Title: Theory / Important / Math / Analysis / Zorich{ / 05, 08 Differentiation / 5 Single variable;
Tags: fromMath mathToDef exampleToQuestion examplesSwitchExample1 fromImplication Copy hasSeenConjDef mathToSee checked closed counterExample duplicateToOtherBook relationHasDefinition

Name: The derivative of \(f\) at \(a\)
Definition: \(\lim_{E\ni x\to a}\frac{f(x)-f(a)}{x-a}\)
Page: 176, 178
Chapter: 5
Section: 1.2
Variables: \(E\subseteq\mathbb R\),
\(a\) a limit point of \(E\)
\(f:E\to\mathbb R\)
Notation: \(f'(a)\)
Index: 1
Kind: Def
Definition2: the only constant \(c\) such that \(f(x)-f(a)=c(x-a)+o(x-a)\).
Notation2: \(\dot f(x)\)
Notation3: \(\frac{df(x)}{dx}\)

Name: A function differentiable at \(x\)
Definition: there exists \(A(x)\) and \(\alpha(x,h)\) such that \(f(x+h)-f(x)=A(x)h+\alpha(x;h)\) and \(\alpha(x;h)=o(h)\) as \(h\to 0\), \(x+h\in E\).
Page: 177, 179
Chapter: 5
Section: 1.2, 1.3
Variables: \(E\subseteq\mathbb R\),
\(x\) a limit point of \(E\)
Index: 2
Kind: Def
Definition2: \(f(x)=c_0+c_1(x-x_0)+o(x-x_0)\) for some \(c_0,c_1\) (assuming \(f\) continuous at \(x\))
Conjdef: A function \(f:E\to\mathbb R\) such that

Name: The increment of the argument
Definition: \((x+h)-x\).
Page: 177, 432
Chapter: 5
Section: 1.2
Notation: \(\Delta x(h)\)
Index: 2
Kind: Def

Name: The increment of the function
Definition: \(f(x+h)-f(x)\).
Page: 177, 432
Chapter: 5
Section: 1.2
Notation: \(\Delta f(x;h)\)
Index: 2
Kind: Def

Name: Differential of \(f\) at \(x\)
Definition: \(h\mapsto f'(x)h\).
Page: 177
Chapter: 5
Section: 1.2
Variables: \(E\subseteq\mathbb R\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R\)
Notation: \(Df(x)\)
Index: 3
Kind: Def
Notation2: \(df(x)\)

Name: The tangeant to the graph of \(f\) at the point \((x_0,f(x_0))\)
Definition: The graph of the function \(y-f(x_0)=f'(x_0)(x-x_0)\).
Page: 179
Chapter: 5
Section: 1.3
Variables: \(E\subseteq\mathbb R\),
\(x_0\) a limit point of \(E\),
\(f:E\to\mathbb R\)
Index: 4
Kind: Def
Assuming: \(f\) is differentiable at \(x_0\)

Name: Two functions having a contact of order at least \(n\) at \(x_0\)
Definition: Two functions \(f,g:E\to\mathbb R\) such that \(f(x)-g(x)=o(x-x_0)^n\) as \(E\ni x\to x_0\).
Page: 181
Chapter: 5
Section: 1.3
Variables: \(E\subseteq\mathbb R\),
\(x_0\) a limit point of \(E\),
\(n\in\mathbb N\)
Kind: Def

Name: Two functions tangeant at \(x_0\)
Definition: Two functions having a contact of order at least \(1\).
Page: 181
Chapter: 5
Section: 1.3
Variables: \(E\subseteq\mathbb R\),
\(x_0\) a limit point of \(E\).
Kind: Def

Question: \(\sin'(x)\)
Answer: equals \(\cos(x)\).
Page: 183
Chapter: 5
Section: 1.5
Index: 1
Kind: Ex

Question: \(\cos'(x)\)
Answer: equals \(-\sin(x)\).
Page: 183
Chapter: 5
Section: 1.5
Index: 2
Kind: Ex

