Title: Theory / Unimportant / Math / Analysis / Complex / Ahlfors/
Tags: fromMath mathToSee mathToDef automaticImplies hasSeenConjDef récupéré derivableBounded duplicateToOtherBook examplesSwitchExample1 interval markedtrigoRelationToQuestion multipleBook

Smaller0: \[\frac{\alpha+i\beta}{\gamma+i\delta} \]
Greater0: \[\frac{(\alpha\gamma+\beta\delta)+i(\beta\gamma-\alpha\delta)}{\gamma^2+\delta^2} \]
Page: 001
Chapter: 1
Variables: \(\alpha,\beta,\gamma,\delta\in \mathbb R\)
Smaller to greater: =
Hide middle: yes

Smaller0: \[\frac1{\alpha+i\beta} \]
Greater0: \[\frac{\alpha-i\beta}{\alpha^2+\beta^2} \]
Page: 002
Chapter: 1
Section: 1.1
Variables: \(\alpha,\beta\in \mathbb R\).
Smaller to greater: =
Hide middle: yes

Definition: [$]
\pm
\left(
\sqrt{
\frac{
a
+
\sqrt{a^2+b^2}
}{2}
}
+
i\frac{
b
}{
\abs{b}
}
\sqrt{
\frac{
-a
+
\sqrt{a^2+b^2}
}2
}
\right)[/$]
Page: 003
Chapter: 1
Section: 2
Variables: \(a,b\in \mathbb R\).
Notation: \(\sqrt{a+ib}\)
Index: 6

Name: The conjugate of \(a+ib\)
Definition: \(a-ib\).
Page: 007
Chapter: 1
Section: 1.4
Variables: \(a,b\in \mathbb R\).
Notation: \(\overline{a+ib}\)

Definition: \(c\) is real and positive.
Page: 009
Chapter: 1
Section: 1.5
Variables: \(c\in\mathbb C\).
Notation: \(c>0\)

\(-|a|\le\text{Im }a, \text{Re }a\le|a|\)
Page: 009
Chapter: 1
Section: 1.5
Variables: \(a\in\mathbb C\).

Smaller0: \(|a+b|\)
Greater0: \(|a|+|b|\)
Page: 009
Chapter: 1
Section: 1.5
Variables: \(a,b\in\mathbb C\).
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Q: [...] implies [...].
A: \(\frac{a_i}{a_j}>0\) for all non-zero \(i,j\) distinct. I.e. they are all colinear with 0 and on the same side of 0; \(|\sum a_i|=\sum |a_i|\)
Page: 009
Chapter: 1
Section: 1.5
Variables: \((a_0,\dots,a_n)\in\mathbb C\).
\((a_i)\) be a finite sequence of complex number.

Name: The extended complex plane
Definition: \(\mathbb C\cup\{\infty\}\)
Page: 018
Chapter: 1
Section: 2.4
Representation: the Rieman sphere of triples \((x,y,z)\) such that \(|x|^2+|y|^2+|z|^2=1\)

Q: [...] can be represented as [...].
A: The extended complex plane; the Rieman sphere of triples \((x,y,z)\) such that \(|x|^2+|y|^2+|z|^2=1\)
Page: 018
Chapter: 1
Section: 2.4

Q: [...] is equal to
[...]
A: The distance between the stereographic projection of two complex \(z\) and \(z'\); \[\frac{2|z-z'|}{\sqrt{(1+|z|^2)(1+|z'|^2)} }\]
Page: 020
Chapter: 1
Section: 2.4
Variables: \(z,z'\in\mathbb C\)

Name: \(f\) have the limit \(A\) as \(x\) tends to \(a\)
Definition: For every \(\epsilon>0\) there exists a number \(\delta>0\) with the property that \(|f(x)-A|<\epsilon\) for all values of \(x\) such that \(|x-a|<\delta\) and \(x\ne a\).
Page: 022
Chapter: 2
Section: 1.1
Variables: \(a,A,x\in\mathbb C\)
\(f:\mathbb C\to\mathbb C\).
Notation: \(\lim_{x\to a}f(x)=A\)
Index: 1
Kind: Def

Smaller0: \(f'\) is defined
Greater0: \(f'=0\)
Page: 023
Chapter: 2
Section: 1.1
Variables: \(f:\mathbb C\to\mathbb R\).

Question: \(z\) can be written
Answer: uniquely as \(x(t)+iy(t)\), with \(x,y:X\to\mathbb R\)
Page: 024
Chapter: 2
Section: 1.1
Variables: \(z:X\to\mathbb C\)

Name: An analytic function
Definition: A complex function which admits a derivative where it is defined.
Page: 024, 25
Chapter: 2
Section: 1.2
Index: ,(6)
Definition2: A function satisfying the Cauchy Riemann differential equation

Smaller0: [$$]\partialderivative{f(x+iy)}{x} [/$$]
Greater0: \[\lim_{k\to0, k\in\mathbb R} \frac{f(x+iy+k)-f(x+iy)}{k}\]
Variables: \(f:\mathbb C\to\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: [$]\partialderivative{f(x+iy)}{y}[/$]
Greater0: \(\lim_{k\to0, k\in\mathbb R} \frac{f(x+i(y+k))-f(x+iy)}{ik}\)
Chapter: 2
Section: 1.1
Variables: \(f:\mathbb C\to\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: [$]\partialderivative{f}{x} [/$]
Greater0: [$]-i\partialderivative{f}{y} [/$]
Page: 025
Chapter: 2
Section: 1.2 (6)
Variables: \(f:\mathbb C\to\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: [$]\partialderivative{u}{y} [/$]
Greater0: [$]-\partialderivative{v}{x} [/$]
Page: 025
Chapter: 2
Section: 1.2 (6)
Variables: \(f(z)=u(z)+iv(z)\) a differentiable complex function.
Smaller to greater: =
Hide middle: yes

Smaller0: \(f'(z)\)
Greater0: [$]\partialderivative{u}{x}+i\partialderivative{v}{y}[/$]
Page: 025
Chapter: 2
Section: 1.1
Variables: \(f(z)=u(z)+iv(z)\) a differentiable complex function.
Index: (4)
Smaller to greater: =
Hide middle: yes

Smaller0: \(|f'(z)|^2 \)
Greater0: [$]\derivative{u}{x}\derivative{ v}{ y}-\derivative{ u}{ y}\derivative{ v}{ x} [/$]
Page: 025
Chapter: 2
Section: 1.1
Variables: \(f(z)=u(z)+iv(z)\) a differentiable complex function.
Smaller to greater: =
Hide middle: yes

Name: Laplace's equation
Definition: [$]\partialderivative[2]{f}{x}+\partialderivative[2]{f}{y}[/$].
Page: 025
Chapter: 2
Section: 1.1
Variables: \(f:\mathbb C\to\mathbb C\).
Notation: \(\Delta f\)

Name: Harmonic function
Definition: \(u:\mathbb C\to\mathbb C\) which satisfies \(\Delta u=0\)
Page: 025
Chapter: 2
Section: 1.2

[$]\partialderivative{u}{x} =\partialderivative{v}{y}[/$], and [$]\partialderivative{v}{\partial x} =-\partialderivative{\partial u}{\partial y} .[/$]
Page: 025
Chapter: 2
Section: 1.2
Variables: \(u:\mathbb C\to\mathbb C\).
Index: (6)
Q: [...]
A: The Cauchy Riemann differential equation

Name: A rational function
Definition: assumed without common factors
Page: 030
Chapter: 2
Section: 1.4
Conjdef: the quotient of two polynomials

Name: The poles of \(u\)
Definition: The zeros of its denominator.
Page: 030
Chapter: 2
Section: 1.4
Variables: \(u\) a rational function.

