Title: Theory / Unimportant / Math / Combinatorics / Analytic - Flajolet Sedgewick+14 / B Complex Asymptotics
Tags: récupéré MissingMedia

Name: First Principle of Coefficient Asymptotics.
Definition: The location of a function’s singularities dictates the exponential growth (\(A^n\)) of its coefficients
Page: 227
Chapter: IV
Section: 1

Name: Second Principle of Coefficient Asymptotics.
Definition: The nature of a function’s singularities determines the associate subexponential factor (\(\theta (n)\)).
Page: 227
Chapter: IV
Section: 1

Name: Riemann's zeta function
Definition: [$]\defFun{\mathbb C}{\mathbb C}{s}{\sum_{n=1}^\infty\frac{1}{n^s}}[/$]
Page: 228
Chapter: IV
Section: 1
Context: Analysis
Notation: \(\zeta\)
Definition2: [$]\defFun{\mathbb C}{\mathbb C}{s}{\prod_{p\text{ prime}}\frac{p^s}{p^s-1}}[/$]
Construction: ?

Smaller0: \([z^n]\frac{h(z)}{1-z}\)
Greater0: \(h(1)\)
Page: 228
Chapter: IV
Section: 1
Variables: \(h\) a power series
Smaller to greater: \(\sim_{n\to\infty}\)
Assuming: \(h(1)\ne 0\),
radius of convergence \(>1\)
Context: Analysis

Smaller0: \([z^n]h(z)\sqrt{1-z}\)
Greater0: \(-\frac{h(1)}{2\sqrt{\pi n^3}}\)
Page: 228
Chapter: IV
Section: 1
Variables: \(h\) a power series
Smaller to greater: \(\sim_{n\to\infty}\)
Assuming: \(h(1)\ne 0\),
radius of convergence \(>1\)

Smaller0: \([z^n]h(z)\log\frac1{1-z}\)
Greater0: \(\frac{h(1)}n\)
Page: 228
Chapter: IV
Section: 1
Variables: \(h\) a power series
Smaller to greater: \(\sim_{n\to\infty}\)
Assuming: \(h(1)\ne 0\),
radius of convergence \(>1\)
Name: Elementary transfer

Name: A simply connected domain
Page: 229
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: Indented disc
Page: 229
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: Annulus
Page: 229
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: Slit complex plane
Page: 229
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: The real part of \(z\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z\in\mathbb C\)
Context: Complex analysis
Notation: \(\mathfrak R(z)\)
Denoted by: \(x\)

Name: The imaginary part of \(z\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z\in\mathbb C\)
Context: Complex analysis
Notation: \(\mathfrak I(z)\)
Denoted by: \(y\)

Name: The argument of \(z\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z\in\mathbb C\)
Context: Complex analysis
Denoted by: \(\theta\)

Name: The norm of \(z\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z\in\mathbb C\)
Context: Complex analysis
Notation: \(\rho\)

Name: Principal determination of complex logarithm
Definition: [$]\defFun{\text{split complex plane}}{\mathbb C}{\rho e^{i\theta}}{\log\rho+i\theta}[/$]
Page: 230
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: Principal determination of square root
Definition: [$]\defFun{\text{split complex plane}}{\mathbb C}{\rho e^{i\theta}}{\sqrt\rho e^{i\theta/2}}[/$]
Page: 230
Chapter: IV
Section: 2.1
Context: Complex analysis

Name: A branch point
Definition: A point where several determinations meet
Page: 230
Chapter: IV
Section: 2.1
Context: Complex analysis
Example: \(0\) for \(\sqrt z\)

Name: An analytic function at \(z_0\)
Definition: A function of the form \(\sum_{n\ge 0}c_n(z-z_0)^n\), defined on a neighborhood of \(z_0\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z_0\in\mathbb C\)
Context: Complex analysis
Index: 1
Kind: Def

Name: An analytic function on \(\Omega\)
Definition: A function analytic at each point of \(\Omega\)
Page: 230
Chapter: IV
Section: 2.1
Variables: \(\Omega\)
Context: Complex analysis
Index: 1
Kind: Def