Smaller0: \(f\) is differentiable at \(x\)
Greater0: \(f\) is continuous at \(x\)
Page: 186, 437
Chapter: 5, 8
Section: 1.5, 2.4
Variables: \(E\subseteq\mathbb R^m\),
\(x\) a limit point of \(E\),
\(f:E\to \mathbb R^n\)
Index: 7
Kind: Ex

Question: The derivative of the absolute function at 0
Answer: does not exists.
Page: 187
Chapter: 5
Section: 1.5
Index: 8
Kind: Ex

Question: \(\exp'\) equals
Answer: \(\exp\).
Page: 187
Chapter: 5
Section: 1.5
Index: 9
Kind: Ex

Question: \(\partial_i(\sum_{l=1}^k\lambda_lf^j_l)(x)\)
Answer: equals \(\sum_{l=1}^k\lambda_l\partial_if_l^j(x)\).
Page: 190, 439, 440
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_k:E\to\mathbb R^m\) differentiable at \(x\)
Index: 1 a, 2, 31'
Kind: Theo, Eq

Question: \(\partial_i(\sum_{l=1}^k\lambda_kf^k)(x)\)
Answer: equals \(\sum_{l=1}^k\lambda_l\partial_i f^j(x)\).
Page: 190, 439, 440
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_k:E\to\mathbb R^m\) differentiable at \(x\)
Index: 1 a, 2, 31'
Kind: Theo, Eq

Question: [$]\dd(\sum_{l=1}^k\lambda_lf_l)(x)[/$]
Answer: equals [$]\sum_{l=1}^k\lambda_l\dd f_l(x)[/$].
Page: 190, 439
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_k:E\to\mathbb R^m\) differentiable at \(x\)
Index: 1 a, 2, 31'
Kind: Theo, Eq

Question: [$]\dd(\prod_l f_l)(x)[/$]
Answer: equals [$]\sum_{l=1}^k(\prod_{p\ne l}f_p\dd f_l)(x)[/$].
Page: 190, 439
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_k:E\to\mathbb R\) differentiable at \(x\)
Index: 1b, 2, 32'
Kind: Theo, eq

Question: \(\partial_i(\prod_l f_l)(x)\)
Answer: equals \(\sum_{l=1}^k(\prod_{p\ne l}f_p\partial_if_l)(x)\).
Page: 190, 440
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_k:E\to\mathbb R\) differentiable at \(x\)
Index: 1b, 2, 32'
Kind: Theo, eq

Question: [$]\dd(\frac fg)(x)[/$]
Answer: equals [$]\frac{1}{g^2(x)}\left(\dd f(x)\cdot g(x)-f(x)\cdot\dd g(x)\right)[/$].
Page: 190, 440
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f,g:E\to\mathbb R^m\) differentiable at \(x\)
Index: 1 c, 2, 33'
Kind: Theo, Eq
Assuming: \(g(x)\ne 0\)
Extra: doublon diff quotient

Question: \(\partial_i(\frac fg)(x)\)
Answer: equals \(\frac{1}{g^2(x)}\left(\partial_i f(x)\cdot g(x)-f(x)\cdot \partial_ig(x)\right)\).
Page: 190, 439
Chapter: 5, 8
Section: 2.1, 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f,g:E\to\mathbb R^m\) differentiable at \(x\)
Index: 1 c, 2, 33'
Kind: Theo, Eq
Assuming: \(g(x)\ne 0\)
Extra: doublon diff quotient

Question: The derivative of a constant function is
Answer: 0.
Page: 191
Chapter: 5
Section: 2.1
Index: 1
Kind: Cor

Question: \((\sum c_if_i)'(x)\)
Answer: is \(\sum c_if_i'(x)\).
Page: 191
Chapter: 5
Section: 2.1
Variables: \(E\subseteq\mathbb R\),
\(x\) a limit point of \(E\),
\(c_1,\dots,c_n\in\mathbb R\),
\(f_1,\dots,f_n:E\to \mathbb R\).
Index: 1
Kind: Cor