Question: The poles of \(u'\) equals
Answer: the poles of the function with order incremented,
Page: 030
Chapter: 2
Section: 1.4
Variables: \(u\) a rational function.

Definition: \(\lim_{x\to 0} R\left(\frac 1x\right)\)
Page: 030
Chapter: 2
Section: 1.4
Variables: \(R\) a function from the complex plane to the extended complex plane
Notation: \(R(\infty)\)

Smaller0: \(R(\frac 1z)\)
Greater0: \(z^{m-n}\frac{\sum_{i=0}^n a_i z^{n-i} }{\sum_{i=0}^m b_i z^{m-i} } \)
Page: 030
Chapter: 2
Section: 1.4
Variables: \(R(z)=\frac{\sum_{i=0}^n a_i z^i}{\sum_{i=0}^m b_i z^i}\).
Smaller to greater: =
Hide middle: yes

Q: Equivalently:
-[...],
-[...].
A: The number of zeros of \(u\), including the one at \(\infty\); its number of poles of \(u\), including the one at \(\infty\)
Page: 030
Chapter: 2
Section: 1.4
Variables: \(u\) a rational function.
Name: The order of the function

Q: [...].
A: The polynomial without constant term such that \(R-G\) is finite at \(\infty\)
Page: 031
Chapter: 2
Section: 1.4
Variables: \(R\) a rational function.
Index: (12)
Name: The singular part of \(R\)
Types: A polynomial whose degree is the order of the pole at \(\infty\)

Smaller0: \(b\) is a pole of \(R\)
Greater0: \(R\left(b+\frac 1\zeta\right)\)
Page: 031
Chapter: 2
Section: 1.4
Variables: \(R\) a rational function and \(b\in\mathbb C\)
Hide middle: yes
Greater1: a pole at \(\infty\)
Connect Greater0: has
Counterexample: contre exemple ?

Smaller0: \[R(z)\]
Greater0: \[G(z)+\sum_{i=1}^q G_i\left(\frac1{z-b_i}\right)\]
Page: 031
Chapter: 2
Section: 1.4
Variables: \(R\) be a rational function with distinct finite poles \(b_1,\dots,b_q\).
\(G\) be its singular part
\(G_i\) be the singular parts of \(R\left(b_i+\frac1\zeta\right)\).
Index: (14)
Smaller to greater: =
Hide middle: yes

Name: The radius of convergence
Definition: maximal such that \(\sum a_iz^i\) converge absolutery for every \(z\) with [$]\abs{z-k}<R[/$]. For \(0\le\rho<R\), the convergence is uniform for [$]\abs{z-k}\le \rho[/$]
Page: 69, 74
Chapter: 2I
Section: 6
Variables: \(\sum a_i(z-k)^i\) a power series.
Index: 2
Kind: Theorem
Definition2: minimal such that, for [$]\abs{z-k}>R[/$] the series diverge
Conjdef: A non-negative real \(0\le R\le\infty\), equivalently
Definition3: maximal such that, for [$]\abs{z-k}<R[/$] the sum of the series is an analytic function.
Extra: Ahlfors, Complex Analysis, Ch 2. 2.1. P 39
Visual complex analysis, CH 2l. 6. P69-74
Pugh, Real analysis, Ch 3. 3. P 197
Intuition: Distance between \(k\) the center of the definition and the nearest singularity
Definition4: [$]\frac{1}{\lim\sup_{k\to\infty}\sqrt[k]{\abs{c_k}} }[/$]
Definition5: [$]\lim_{k\to\infty}\frac{1}{\sqrt[k]{\abs{c_k}} }[/$] if it exists
Definition6: [$]\lim_{n\to\infty}\abs{\frac{c_n}{c_{n+1}}}[/$] if it exists

Question: The derivation of \(s\) can be obtained
Answer: by termwise differentiation
Page: 034
Chapter: 2
Section: 2.1
Variables: \(s=\sum a_iz^i\) a series.
Kind: Theorem

Question: The radius of convergence of the derivation of \(s\)
Answer: is the same than the one of \(s\).
Page: 034
Chapter: 2
Section: 2.1
Variables: \(s=\sum a_iz^i\) a series.
Kind: Theorem

Smaller0: \(f^{(k)}(z)\)
Greater0: \(\sum_{i\in\mathbb N}\frac{(k+i)!}{i!}a_{k+i}z^i \)
Page: 041
Chapter: 2
Section: 2.4
Variables: \(f(z)=\sum_{i\in\mathbb N} a_iz^i\)
Smaller to greater: =
Hide middle: yes

Q: \([...]=\sum f^{(i)}(0) [...]\)
A: f(z); \frac{z^i }{i!}
Page: 041
Chapter: 2
Section: 2.4
Variables: \(f(z)=\sum a_iz^i\)

Smaller0: \(\sum a_n\) converges,
Greater0: \(f(z)=\sum a_nz^n\)
Page: 042
Chapter: 2
Section: 2.4
Index: 3
Kind: Theorem
Name: Abel's limit theorem
Greater1: \(\frac{|1-z|}{1-|z|}\) remains bounded
Connect Greater0: tends to \(f(1)\) in such a way that

Question: \(|e^{iy}|=\)
Answer: \(1\)
Page: 044
Chapter: 2
Section: 3.1

Question: \(|e^{x+iy}|=\)
Answer: \(e^x\)
Page: 044
Chapter: 2
Section: 3.1

Smaller0: \[\sin z\]
Greater0: \[\frac{e^{iz}-e^{-iz} }{2i} \]
Page: 044
Chapter: 2
Section: 3.2
Index: (25)
Smaller to greater: =
Hide middle: yes
Greater1: \[\sum_k\frac{(-1)^{k}z^{2k+1} }{(2k+1)!} \]
Connect Greater0: \[=\]

Smaller0: \(e^{iz}\)
Greater0: \(\cos z+i\sin z\)
Page: 044
Chapter: 2
Section: 3.2
Smaller to greater: =
Hide middle: yes

Question: \[\sin'=\]
Answer: \[\cos\]
Page: 044
Chapter: 2
Section: 3.2
Index: (28)

Smaller0: \(\cos(a+b)\)
Greater0: \(\cos a\cos b-\sin a\sin b\)
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

Smaller0: \[\tan z\]
Greater0: \[-i \frac{e^{iz}-e^{iz} }{e^{iz}+e^{-iz} } \]
Page: 045
Chapter: 2
Section: 3.2
Smaller to greater: =
Hide middle: yes
Greater1: \[\frac{\sin z}{\cos z} \]
Connect Greater0: \[=\]

Name: A function with the period \(c\)
Definition: \(f\) such that \(f(z+c)=f(z)\).
Page: 045
Chapter: 2
Section: 3.3
Variables: \(c\in\mathbb C\).