Name: The disc of convergence of \(f\)
Definition: The set of values at which \(f\) is defined
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z_0,(c_i)_{i\in\mathbb N}\in\mathbb C\)
\(f(z)=\sum_{n\in\mathbb N}c_i(z-z_0)^i\)
Context: Complex analysis
Index: 1
Kind: Def

Name: The radius of convergence of \(f\)
Definition: The radius of the disc of convergence
Page: 230
Chapter: IV
Section: 2.1
Variables: \(z_0,(c_i)_{i\in\mathbb N}\in\mathbb C\)
\(f(z)=\sum_{n\in\mathbb N}c_i(z-z_0)^i\)
Context: Complex analysis
Index: 1
Kind: Def
Definition2: The maximum of the values \(x\ge0\) such that \(f(z_0+x)\) is bounded
Abbreviation: \(\mathrm{R}_{conv}(f;z_0)\)

Name: Elementary transfers
Definition: The relations between \([z^n]h(z)\) and \(f(h(1),n)\) for some function \(f\)
Page: 228
Chapter: IV
Variables: \(h\) analyticy

Name: A function analytic at \(z_0\)
Definition: a function such that \(f(z)=\sum_{n\ge0}c_n(z-z_0)^n\) on a neighborhood of \(z_0\)
Page: 230
Chapter: ID
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)

Name: A function analytic on \(\Omega\)
Definition: a function analytic on each point of \(\Omega\)
Page: 230
Chapter: IV
Section: 2
Variables: \(\Omega\) a region

Name: the radius of convergence of \(f\)
Definition: the supremum of all values of \(x\ge0\) such that \(\sum_{n\ge0}c_n(x-z_0)^n\) is bounded
Page: 230
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
\(f(z)=\sum_{n\ge0}c_n(z-z_0)^n\) analytic at \(z_0\)
Notation: \(\mathrm{R_{conv}}(f;z_0)\)
Extra: The disc of convergence of \(f\)

Smaller0: [$]\abs{z_0-x}[/$] is less than the radius of convergence
Greater0: \(f\) converges on \(x\)
Page: 230
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
\(f(z)=\sum_{n\ge0}c_n(z-z_0)^n\) analytic at \(z_0\)
\(x\in\Omega\)

Smaller0: [$]\abs{z_0-x}[/$] is greater than the radius of convergence
Greater0: \(f\) does not converge on \(x\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
\(f(z)=\sum_{n\ge0}c_n(z-z_0)^n\) analytic at \(z_0\)
\(x\in\Omega\)

Name: A function complex-differentiable on \(\Omega\)
Definition: a function complex-differentiable at every \(z_0\in\Omega\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
Name2: A function holomorphic on \(\Omega\)

Name: A function complex-differentiable at \(z_0\)
Definition: a function \(f:\Omega\to\mathbb C\) such that \(f'(z_0)\) exists.
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
Name2: A function holomorphic at \(z_0\)
Extra: analytic?

Definition: \(\lim_{\delta\to0}\frac{f(z_0+\delta)-f(z_0)}{\delta}\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
\(f:\Omega\to\mathbb C\)
Notation: \(f'(z_0)\)
Notation2: [$]\left.\pdv{}{z}f(z)\right|_{z_0}[/$]
Notation3: \(\partial_zf(z_0)\)

Smaller0: \(f\) is analytic in \(\Omega\)
Greater0: \(f\) is complex-differentiable in \(\Omega\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)
Smaller to greater: iff

Name: Meromorphic function at \(z_0\)
Definition: \(f(z)=g(z)/h(z)\) with \(f,h\) analytic at \(z_0\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(z_0\in\Omega\)
Definition2: \(f(z)=\sum_{n\ge-M}a_n(z-n_0)^n\) for some \(M\in\mathbb N\)
Conjdef: A function \(f:\Omega\to\mathbb C\) such that, on the neighborhood of \(z_0\), equivalently

Name: A pole of \(f\)
Definition: some \(z_0\in\Omega\) such that \(f(z)=\sum_{n\ge -M}a_n(z-z_0)^n\) with \(M\ge 1\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)