Question: \((\Pi f_i)'(x)\)
Answer: is \(\sum_{j=1}^n f'_j(x)\Pi_{i\ne j}f_i(x)\).
Page: 192
Chapter: 5
Section: 2.1
Variables: \(E\subseteq\mathbb R\),
\(x\) a limit point of \(E\),
\(f_1,\dots,f_n:E\to \mathbb R\).
Index: 2
Kind: Cor

Smaller0: [$]\dd (g\circ f)[/$]
Greater0: [$]\dd g(f(x))\circ \dd f(x)[/$]
Page: 195, 441
Chapter: 5, 8
Section: 2.1, 3.2
Variables: \(X_i\subseteq \mathbb R^{m_i}\)
\(f_i:X_i\to X_{i+1}\)
\(x\) a limit point of \(X\),
Index: 4
Kind: Cor, Theo
Smaller to greater: equals
Hide middle: yes
Assuming: \(f\) is differentiable at \(x\), \(g\) differentiable at \(f(x)\), a limit point of \(Y\)

Smaller0: \(\partial_i (g\circ f)\)
Greater0: \(\sum_j\partial_jg(f(x))\cdot\partial_if^j(x)\)
Page: 195, 441
Chapter: 5, 8
Section: 2.1, 3.2
Variables: \(m,n\in\mathbb N\)
\(X\subseteq\mathbb R^{m}, Y\subseteq\mathbb R^n\)
\((f^j:\mathbb X\to\mathbb R)_{1\le j\le n}\), with \((f^1,\dots,f^n):X\to Y\)
\(g:Y\to\mathbb R\)
\(x\) a limit point of \(X\),
Index: 4
Kind: Cor, Theo
Smaller to greater: equals
Hide middle: yes
Assuming: \(f\) is differentiable at \(x\), \(g\) differentiable at \(f(x)\), a limit point of \(Y\)

Smaller0: \((f_n\circ \dots \circ f_0)'(x)\)
Greater0: \(\Pi_{i=0}^nf'_i(f_{i-1}\circ\dots\circ f_1(x))\)
Page: 194
Chapter: 5
Section: 2.1
Variables: \(f_0,\dots,f_n\) a sequence of real functions and \(x\in\mathbb R\)
Kind: Cor
Smaller to greater: equals
Hide middle: yes

Smaller0: \((f^{-1})'(f(x_0))\)
Greater0: \(\frac{1}{f'(x_0)}\)
Page: 197
Chapter: 5
Section: 2.3
Variables: \(X,Y\subseteq\mathbb R\),
\(x_0\in X\),
\(f:X\to Y\) with inverse \(f^{-1}\)
Index: 3
Kind: Theo
Smaller to greater: equals
Hide middle: yes
Assuming: this expression is well defined

Question: \(\sinh(-x)\)
Answer: equals \(-\sinh (x)\).
Page: 199
Chapter: 5
Section: 2.3
Variables: \(x\in\mathbb C\)
Index: 14
Kind: Ex

Question: \(\cosh(-x)\)
Answer: equals \(\cosh (x)\).
Page: 199
Chapter: 5
Section: 2.3
Variables: \(x\in\mathbb C\)
Index: 14
Kind: Ex

Question: \(\cosh'\)
Answer: equals \(\sinh\)
Page: 200
Chapter: 5
Section: 2.3

Question: \(\sinh'\)
Answer: equals \(\cosh\)
Page: 200
Chapter: 5
Section: 2.3

Name: Hyperbolic tangeant
Definition: \(\frac\sinh\cosh\)
Page: 200
Chapter: 5
Section: 2.3
Abbreviation: \(\tanh\)

Name: Hyperbolic cotangeant
Definition: \(\frac{\text{cosh} }{\text{sinh} }\)
Page: 200
Chapter: 5
Section: 2.3
Abbreviation: \(\text{cotanh}\)