Question: \(e^{\pi i/2}=\)
Answer: \(i\)
Page: 046
Chapter: 2
Section: 3.3

Question: \(e^{\pi i}=\)
Answer: \(-1\)
Page: 046
Chapter: 2
Section: 3.3

Question: \(e^{2\pi i}=\)
Answer: \(1\)
Page: 046
Chapter: 2
Section: 3.3

Q: The exponential function is [...] between [...] of real numbers and [...]
A: an homomorphism; the additive group; the multiplicative group of complex numbers with absolute value 1.
Page: 046
Chapter: 2
Section: 3.3

Q: [...]. It is interpreted [...]
A: The imaginary part of \(\log w\); geometrically as the angle, mesured in radians between the positive real axis and the half line from 0 to \(w\)
Page: 046
Chapter: 2
Section: 3.4
Variables: \(w\in\mathbb C\).
Name: The argument of \(w\)
Notation: arg \(w\)

Definition: \(\log |w|+i\text{arg }w\)
Page: 047
Chapter: 2
Section: 3.4
Variables: \(w\in\mathbb C\).
Notation: \(\log(w)\)

Question: \(a^b\) is defined as
Answer: \(\exp(b\log a)\).
Page: 047
Chapter: 2
Section: 3.4
Variables: \(a,b\in\mathbb C\) with \(a>0\).

Smaller0: \(\text{ arc sin }w\)
Greater0: \(\frac \pi 2- \text{arc cos }w \)
Page: 048
Chapter: 2
Section: 3.4
Smaller to greater: =
Hide middle: yes

Definition: The complement of \(X\).
Page: 050
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)
Notation: \(\sim X\)

Name: The closure of \(X\)
Definition: The set of points \(x\) such that all neighboorhod intersects \(X\).
Page: 053
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)
Notation: \(X^-\)
Index: ii
Abbreviation: \(Cl X\)

Q: [...] is [...]
A: The exterior of a set \(X\); \(\sim Int X\)
Page: 053
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)
Index: iv

Q: A point \(x\in X\) is [...] if [...].
A: an isolated point; it admits a neighboorhod \(N\) such that \(N\cap X=\{x\}\)
Page: 053
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)

Name: An accumulation point
Definition: A point of \(X^-\) which is not an isolated point
Page: 053
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)

Smaller0: connected
Greater0: any two points can be joined by a polygon which lies in the set
Page: 056
Chapter: 3
Section: 1.3
Variables: \(X\subseteq \mathbb C\)
Kind: Theorem
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: \(X\) is

Q: \(X\) has a [...] decomposition into components.
A: unique
Page: 057
Chapter: 3
Section: 1.3
Variables: \(X\subseteq \mathbb C\)
Index: 4
Kind: Theo

Name: A component of \(X\)
Definition: A maximal connected subset of \(X\).
Page: 057
Chapter: 3
Section: 1.3
Variables: \(X\subseteq \mathbb C\)

Smaller0: \(X\) is open
Greater0: open
Page: 058
Chapter: 3
Section: 1.3
Variables: \(X\subseteq\mathbb R^n\)
\(S\) a component of \(X\)
Index: 5
Kind: Theo
Prefix Greater: \(S\) is

Name: A totally bounded subset of \(X\)
Definition: A set \(X\subseteq\mathbb C\) such that for every \(\epsilon>0\), \(X\) can be covered by finitely many \(\epsilon\)-neighborhood
Page: 061
Chapter: 3
Section: 1.3
Variables: \(X\subseteq \mathbb C\).
Index: 7
Kind: Def

Name: A limit point of \(X\)
Definition: A limit point of a sequence of elements of \(X\) with possible repetitions.
Page: 062
Chapter: 3
Section: 1.3
Variables: \(X\subseteq \mathbb C\)
Name2: A cluster value \(X\)

A metric space is compact iff
-every infinite sequence has a limit point (Bolzano-Weierstrass theorem)
-it is complete and totally bounded.
Page: 062
Chapter: 3
Section: 1.4
Index: 6, 7
Kind: Theorem

Q: Under [...], [...] is [...].
A: a continuous mapping; the image of every compact set; compact
Page: 065
Chapter: 3
Section: 1.5
Index: 8
Kind: Theorem

Smaller0: \(f\) is continuous, \(S\) is connected
Greater0: \(f(S)\) is connected
Page: 065
Chapter: 3
Section: 1.5
Variables: \(X,Y\) topological sets, \(f:X\to Y\) and \(S\subseteq X\).
Index: 9
Kind: Theorem
Hide middle: yes

Name: A Hausdorff space
Definition: A topology in which any two distinct points are contained in disjoint open sets.
Page: 067
Chapter: 3
Section: 1.6
Index: 9
Kind: Def

Q: [...] is [...].
A: An arc; a complex function whose domain is \([a,b]\)
Page: 068
Chapter: 3
Section: 2.1

Smaller0: the derivative of \(z(t)\) exists and is non null
Greater0: the tangeant of \(z(t)\)
Page: 069
Chapter: 3
Section: 2.1
Variables: \(z:[a,b]\to\mathbb C\) an arc
Greater1: is determined by \(arg (z'(t))\)

Name: Differentiable arc
Definition: An arc whose derivative exists and is continuous.
Page: 069
Chapter: 3
Section: 2.1
Variables: \(z:[a,b]\to\mathbb C\) an arc

Name: Piecewise differentiable
Definition: An arc which is differentiable, except for a finite number of values \(t\) where left and right limits of derivative still exists
Page: 069
Chapter: 3
Section: 2.1
Name2: Piecewise regular

Name: A differentiable change of parameter
Definition: A change of parameter \(t=\phi(\tau)\) such that \(\phi'(\tau)\) is continuous.
Page: 069
Chapter: 3
Section: 2.1
Extra: régular ?

Q: \(z\) is [...] or [...], if [...].
A: simple; Jordan arc; \(z(t_1)=z(t_2)\) only for \(t_1=t_2\)
Page: 069
Chapter: 3
Section: 2.1
Variables: \(z:[a,b]\to\mathbb C\) an arc

Name: A closed curve
Definition: An arc whose endpoints coincide.
Page: 069
Chapter: 3
Section: 2.1
Variables: \(z:[a,b]\to\mathbb C\) an arc
Extra: loop ?

Name: A point curve
Definition: A constant arc \([a,b]\to\mathbb C\)
Page: 069
Chapter: 3
Section: 2.1
Context: Curve

Q: \(z\) is [...] if [...].
A: a circle; \(z(t)=a+re^{it}\) and \([a,b]=[0,2\pi]\)
Page: 069
Chapter: 3
Section: 2.1
Variables: \(z:[a,b]\to\mathbb C\) an arc

Q: [...] is [...].
A: The direction of the tangent of \(f\circ z\) at \(t_0\); \(\text{arg }f'(z(t_0))+\text{arg }z'(t_0)\)
Page: 073
Chapter: 3
Section: 2.3
Variables: \(z:[a,b]\to\mathbb C\) an arc
\(f:\mathbb C\to\mathbb C\)
Assuming: This value is defined

Q: \([...]\) is equal to [$]\abs{f'(z_0)}[/$]
A: \lim_{z\to z_0}\frac{|f(z)-f(z_0)|}{|z-z_0|}
Page: 074
Chapter: 3
Section: 2.3
Variables: \(z_0\in\mathbb C\),
\(f:\mathbb C\to\mathbb C\).
Q: [...]
A: the linear change of scale at \(z_0\) effected by the tranformation \(f\) is independent of the direction.