Name: The order of the pole \(z_0\)
Definition: the least \(M\) such that \(f(z)=\sum_{n\ge -M}a_n(z-z_0)^n\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)
\(z_0\) a pole of \(f\)

Name: The residue of \(f\) at \(z_0\)
Definition: the value \(a_{-1}\) such that \(f(z)=\sum_{n\ge -M}a_n(z-z_0)^n\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)
\(z_0\) a pole of \(f\)
Notation: \(\mathrm{Res}[h(z);z=z_0]\)

Name: A function meromorphic on \(\Omega\)
Definition: A function meromorphic at each \(z_0\in\Omega\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region

Name: A path in \(\Omega\)
Definition: The image of a continuous function \([0,1]\to\Omega\)
Page: 231
Chapter: IV
Section: 2
Variables: \(\Omega\) a region
Denoted by: \(\gamma\)

Name: A parametrization of \(\gamma\)
Definition: A continuous function \([0,1]\to\Omega\) whose image is \(\gamma\)
Page: 231
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(\gamma\) path in \(\Omega\)

Name: Homotopic paths in \(\Omega\)
Definition: can be continuously transformed from one to the other
Page: 231
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
Definition2: have the same endpoints
Conjdef: The equivalence relation on paths which

Name: A closed path
Definition: A path parametrized by a function \(\gamma\) such that \(\gamma(0)=\gamma(1)\)
Page: 231
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region

Name: A simple path
Definition: A path with an injective parameter function
Page: 231
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region

Name: A loop
Definition: A path homotopic to a single point
Page: 231
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
Name2: A path homotopic to 0

Definition: [$]\int_0^1f(\gamma(t))\gamma'(t)\dd t[/$]
Page: 234
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(\gamma\) a path
\(f:\Omega\to\mathbb C\)
Notation: [$]\int_\gamma f(z)\dd z[/$]

Smaller0: \(\gamma\) simple loop
Greater0: \(\int_\gamma f=0\)
Page: 234
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(\gamma\) a path
\(f:\Omega\to\mathbb C\)
Smaller1: \(f\) analytic
Name: Null integral property

Smaller0: \(\gamma\) and \(\gamma'\) homotopic
Greater0: \(\int_\gamma f=0\)
Page: 234
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(\gamma,\gamma'\) a path
\(f:\Omega\to\mathbb C\)
Smaller1: \(f\) analytic

Smaller0: \(\gamma\) positively oriented simple loop it \(\Omega\)
Greater0: equals \(2i\pi\sum_{s}\mathrm{Res}[h(z);z=s]\) for each pole \(s\) enclosed by \(\gamma\)
Page: 234
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(\gamma\) a path
\(f:\Omega\to\mathbb C\)
Prefix Greater: [$]\int_\gamma f(z)\dd z[/$]
Smaller1: \(f\) meromorphic

Smaller0: [$]\frac{k!}{2i\pi}\int_\lambda f(\xi)\frac{\dd\xi}{\xi-z_0}[/$] for \(\lambda\) a positive loop around \(z_0\)
Greater0: \(f^{(k)}(z_0)\)
Page: 236
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)
\(z_0\in\Omega\)
\(k\in\mathbb N\)
Smaller to greater: =
Assuming: \(f\) analytic in \(\omega\)
Construction: Derivating Cauchy's coefficient formula under the integral sign

Smaller0: [$]\frac{1}{2i\pi}\int_\lambda f(\xi)\frac{\dd\xi}{\xi-z_0}[/$] for \(\lambda\) a positive loop around \(z_0\)
Greater0: \(f(z_0)\)
Page: 236
Chapter: IV
Section: 2.2
Variables: \(\Omega\) a region
\(f:\Omega\to\mathbb C\)
\(z_0\in\Omega\)
Smaller to greater: =
Assuming: \(f\) analytic in \(\Omega\)
Smallest: \(\mathrm{Res}[\frac{f(\xi)}{\xi-z_0}; \xi=z_0]\)