Question: \((g\circ f^{-1})'=\)
Answer: [$]\defFun{V}{\mathbb R}{t}{\frac{g'(t)}{f'(t)}}[/$].
Page: 202
Chapter: 5
Section: 2.5
Variables: \(t\in \mathbb R\),
\(U\subseteq \mathbb R\) a neighborhood of \(t\),
\(f,g:U\to\mathbb R\), with \(f\) inversible on some neighborhood \(V\) of \(f(t)\)
Assuming: \(f'(t)\ne0\)

Name: The \(n\)-th derivative of \(f\) at \(x\)
Definition: If \(n=0\) then \(f(x)\), otherwise \((f^{(n-1)})'(x)\)
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R\)
Notation: \(f^{(n)}(x)\)
Notation2: \(\frac{d^n f(x)}{dx^n}\)
Notation3: \(f^{\prime\dots\prime}(x)\)

Definition: The set of real function with domain \(E\) which have a continuous \(n\)-th derivative.
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R\),
Notation: \(C^n(E)\), \(C^n(E,\mathbb R)\), \(C^{(n)}(E)\), \(C^{(n)}(E,\mathbb R)\)

Definition: \(C^0(E)\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R\),
Notation: \(C(E)\)

Question: \(\frac{d a^x}{dx^n}\)
Answer: equals \(a^x(\ln a)^n\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Question: \(\frac{d e^x}{dx^n}\)
Answer: equals \(e^x\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Question: \(\frac{d \sin x}{dx^n}\)
Answer: equals \(\sin(x+n\pi/2)\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Question: \(\frac{d \cos x}{dx^n}\)
Answer: equals \(\cos(x+n\pi/2)\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Question: \(\frac{d (1+x)^\alpha}{dx^n}\)
Answer: equals \(\alpha^{\underline{n}}(1+x)^{\alpha-n}\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex
Extra: alpha ?
1+x ?

Question: \(\frac{d x^\alpha}{dx^n}\)
Answer: equals \(\alpha(\alpha-1)\dots(\alpha-n+1)x^{\alpha-n}\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Question: [$]\frac{d \ln\abs{x}}{dx^n}[/$]
Answer: equals \((-1)^{n-1}(n-1)!x^{-n} \).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Smaller0: \((uv)^{(n)}\)
Greater0: \(\sum_{m=0}^n{n\choose m}u^{(n-m)}v^{(m)}\)
Page: 207
Chapter: 5
Section: 2.6
Variables: \(E\subseteq\mathbb R\)
\(u,v\in C^{(n)}(E)\)
Index: 24
Kind: Ex
Smaller to greater: =
Hide middle: yes
Name: Leibniz' formula
Extra: duplicate, for n=1
Context: Derivative
Intuition: Newton’s binomial formula for derivative

Name: A local minimum of \(f\)
Definition: A value \(x\in E\) such that there exists a neighborhood \(U_E(x)\) such that for all \(y\in U_E(x)\), \(f(y)\ge f(x)\).
Page: 211, 463
Chapter: 5, 8
Section: 3.1, 4.5
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
Index: 1
Kind: Def

Name: A local maximum of \(f\)
Definition: A value \(x\in E\) such that there exists a neighborhood \(U_E(x)\) such that for all \(y\in U_E(x)\), \(f(y)\le f(x)\).
Page: 211, 463
Chapter: 5, 8
Section: 3.1, 4.5
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
Index: 1
Kind: Def

Name: A strict local minimum of \(f\)
Definition: A value \(x\in E\) such that there exists a deleted neighborhood \(\mathring U_E(x)\) such that for all \(y\in U_E(x)\), \(f(y)>f(x)\).
Page: 211, 463
Chapter: 5, 8
Section: 3.1, 4.5
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
Index: 2
Kind: Def

Name: A strict local maximum of \(f\)
Definition: A value \(x\in E\) such that there exists a deleted neighborhood \(\mathring U_E(x)\) such that for all \(y\in U_E(x)\), \(f(y)< f(x)\).
Page: 211, 463
Chapter: 5, 8
Section: 3.1, 4.5
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
Index: 2
Kind: Def