Q: The [...] associates [...] to [...].
A: stereographic projection; \(a+ib\in\mathbb C\) (respectively \(\infty\)); \((\frac{2a}{1+a^2+b^2}, \frac{2b}{1+a^2+b^2},\frac{a^2+b^2-1}{1+a^2+b^2})\) respectively \((0,0,1)\)
Page: 018
Chapter: 1
Section: 2.4

Q: The stereographic projection associates [...] to[...]
A: \((x,y,z)\ne(0,0,1)\); \(\frac{x+iy}{1-z}.\)
Page: 018
Chapter: 1
Section: 2.4

Smaller0: Let \((a,b,0)\) and \((x,y,z)\) linked by stereographical projection. The line they generate
Greater0: \((0,0,1)\)
Page: 018
Chapter: 1
Section: 2.4
Smaller to greater: contains
Hide middle: yes
Greater to smaller: belongs to

Name: A function analytic on \(\Omega\)
Definition: A function defined on \(\Omega\), which has a derivative at each point of \(\Omega\).
Page: 070
Chapter: 3
Section: 2.2
Variables: \(\Omega\subseteq\mathbb C\) a region and
\(f:\Omega\to\mathbb C\).
Index: 10
Kind: Definition

Name: A function which is analytic on \(\Omega\)
Definition: A function which is analytic on a region containing \(\Omega\).
Page: 070
Chapter: 3
Section: 2.2
Variables: \(\Omega\subseteq\mathbb C\) a point set and \(f:\Omega\to\mathbb C\).
Index: 11
Kind: Definition

Q: [...] and [...] are implicitely assumed to be defined on [...].
A: \(\sqrt x\); \(\log x\); \(\{x+iy\mid y\ne 0\text{ or } x>0\}\)
Page: 070
Chapter: 3
Section: 2.2
Variables: \(x\in\mathbb C\)

Name: A degenerate analytic function in \(\Omega\)
Definition: A function which reduces to a constant.
Page: 073
Chapter: 3
Section: 2.3
Variables: \(\Omega\subseteq \mathbb C\).

\(f'=0\), \(Im f\) is constant, \(Re f\) is constant, \(| f|\) is constant or \(arg f\) is constant implies \(f\) degenerates.
Page: 073
Chapter: 3
Section: 2.2
Variables: \(\Omega\subseteq \mathbb C\)
\(f:\Omega\to\mathbb C\).

Name: A complete metric space
Definition: A metric space such that every Cauchy sequence is convergent.
Page: 059
Chapter: 3
Section: 1.4
Index: 5
Kind: Definition

Smaller0: \[Re a\]
Greater0: \[\frac{a+\overline a}2\]
Page: 007
Chapter: 1
Section: 1.4
Variables: \(a\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: \[\text{Im }a\]
Greater0: \[\frac{a-\overline a}2\]
Page: 007
Chapter: 1
Section: 1.4
Variables: \(a\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: \[\overline {a+b} \]
Greater0: \[\overline{a}+\overline{b} \]
Page: 007
Chapter: 1
Section: 1.4
Variables: \(a,b\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: \[\overline {ab} \]
Greater0: \[\overline{a}\overline{b} \]
Page: 007
Chapter: 1
Section: 1.4
Variables: \(a,b\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: \[a\overline {a} \]
Greater0: \[|a|^2 \]
Page: 008
Chapter: 1
Section: 1.4
Variables: \(a\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Smaller0: \[|ab| \]
Greater0: \[|a||b| \]
Page: 008
Chapter: 1
Section: 1.4
Variables: \(a,b\in\mathbb C\).
Smaller to greater: =
Hide middle: yes

Q: Geometrically, [...] are [...].
A: \(a+b\) and \(a-b\); the diagonals of the parallelogram, with side \(a\) and \(b\)
Page: 011
Chapter: 1
Section: 2.1
Variables: \(a,b\in\mathbb C\).

Q: Geometrically, [...] is [...].
A: the function \(\overline \cdot\); the symmetry with respect to the real axis
Page: 011
Chapter: 1
Section: 2.1

Q: The polar coordinates is [...], equal to [...].
A: \(re^{i\phi}\); \(r(\cos\phi+i\sin \phi)\)
Page: 013
Chapter: 1
Section: 2.1
Variables: \(r,\phi\in\mathbb R\).

Smaller0: \((r(cos\phi+i\sin \phi))^n\)
Greater0: \(r^n(\cos n\phi+i\sin n\phi)\)
Page: 015
Chapter: 1
Section: 2.2
Variables: \(r,\phi\in\mathbb R\).
Smaller to greater: =
Hide middle: yes

Smaller0: \(a=r(cos\phi+i\sin \phi)\)
Greater0: \(a^{-1} \)
Page: 006
Chapter: 1
Section: 2.2
Variables: \(r,\phi\in\mathbb R\).
Greater1: \(r^{-1}(\cos (-\phi)+i\sin (-\phi))\)
Connect Greater0: \(=\)

Q: \(z\) admits [...] \(n\)th root. They have [...] and [...].
A: \(n\); the same modulus; their argument are equally spaced
Page: 016
Chapter: 1
Section: 2.2
Variables: \(n\in\mathbb N\) and \(z\in\mathbb C\setminus \{0\}\).

Q: The algebraic equation [...] are [...] or [...].
A: of a circle; \((z-a)(\overline z-\overline a)=r^2\); \(|z-a|=r\)
Page: 017
Chapter: 1
Section: 2.3

Q: The equation [...] is [...].
A: of a line; \(a+bt\) where \(a,b\in\mathbb C\) and \(t\) runs through all real values
Page: 017
Chapter: 1
Section: 2.3

Q: They are [...] if [...]
A: parallel; \(b'\) is a real multiple of \(b\)
Page: 017
Chapter: 1
Section: 2.3
Variables: \(a,b\in\mathbb C\). Let \(a+bt\) and \(a'+b't\) be two lines.

Smaller0: equally oriented
Greater0: they are parallel and \(b'\) is a positive multiple of \(b\)
Page: 017
Chapter: 1
Section: 2.3
Variables: \(a,b\in\mathbb C\)
\(a+bt\) and \(a'+b't\) be two lines.
Smaller to greater: iff
Prefix Smaller: they are

Q: Its [...] is [...].
A: direction; the argument of \(b\)
Page: 017
Chapter: 1
Section: 2.3
Variables: \(a,b\in\mathbb C\). Let \(a+bt\) be a line.

Q: They are [...] if [...].
A: orthogonals; \(b/b'\) is purely imaginary
Page: 017
Chapter: 1
Section: 2.3
Variables: \(a,b\in\mathbb C\)
\(a+bt\) and \(a'+b't\) be two lines.

Question: \(a+\infty=\)
Answer: \(\infty\)
Page: 018
Chapter: 1
Section: 1.5
Variables: \(a\in\mathbb C\).

Question: \(b\infty=\)
Answer: \(\infty\)
Page: 018
Chapter: 1
Section: 1.5
Variables: \(b\in\mathbb C\setminus\{0\}\)

Question: \(\frac b0=\)
Answer: \(\infty\)
Page: 018
Chapter: 1
Section: 1.5
Variables: \(b\in\mathbb C\setminus\{0\}\)

Question: \(\frac c\infty=\)
Answer: \(0\)
Page: 018
Chapter: 1
Section: 1.5
Variables: \(c\in\mathbb C\)

Q: The stereographical projection sends [...] into [...].
A: straight lines in the \(z\) plane; a circle on \(S\) which passes through the pole (0,0,1)
Page: 020
Chapter: 1
Section: 2.4

Q: They are [...] if [...]
A: equal; \((a-a')\) and \(b'\) are real multiples of \(b\)
Page: 017
Chapter: 1
Section: 2.3
Variables: \(a+bt\) and \(a'+b't\) be two lines.

