1}\abs{g-a_2}[/$] is constant for all \(g\in G\)
Page: 60, 61b
Chapter: 2
Section: II.3
Variables: \(G\) a giovanni Cassini

Name: An ellipse with foci \(a_1\) and \(a_2\)
Definition: the set of \(z\) such that [$]\abs{z-a_1}+\abs{z-a_2}=l[/$], for some [$]l>2\abs{a_1-a_2}[/$]
Page: 61
Chapter: 2
Section: II.3
Variables: \(a_1,a_2\in\mathbb C\)

Name: A foci of \(L\)
Definition: some \(a_1\) such that there exists \(a_2\) such that [$]\abs{l-a_1}+\abs{l-a_2}[/$] is constant for each \(l\in L\)
Page: 61
Chapter: 2
Section: II.3
Variables: \(L\) an ellipse

Name: A Giovanni Cassini with foci \(a_1\) and \(a_2\)
Definition: the set of \(z\) such that [$]\abs{z-a_1}\cdot\abs{z-a_2}=k^2[/$] for some \(k\).
Page: 60, 61b
Chapter: 2
Section: II.3
Variables: \(a_1,a_2\in\mathbb C\)
Definition2: the set of solution of the quadratic polynomial \((z-a_1)(z-a_2)=k\)

Name: the \(n\)-th partial sum
Definition: \(z\mapsto\sum_{j=0}^nc_ij^i\)
Page: 67
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series
Notation: \(P_n\)

Name: A point of convergence of \(P\)
Definition: Some \(a\in\mathbb C\) such that \(P_n(a)\) converges
Page: 67
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series

Name: A point of divergence of \(P\)
Definition: Some \(a\in\mathbb C\) such that \(P_n(a)\) diverges
Page: 67
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series

Name: A point of absolute convergence of \(P\)
Definition: Some \(a\in\mathbb C\) such that \(\tilde P_n(a)\) converges
Page: 67
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series
Extra: [$]\tilde P_n(a)=\sum_j\abs{c_jz^j}[/$]

Definition: the power series [$]\sum_{j}\abs{c_jz^j}[/$]
Page: 68
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series
Context: complex analysis
Notation: \(\tilde P\)

Smaller0: \(P\) converges absolutely at \(a\)
Greater0: [$]r<\abs{a}[/$]
Page: 68, 70
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series
\(a,b\in\mathbb C\)
Greater1: \(P\) is uniformly and absolutely convergent of the closed disc of radius \(r\),
Connect Greater0: implies
Greatest: \(P\) converges at \(a\)

Name: circle of convergence of \(P\)
Definition: The circle, centered at 0, whose radius is the radius of convergence of \(P\)
Page: 69
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series

Name: disk of convergence of \(P\)
Definition: The interior of the circle of convergence of \(P\)
Page: 69
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series

Definition: [$]\defFun{\mathbb C}{\mathbb C}{z}{\abs{P(z)-P_m(z)}}[/$]
Page: 70
Chapter: 2I
Section: 6
Variables: \(P(z)=\sum_j c_jz^j\) a power series
\(m\in\mathbb N\)
Notation: \(E_m\)

Smaller0: \(\cosh(a+b)\)
Greater0: \(\cosh a\cosh b+\sinh a\sinh b\)
Page: 86
Chapter: 2V
Section: 2
Variables: \(a,b\in\mathbb C\)
Smaller to greater: =
Hide middle: yes

Smaller0: \(\sinh(a+b)\)
Greater0: \(\cosh a\sinh b+\cosh a\sinh b\)
Page: 86
Chapter: 2V
Section: 2
Variables: \(a,b\in\mathbb C\)
Smaller to greater: =
Hide middle: yes

Question: \(\sin(z+\frac\pi2)\)
Answer: equals \(\cos(z)\)
Page: 88
Chapter: 2V
Section: 3
Variables: \(a\in\mathbb C\)

Question: \(\cos(x+iy)\)
Answer: equals \(\cos x\cosh y-i\sin x\sinh y\)
Page: 89
Chapter: 2V
Section: 3
Variables: \(x,y\in\mathbb R\)

Name: many-valued function
Definition: a function which associate multiple value to each element of the domain
Page: 90
Chapter: 2VI
Section: 1
Context: Complex analysis
Example: \(\sqrt[3]z\) is a 3-valued function
Abbreviation: multifunction

Name: \(n\)-valued function
Definition: a function which associate \(n\)-values to each element of the domain
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
Context: Complex analysis

Question: Why is the function \(re^{i\theta}\mapsto {\sqrt[3]re^{i(\theta/3)}}\) ambiguous ?
Answer: Because \(\theta\) can takes multiple values
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural

Name: A branch point of \(f\)
Definition: A point \(z\) such that any small loop circling around \(z\) exactly once can not have a single valued restriction of \(f\) defined on it
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
\(f\) a \(n\)-valued function
Example: A branch point of \(\sqrt z\) is 0

Name: the order of \(z\)
Definition: the least number \(n\) such that, a loop circling \(n+1\) times around \(z\) can have a single valued restriction of \(f\) defined on the part where it does not cross itself
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
\(f\) a \(n\)-valued function
\(z\in\mathbb C\) a branch point
Context: Branch point

Name: An algebraic branch point of \(f\)
Definition: A branch point of finite order
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
\(f\) a \(n\)-valued function

Name: A simple branch point of \(f\)
Definition: A branch point of \(f\) of order 1
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
\(f\) a \(n\)-valued function

Name: A logarithmic branch point of \(f\)
Definition: A branch point which is not algebraic
Page: 90
Chapter: 2VI
Section: 1
Variables: \(n\) natural
\(f\) a \(n\)-valued function