Name: The (strict) local maximum value of \(f\) associated to \(x\)
Definition: \(f(x)\).
Page: 211
Chapter: 5
Section: 3.1
Variables: \(E\subseteq\mathbb R\)
\(f:E\to\mathbb R\),
\(x\) a (strict) local maximum of \(f\)
Index: 2
Kind: Def

Name: A local extrema (value) of \(f\)
Definition: A local maximum (value) or a local minimum (value) of \(f\).
Page: 211, 463
Chapter: 5, 8
Section: 3.1, 4.5
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
Index: 3
Kind: Def

Name: An interior extremum of \(f\)
Definition: An extremum \(x_0\in E\) of \(f\) such that \(x_0\) is a limit point of \(\{x\in E\mid x<x_0\}\) and of \(\{x\in E\mid x>x_0\}\).
Page: 211
Chapter: 5
Section: 3.1
Variables: \(E\subseteq\mathbb R\)
\(f:E\to\mathbb R\)
Index: 4
Kind: Def

Smaller0: \(x\) is an interior extremum
Greater0: \(\partial_if(x_0)=0\) for each \(1\le i\le m\) (assuming \(f\) is derivable)
Page: 212, 463
Chapter: 5, 8
Section: 3.1, 4
Variables: \(E\subseteq\mathbb R^m\)
\(f:E\to\mathbb R\)
\(x\in E\)
Index: 1, 5
Kind: Lem, Theo
Hide middle: yes
Extra: if it exists

Smaller0: \(f\) is continuous and defined on \([x,x+h]\)
Greater0: there exists \(\xi\in]x,x+h[\) such that \(f(x)-f(x+h)=f'(\xi)h\)
Page: 213, 454
Chapter: 5, 8
Section: 3.2, 4.1
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x,x+h\in \mathbb R^m\)
\(f:G\to\mathbb R\)
Index: 1
Kind: Theo
Hide middle: yes
Smaller1: \(f\) is differentiable on \((x,x+h)\)
Name: Lagrange's mean-value theorem
Extra: Similar to Cauchy's finite-increment theorem, with \(g(x)=x\)
Construction: Let [$]F=\defFun{[a,b]}{\mathbb R}{x}{f(x)-\frac{f(b)-f(a)}{b-a}(x-a)}[/$], \(F(a)=F(b)=f(a)\), by Rolles, it has a derivative equal to 0.
Etymology: There is a mean between \(x\) and \(x+h\) such that the speed at this mean is the average speed

Smaller0: \(f'\) exists and is nonnegative
Greater0: \(f\) is nondecreasing
Page: 215
Chapter: 5
Section: 3.2
Variables: \(a<b\in \mathbb R\)
\(f:[a,b]\to\mathbb R\)
Index: 1
Kind: Cor
Hide middle: yes

Smaller0: \(f'\) exists and is positive
Greater0: \(f\) is increasing
Page: 215
Chapter: 5
Section: 3.2
Variables: \(a<b\in \mathbb R\)
\(f:[a,b]\to\mathbb R\)
Index: 1
Kind: Cor
Hide middle: yes

Smaller0: \(f'\) is 0 on \(G\)
Greater0: \(f\) is constant
Page: 216, 455
Chapter: 5, 8
Section: 3.2, 4.1
Variables: \(m,n\in\mathbb N\)
\(G\subseteq\mathbb R^m\) a domain
\(f:G\to\mathbb R\)
Index: 2
Kind: Cor
Smaller to greater: iff
Context: Real function