Name: A linear (fractional) transformation
Definition: \(f(x)=\frac{az+b}{cz+d}\) with \(ad\ne bc\).
Page: 076
Chapter: 3
Section: 3.1

Name: A normalized rational fraction function
Definition: \(f(x)=\frac{az+b}{cz+d}\) with \(ad-bc=1\).
Page: 076
Chapter: 3
Section: 3.1

Question: The number of normalized representation of this function is
Answer: 2.
Page: 076
Chapter: 3
Section: 3.1
Variables: \(f(x)=\frac{az+b}{cz+d}\).

Question: The matrix representation of \(f\) is
Answer: \(\left(\begin{array}{ll}a&b\\c&d\end{array}\right)\).
Page: 076
Chapter: 3
Variables: \(f(x)=\frac{az+b}{cz+d}\).

Definition: The set of linear transformation.
Page: 077
Chapter: 3
Typ: A group under which opération?

Smaller0: the function sending those variables to 1,0, \(\infty\)
Greater0: \[\frac{(z-z_3)(z_2-z_4)}{(z-z_4)(z_2-z_3)}\] if no variables are \(\infty\), otherwise, the parenthesis with the variable at \(\infty\) are removed.
Page: 078
Chapter: 3
Section: 3.2
Variables: \(z_2,z_3,z_4\) distinct in the extended complex plane.
Smaller to greater: is equal to
Hide middle: yes

Question: \((Tz_1,Tz_2,Tz_3,Tz_4)=\)
Answer: \((z_1,z_2,z_3,z_4)\).
Page: 078
Chapter: 3
Section: 3.2
Variables: \(z_1,z_2,z_3,z_4\) in the extended complex plane
\(z_2,z_3,z_4\) distinct
\(T\) a linear tranformation
Index: 12
Kind: Theo

Smaller0: \((z_1,z_2,z_3,z_4)\) is real
Greater0: the four points lie on a circle or on a straight line
Page: 078
Chapter: 3
Section: 3.2
Variables: \(z_1,z_2,z_3,z_4\) in the extended complex plane
Index: 13
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Assuming: \(z_2,z_3,z_4\) distinct

Question: Linear transformation preserves
Answer: circles of the extended plane.
Page: 079
Chapter: 3
Section: 3.2
Index: 14
Kind: Theo

Q: Then \(T\) transform [...] into [...].
A: pairs of symmetric points with respect to \(C_1\); pairs of symmetric points with respect to \(C_2\)
Page: 082
Chapter: 3
Section: 3.3
Variables: \(C_1,C_2\) circles of the extended planes and \(T\) transforming \(C_1\) into \(C_2\).
Index: 15
Kind: Theo
Name: The symmetry principle
Extra: reformuler

Name: A complex function conformal (at \([x]\))
Definition: A complex function which preserve angles (at \(x\)).
Page: 074
Chapter: 3
Section: 2.3
Example: \(w=z^\alpha\) with \(\alpha>0\)

Name: A complex function indectly conformal (at \([x]\))
Definition: A complex function which inverses angles (at \(x\)).
Page: 074
Chapter: 3
Section: 2.3

Q: \[S'(z)=[...].\]
A: \frac{ad-bc}{(cz+d)^2}
Page: 083
Chapter: 3
Section: 3.4
Variables: \(Sz=\frac{az+b}{cz+d}\)

Q: [...] is [...].
A: The right side of \(C\); the set of points \(z\) such that \(Im(z,z_1,z_2,z_3)>0\)
Page: 083
Chapter: 3
Section: 3.4
Variables: \(z_1,z_2,z_3\) three distinct points in the extended plane on a circle \(C\).

Q: [...] is [characterization].
A: The right side of \(C\); the side containing \(\infty\)
Page: 084
Chapter: 3
Section: 3.4
Variables: \(z_1,z_2,z_3\) four distinct points in the complex plane on a circle \(C\).

Name: \(z\)-plane
Definition: The complex plane as the domain of the function
Page: ?

Definition: The solutions of \(\left|\frac{z-a}{z-b}\right|=\frac \rho{|k|}\) with \(\rho\) the radius.
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\)
Notation: \(C_2\)

Name: Circular net \(a\) and \(b\)
Definition: The union of \(C_1\) and \(C_2\)
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\)
Name2: Steiner circles \(a\) and \(b\)

Definition: The circles through \(a\) and \(b\).
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\)
Context: Complex analysis
Notation: \(C_1\)

Q: The number of [...] through \(c\) is [...].
A: \(C_1/C_2\) cercles; exactly one
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b,c\in\mathbb C\) distinct

Question: \(C\) and \(C'\) have the property
Answer: to meet at right angle.
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\) distinct. Let \(C\in C_1\), \(C'\in C_2\)

Q: [Property of reflexion].
A: The reflexion through a \(C_i\) sends a \(C_i\) to itself and a \(C_{i-1}\) to another \(C_{i-1}\)
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\) distinct.

Smaller0: sends \(a\) to \(b\)
Greater0: \(C\) is a \(C_2\)
Page: 85
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\) distinct and \(C\) a circle
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: the symmetry through \(C\)

Smaller0: \(T\) sends \(a\) to \(a'\) and \(b\) to \(b'\)
Greater0: it is of the form \(\frac{w-a'}{w-b'}=k\cdot\frac{z-a}{z-b}\)
Page: 86
Chapter: 3
Section: 3.5
Variables: \(a,b,a',b'\in\mathbb C\) and \(T\) a linear transformation.
Smaller to greater: iff
Hide middle: yes

Q: [...].
It [property].
A: \(\frac{w-a'}{w-b'}=k\cdot\frac{z-a}{z-b}\) with \(k>0\); sends \(C_1\) (for \(a,b\)) to \(C_1\) (for \(a',b'\))
Page: 86
Chapter: 3
Section: 3.5
Variables: \(a,b\in\mathbb C\) and \(T\) a linear transformation \(\frac{w-a'}{w-b'}=k\cdot\frac{z-a}{z-b}\).
Name: Hyperbolic linear transformation

Name: Parabolic linear transformation
Definition: A linear transformation which has a single fixed point in the extended complex plane.
Page: 86
Chapter: 3
Section: 5
Variables: \(a,b\in\mathbb C\)
\(T\) a linear transformation \(\frac{w-a'}{w-b'}=k\cdot\frac{w-a}{z-b}\).

Smaller0: \(a,b\) are the distinct fixed point of \(T\)
Greater0: \(\frac{w-a}{w-b}=k\cdot\frac{z-a}{z-b}\)
Page: 86
Chapter: 3
Section: 5
Variables: \(T\) a linear fractional transformation and \(a,b\in\mathbb C\)
Hide middle: yes
Prefix Greater: \(T\) is of the form
Extra: sauté au à section 4 par manque de compréhension

Name: A level curve of \(u\)
Definition: A curve of the form \(u(z)=w\) for some \(w\) fixed.
Page: 89
Chapter: 3
Section: 4.1
Variables: \(S\subseteq\mathbb C\),
\(T\subseteq\mathbb R\)
\(u: S\to T\)

Smaller0: \(u+iv\) is conformal
Greater0: the level curves of \(u\) and \(v\) are orthogonal
Page: 89
Chapter: 3
Section: 4.1
Variables: \(S\subseteq \mathbb C\)
\(T\subseteq\mathbb R\) two sets
\(u,v: S\to T\)

Name: A sector
Definition: \(\{z\in\mathbb C\mid \phi_1<\text{arg} z<\phi_2\}\).
Page: 90
Chapter: 3
Section: 4.1
Variables: \(\phi_1,\phi_2\in\mathbb R\)
\(0<\phi_2-\phi_1\le 2\pi\)
Notation: \(S(\phi_1,\phi_2)\)
Extra: Saute chapitre 4, je comprend rien

Definition: [$]\int_a^b u(t)\dd t+i\int_a^b v(t)\dd t[/$].
Page: 102
Chapter: 4
Section: 1.1
Variables: \(u,v:\mathbb R\to\mathbb R\)
\(a,b\in\mathbb R\).
Notation: [$]\int_a^b(u+iv)(t)\dd t[/$]
Index: (1)

Smaller0: [$]|\int_a^bf(t)\dd t|[/$]
Greater0: [$]\int_a^b|f(t)|\dd t[/$]
Page: 102
Chapter: 4
Section: 1.1
Variables: \(f(t):\mathbb R\to\mathbb C\)
\(a<b\in\mathbb R\)
Index: (3)
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Definition: [$]\int_a^bf(z(t))z'(t)\dd t[/$].
Page: 102
Chapter: 4
Section: 1.1
Variables: \(f(t):\mathbb C\to\mathbb C\)
\(\gamma\) a piecewise differentiable arc with equation \(z=z(t)\)
\(a\le t\le b\).
Notation: [$]\int_\gamma f(t)\dd t[/$]
Index: (4)

Smaller0: \(z,t\) piecewise differentiable
Greater0: [$]\int_a^bf(z(t))z'(t)\dd t[/$]
Page: 103
Chapter: 4
Section: 1.1
Variables: \(f(t):\mathbb C\to\mathbb C\)
\(z:[a,b]\to\mathbb C\) a curve
\(t:[\alpha,\beta]\to[a,b]\).
Hide middle: yes
Smaller1: \(t\) increasing
Greater1: [$]\int_\alpha^\beta f(z(t(\tau)))z'(t(\tau))t'(\tau)\dd\tau[/$]
Connect Greater0: \(=\)

Definition: [$]\overline{\int_{\gamma}\overline{f}\dd z}[/$].
Page: 103
Chapter: 4
Section: 1
Variables: \(f(t):\mathbb C\to\mathbb C\)
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Notation: [$]\int_{\gamma} f\overline{\dd z}[/$]

Definition: [$]\frac12\left(\int_\gamma f\dd z+\int_\gamma f\overline{\dd z}\right)[/$].
Page: 103
Chapter: 4
Section: 1
Variables: \(f(x+iy):\mathbb C\to\mathbb C\)
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Notation: [$]\int_{\gamma} f\dd x[/$]

Definition: [$]\frac 1{2i}\left(\int_{\gamma}f\dd z-\int_\gamma f\overline{\dd z}\right)[/$].
Page: 103
Chapter: 4
Section: 1
Variables: \(f(x+iy):\mathbb C\to\mathbb C\),
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Notation: [$]\int_{\gamma} f\dd y[/$]

Name: Integration with respect to arc length
Definition: \(\int_\gamma f(z(t))|z'(t)|\ dt\)
Page: 104
Chapter: 4
Section: 1.1
Variables: \(f(z):\mathbb C\to\mathbb C\)
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Notation: [$]\int_{\gamma} f\abs{\dd z}[/$]
Definition2: \(\lim\sum_{k=1}^n f(z(t_k))|z(t_k)-z(t_{k-1})|\), limit taken on all \(a= t_0<\dots<t_n=b\).
Notation2: [$]\int_\gamma f\dd s[/$]

Question: [$]\int_{-\gamma} f\abs{\dd z}[/$]=
Answer: [$]\int_{\gamma} f\abs{\dd z}[/$]
Page: 104
Chapter: 4
Section: 1.1
Variables: \(f(z):\mathbb C\to\mathbb C\), \(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.

Smaller0: \(\left|\int_{\gamma} f dz\right|\)
Greater0: [$]\int_{\gamma} |f||\dd z|[/$]
Page: 104
Chapter: 4
Section: 1.1
Variables: \(f(z):\mathbb C\to\mathbb C\)
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Name: A rectifiable arc
Definition: A \(\gamma:[a,b]\to\mathbb C\) such that \(\sum_{i=0}^{n-1}|z(t_{i+1})-z(t_i)|\) is bounded, for all finite sequences \(a=t_0<\dots<t_n=b\).
Page: 104
Chapter: 4
Section: 1.2

Q: [...] if [...].
A: \(S'\) is a neighborhood of \(x\); there is \(\delta>0\) such that \(B(x,\delta)\subseteq S'\)
Page: 52
Chapter: 3
Section: 1
Variables: \(S\) a metric space, \(x\in S\), \(S'\subseteq S\).

Q: [...] [...] [rectifiable and piecewise differentiable].
A: \(z\) is piecewise differentiable; implies; \(z\) is rectifiable
Page: 105
Chapter: 4
Section: 1.2
Variables: \(z:[a,b]\to\mathbb C\)

Name: A function of bounded variation
Definition: A function \(x:[a,b]\to\mathbb R\) such that \(\sum_{i=1}^n |x(t_i)-x(t_{i-1})|\) is bounded for all subdivisions \(a=t_0< \dots< t_{n}=b\).
Page: 105
Chapter: 4
Section: 1.2

Smaller0: \(x\) and \(y\) are of bounded variations
Greater0: \(x+iy\) is rectifiable
Page: 105
Chapter: 4
Section: 1.2
Variables: \(x,y:[a,b]\to\mathbb R\).
Smaller to greater: iff

Smaller0: [$]p\dd x+q\dd y[/$] is an exact differential
Greater0: [$]\int_\gamma p\dd x+ q\dd y[/$] depends only on the end points of \(\gamma\)
Page: 106
Chapter: 4
Section: 1.3
Variables: \(\Omega\subseteq\mathbb C\)
\(p,q:\Omega\to\mathbb C\)
\(\gamma\) a rectifiable arc included in \(\Omega\).
Index: 1
Kind: Theo
Hide middle: yes

Name: An exact differential
Definition: \(p\ dx+q\ dy\) such that there exists \(U:\Omega\to\mathbb C\) such that \(p=\partial U/\partial x\) and \(q=\partial U/\partial y\).
Page: 107
Chapter: 4
Section: 1.3
Variables: \(\Omega\subseteq\mathbb C\).

Definition: \((\partial U/\partial x)dx+(\partial U/\partial y)dy\).
Page: 107
Chapter: 4
Section: 1.3
Variables: \(\Omega\subseteq\mathbb C\)
\(U:\Omega\to\mathbb C\)
Notation: \(dU\)

Smaller0: \(n\) is (not) \(-1\)
Greater0: 0 (\(2k\pi n\), with \(k\) the number of turn around \(a\))
Page: 107
Chapter: 4
Section: 1.3
Variables: \(a\in\mathbb C\)
\(n\in\mathbb Z\)
\(\gamma\) a closed curve.
Hide middle: yes
Prefix Greater: \(\int_\gamma(z-a)^n\) is
Extra: Todo

Definition: The closed curve \((a,c), (b,c), (b,d), (a,d)\).
Page: 109
Chapter: 4
Section: 1.3
Variables: \(R=\{[a,b]\otimes [c,d]\}\)
Notation: \(\partial R\)

Smaller0: \(\gamma\) is closed
Greater0: [$]\int_{\gamma}f(z)\dd z=0[/$]
Page: 113
Chapter: 4
Section: 1.3
Variables: \(\Delta\) an open disk
\(f:\Delta\to\mathbb C\)
\(\gamma\) a curve in \(\Delta\).
Index: 5
Kind: Theo
Hide middle: yes
Smaller1: \(f\) is analytic, apart on a finite number of removable points

Name: The index of \(a\) with respect to \(\gamma\)
Definition: [$]\frac{1}{2\pi i}\int_\gamma\frac{\dd z}{z-a}[/$].
Page: 115
Chapter: 4
Section: 2.1
Variables: \(a\in\mathbb C\)
\(\gamma\) a closed curve.
Notation: \(n(\gamma,a)\)
Name2: winding number of \(\gamma\) with respect to \(a\)

Smaller0: \(n(-\gamma,a)\)
Greater0: \(-n(\gamma,a)\)
Page: 115
Chapter: 4
Section: 2.1
Variables: \(a\in\mathbb C\), \(\gamma\) a closed curve.
Smaller to greater: =
Hide middle: yes

Smaller0: \(\gamma\) lies inside of a circle, \(a\) outside of it
Greater0: \(n(\gamma,a)=0\)
Page: 116
Chapter: 4
Section: 2.1
Variables: \(a\in\mathbb C\)
\(\gamma\) a closed curve.
Counterexample: multiple if

Smaller0: \(\gamma\)
Greater0: the regions of \(\mathbb C\setminus\gamma\)
Page: 116
Chapter: 4
Section: 2.1
Variables: \(\gamma\) a closed curve
Smaller to greater: determines
Greater to smaller: are determined by

Question: The number of unbounded region determined by \(\gamma\) is
Answer: 1.
Page: 116
Chapter: 4
Section: 2.1
Variables: \(\gamma\) a closed curve

Smaller0: \(a,a'\) belongs to the same region determined by \(\gamma\)
Greater0: \(n(\gamma,a)=n(\gamma,a')\)
Page: 116
Chapter: 4
Section: 2.1
Variables: \(\gamma\) a closed curve.
\(a,a'\in\mathbb C\) points

Q: \(n(\gamma,a)\) is [...] in [...].
A: 0; the unbounded region determined by \(\gamma\)
Page: 116
Chapter: 4
Section: 2.1
Variables: \(\gamma\) a closed curve

Smaller0: \(\gamma=\gamma_1\cup\gamma_2\), with \(\gamma_1\) (\(\gamma_2\) from lower-half to top-half (and vice-versa) not passing through the negative (positive) real axis
Greater0: \(n(\gamma,0)=1\)
Page: 116
Chapter: 4
Section: 2.1
Variables: \(\gamma\) a closed curve
Index: 2
Kind: Lem
Hide middle: yes
Suffix Smaller: ,

Smaller0: \(n(\gamma,a)\cdot f(a)\)
Greater0: [$$]\frac1{2\pi i}\int_\gamma\frac{f(z)\dd z}{z-a} [/$$]
Page: 119
Chapter: 4
Section: 2.2
Variables: \(\Delta\) an open disk,
\(f:\Delta\to\mathbb C\),
\(a\in \Delta\),
\(\gamma\) a closed curve included in \(\Delta\setminus\{a\}\).
Index: 6
Kind: Theo
Smaller to greater: =
Hide middle: yes
Assuming: \(f\) is analytic in \(\Delta\), apart from a finite number of removable point

Smaller0: \(f\) is analytic on \(\Delta\), apart on a finite number of removable point
Greater0: [$]\frac{1}{2\pi i}\int_\gamma\frac{f(\xi)\dd \xi}{\xi-z} [/$]
Page: 119
Chapter: 4
Section: 2.2
Variables: \(\Delta\) an open disk,
\(f:\Delta\to\mathbb C\),
\(\gamma\) a closed curve included in \(\Delta\)\(z\in\Delta\)
Index: (22)
Hide middle: yes
Prefix Greater: \(f(z)=\)
Smaller1: \(n(\gamma,z)=1\)
Name: Cauchy integral formula for \(f\)
Connect Smaller0: and

Definition: [$]\int_\gamma\frac{f(\xi)\dd \xi}{(\xi-z)^n}[/$]
Page: 121
Chapter: 4
Section: 2.2
Variables: \(\Delta\) an open disk,
\(\gamma\) a closed curve included in \(\Delta\setminus\{a\}\),
\(f:\gamma \to\mathbb C\)
\(n\in\mathbb N\)
Notation: \(F_n(z)\)
Index: 3
Kind: Lem
Assuming: \(f\) is continuous on \(\gamma\)

Smaller0: \(f\) is continuous on \(\gamma\)
Greater0: \(F'_n(z)\)
Page: 121
Chapter: 4
Section: 2.2
Variables: \(\Delta\) an open disk,
\(f:\Delta\to\mathbb C\),
\(\gamma\) a closed curve included in \(\Delta\setminus\{a\}\) and
\(n\in\mathbb N\)
Index: 3
Kind: Lem
Hide middle: yes
Greater1: \(nF_{n+1}(z)\)
Connect Greater0: \(=\)

Smaller0: \(f\) is continuous on \(\gamma\)
Greater0: analytic
Page: 121
Chapter: 4
Section: 2.2
Variables: \(\Delta\) an open disk, \(\gamma\) a closed curve included in \(\Delta\), \(f:\gamma\to\mathbb C\), and \(n\in\mathbb N\)
Index: 3
Kind: Lem
Hide middle: yes
Prefix Greater: \(F_n(z)\) is
Greater1: each region determined by \(\gamma\)
Connect Greater0: in

Smaller0: \(f\) is analytic and bounded
Greater0: \(f\) is constant
Page: 122
Chapter: 4
Section: 2.2
Variables: \(f:\mathbb C\to\mathbb C\).
Hide middle: yes
Name: Liouville's theorem

Smaller0: \(f\) is analytic and \(a\) is a removable singularity
Greater0: \(f\) admits a unique analityc extension on \(\Omega\)
Page: 124
Chapter: 4
Section: 3.1
Variables: \(\Omega\subseteq \mathbb C\) a region,
\(a\in\Omega\)
\(f:\Omega\setminus\{a\}\to\mathbb C\).
Index: 7
Kind: Theo

Smaller0: \(f\) is analytic
Greater0: \(\sum_{i=0}^{n-1}\frac{f^{(i)}(a) }{i!}(z-a)^i+f_n(z)(z-a)^n\)
Page: 125
Chapter: 4
Section: 3.1
Variables: \(\Omega\) a region,
\(\Delta\subseteq\Omega\) an open disk of contour \(C\),
\(f:\Omega \to\mathbb C\) and
\(n\in\mathbb N\)
Prefix Greater: \(f(z)\) is equal to
Name: Taylor Theorem, finite sum

Q: [...], with \(z\in[...]\).
A: [$]\frac{1}{2\pi i}\int_C\frac{ f(\xi)\dd \xi}{(\xi-a)^n(\xi -z)}[/$]; \Delta
Page: 126
Chapter: 4
Section: 3.1
Variables: \(\Omega\) a region, \(\Delta\subseteq\Omega\) an open disk of contour \(C\), \(f:\Omega \to\mathbb C\) and \(n\in\mathbb N\)
Notation: \(f_n(z)\)
Types: Analytic on \(\Delta\)

Smaller0: \(f\) is analytic on \(\Omega\) and \(f^i(a)=0\) for each \(i<n\)
Greater0: \(f_n(z)(z-a)^n\)
Page: 126
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region,
\(\Delta\subseteq\Omega\) an open disk of contour \(C\),
\(a\in \Delta\),
\(f:\Omega \to\mathbb C\)
\(n\in\mathbb N\)
Hide middle: yes
Prefix Greater: \(f(z)=\)

Smaller0: \(f\) is analytic on \(\Omega\), and \(0\) on \(\Delta\)
Greater0: \(f(z)=0\) on \(\Omega\)
Page: 127
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region
\(\Delta\subseteq\Omega\) a region
\(f:\Omega \to\mathbb C\)
Hide middle: yes

Name: A zero of order \(n\) of \(f\)
Definition: \(a\in\Omega\) such that \(f^{(i)}(a)=0\) for all \(i<n\) and \(f^{(n)}\ne0\).
Page: 127
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region
\(f:\Omega \to\mathbb C\)
\(n\in\mathbb N\)

Name: A pole of \(f\) of order \(n\)
Definition: An element \(a\in\Omega\) such that \(\lim_{z\to a}\frac{f(z)}{(z-a)^i}=\infty\) for all \(i<n\) and \(\lim_{z\to a}\frac{f(z)}{(z-a)^n}\in\mathbb C\).
Page: 127
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region,
\(f:\Omega\setminus\{a\} \to\mathbb C\) continuous
\(n\in\mathbb N\)

Q: Assuming [...]: \(a\) is [name].
A: \(f\) is analytic; an isolated singularity
Page: 127
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region, \(a\in\Omega\) a region, \(f:\Omega\setminus\{a\} \to\mathbb C\) continuous

Name: A regular change of parameter
Definition: A differentiable change of parameter such that \(\phi'(\tau)\ne0\).
Page: 069
Chapter: 3
Section: 2.1
Variables: \(t=\phi(\tau)\) a change of parameter.

Question: A differentiable change of parameter
Answer: leaves the arc invariant.
Page: 069
Chapter: 3
Section: 2.1

Smaller0: \(f\) is analytic on \(\Omega\) and \(f^i(a)=0\) for each \(i\in\mathbb N\)
Greater0: \(f(z)=0\) on \(\Omega\)
Page: 126
Chapter: 4
Section: 3.2
Variables: \(\Omega\) a region,
\(\Delta\subseteq\Omega\) an open disk of contour \(C\),
\(a\in \Delta\),
\(f:\Omega \to\mathbb C\).

Smaller0: \(\overline{T(\overline z)}=T(z)\)
Greater0: \(T\) has real coefficients
Page: 079
Chapter: 3
Section: 3.3
Variables: \(T\) a linear transformation.
Smaller to greater: implies
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Smaller0: \(f\) is derivable
Greater0: \(f\) is continuous
Page: 149, , 156
Chapter: 3, 2, 4
Section: 1, 1.1, 2.1
Variables: \(a<x<b\) three reals
\(f:(a,b)\to\mathbb R\).
Index: 1a, Complex Ahlfor, 1 1 Zorich
Kind: , , Theo
Greatest: \(f\) is locally bounded

Name: The Boundary of \(X\)
Definition: \(Cl X\setminus Int X\).
Page: 053
Chapter: 3
Section: 1.1
Variables: \(X\subseteq \mathbb C\)
Notation: \(\partial X\)
Index: (iii)
Abbreviation: \(Bd\,X\)

Smaller0: The radius of convergence of the series \(s\)
Greater0: \(\frac{1}{\text{lim sup}_{n\to\infty}\sqrt[n]{|a_i|} }\)
Page: 039
Chapter: 2
Section: 2.4
Variables: \(s(z)=\sum_{i\in\mathbb N} a_iz ^i\) a series
Smaller to greater: equals
Hide middle: yes
Name: Hadamard's formula

Smaller0: \(|f^{(n)}(a)|\)
Greater0: \(\left(\max\{|f(a+re^{ix})|\mid x\in\mathbb R\}\right)n!r^{-n} \)
Page: 122
Chapter: 4
Section: 2.2
Variables: \(f:\mathbb C\to\mathbb C\),
\(a\in\mathbb C\),
\(n\in\mathbb N\)
\(r>0\).
Smaller to greater: \(\le\)
Assuming: \(f\) is analytic on the sphere of radius \(r\) around \(a\)
Name: Cauchy's estimate
Greater to smaller: \(\ge\)

Question: The inverse of \(x\mapsto\frac{az+b}{cz+d}\) is
Answer: \(x\mapsto \frac{dx-b}{-cx+a}\).
Page: 076
Chapter: 3
Section: 3.1
Assuming: \(ad-bc\ne0\)

Smaller0: all zeros of \(u\) lie in \(C\)
Greater0: all zeros of \(u'\) lie in \(C\)
Page: 029
Chapter: 2
Section: 1.3
Variables: \(u:\mathbb C\to\mathbb C\) polynomial and
\(C\) a half-plane.
Index: 1
Kind: Theo
Hide middle: yes
Name: Lucas's theorem

Question: The zeros of \(u'\)
Answer: lie in the convex hull of the zeros of \(u\).
Page: 029
Chapter: 2
Section: 1.3
Variables: \(u:\mathbb C\to\mathbb C\) polynomial
\(C\) a half-plane.
Index: 1
Kind: Theo

Question: \(\cos'=\)
Answer: \(-\sin\)
Page: 044
Chapter: 2
Section: 3.2
Index: (28)

Definition: [$]\int_{-b}^{-a}f(\gamma(-t))\left(-\gamma'(-t)\right)\dd t[/$].
Page: 103
Chapter: 4
Section: 1.1
Variables: \(f(t):\mathbb C\to\mathbb C\),
\(\gamma:[a,b]\to\mathbb C\) a piecewise differentiable curve.
Notation: [$]\int_{-\gamma}f(t)\dd t[/$]
Definition2: [$]-\int_\gamma f(z)\dd z[/$]

Definition: the greatest \(n\) such that \((x-\lambda)^n\) divides \(p\)
Page: 029
Chapter: 2
Section: 1.3
Variables: \(f(z)=u(z)+iv(z)\) a differentiable complex function.
\(p\) be a polynomial.
Notation: The order of the zero \(\lambda\)
Definition2: the least integer \(n\) such that \(p^{(n)}(\lambda)\ne 0\).

Smaller0: \(f\) is continuous and for all closed curve \(\gamma\subseteq\Omega\), [$]\int_\gamma f\dd z=0[/$]
Greater0: \(f\) is analytic in \(\Omega\)
Page: 122
Chapter: 4
Section: 2.2
Variables: \(\Omega\subseteq\mathbb C\) a region,
\(f:\Omega\to\mathbb C\).
Hide middle: yes
Name: Morera's theorem
Extra: counter example contra ?

Smaller0: The distance between the stereographic projection of a complex \(z\) and \(\infty\)
Greater0: \(2\sqrt{1+|z|^2}\)
Page: 020
Chapter: 1
Section: 2.4
Variables: \(z\in\mathbb C\).
Smaller to greater: equals
Hide middle: yes

Question: \(z\mapsto\sum a_iz^i\) is uniformly convergent on
Answer: the circle of radius \(\rho\), for any \(\rho\) strictly less than the radius of convergence
Page: 39
Chapter: 2
Section: 2.1
Variables: \(\sum a_iz^i\) a power series.
Kind: Theo