Smaller0: \(f,g\) are differentiable on \(]a,b[\)
Greater0: \(f'(x)(g(b)-g(a))=g'(x)(f(b)-f(a))\) for some \(x\in]a,b[\)
Page: 216
Chapter: 5
Section: 3.2
Variables: \(a<b\in \mathbb R\)
\(f,g:[a,b]\to\mathbb R\)
Index: 2
Kind: Cor
Smaller1: \(f,g\) are continuous on \([a,b]\)
Name: Cauchy's finite-increment theorem
Extra: Similar to Lagrange's finite-increment theorem, with a second function
Construction: \(F(x)=f(x)(g(b)-g(a))-g(x)(f(b)-f(a))\) satisfies Rolle's theoreme
\(\begin{array}{ll}&f(b)(g(b)-g(a))-g(b)(f(b)-f(a))\\=&f(b)g(b)-f(b)g(a)-g(b)f(b)+g(b)f(a)\\=&g(b)f(a)-f(b)g(a)\\=&f(a)g(b)-g(a)f(b)\\=&f(a)g(b)-f(a)g(a)-g(a)f(b)-g(a)f(a)\\=&f(a)(g(b)-g(a))-g(a)(f(b)-f(a))\end{array}\)
thus there exists \(x\in[a,b]\) such that \(F'(x)=f'(x)(g(b)-g(a))-g'(x)(f(b)-f(a))=0\) hence the equality

Name: The Taylor polynomial of order \(n\) of \(f(x)\) at \(x_0\)
Definition: \(\sum_{i=0}^n\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i\).
Page: 217
Chapter: 5
Section: 3.3
Variables: \(n\in\mathbb N\),
\(x\in\mathbb R\),
\(U\) a neighborhood of \(x\),
\(f\in C^n(U)\)
Notation: \(P_n(x_0;x)\)
Kind: Def

Name: The \(n\)-th remainder (term in Taylor's formula)
Definition: \(f(x)-P(x_0;x)\).
Page: 217
Chapter: 5
Section: 3.3
Variables: \(n\in\mathbb N\),
\(x\in\mathbb R\),
\(U\) a neighborhood of \(x\),
\(f\in C^n(U)\)
Notation: \(r_n(x_0;x)\)
Kind: Def

Smaller0: -\(f\) is differentiable \(n\) times in \([x_0,x]\)
Greater0: \((x_0,x)\)
Page: 218
Chapter: 5
Section: 3
Variables: \(n\in\mathbb N\),
\(x_0<x\in\mathbb R\),
\(f:[x_0,x]\to\mathbb R\),
\(\phi:[x_0,x]\to\mathbb R\)
Index: 2
Kind: Theo
Prefix Greater: -there exists a real \(\xi\) belonging to
Smaller1: -\(f\) is differentiable \(n+1\) times in \((x_0,x)\)
Greater1: \[\sum_{i=0}^n\frac{f^{(i)}(x_0)}{i!}(x-x_0)+\frac{\phi(x)-\phi(x_0)}{\phi'(\xi)n!}(x-\xi)^nf^{(n+1)}(\xi)\]
Connect Greater0: such that \(f(x)\) equals
Extra: checked
Construction: apply Cauchy theorem to \(f(x)-\sum_{i=0}^n\frac{f^{(i)(x_0)}}{i!}(x-x_0)\) and \(\phi\)
Smaller2: -\(\phi\) is continous on \([x_0,x]\)
Smaller3: -\(\phi'\) exists and is non-nul in \((x_0,x)\)

Name: Taylor series of \(f\)
Definition: \(\sum_n \frac{f^{(n)}(x_0)(x-x_0)^n}{n!}\).
Page: 222
Chapter: 5
Section: 3
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x\),
\(f:U\to \mathbb R\)
Index: 1, 2
Kind: Cor
Assuming: \(f\) infinite derivable on \(U\)

Name: Local Taylor formula with the Peano form of the remainder
Definition: \(f(x)=_{E\ni x\to x_0}\sum_{i=0}^n\frac{f^{(i)}(x_0)}{i!} +o((x-x_0)^n)\)
Page: 224, 225
Chapter: 5
Section: 3
Variables: \(E\) a closed interval,
\(x_0\) an endpoint,
\(n\in\mathbb N\),
\(\phi:E\to\mathbb R\)
Index: 4
Kind: Prop
Assuming: \(f\) is \(n\)-th derivable on \(E\)'s interior

Smaller0: (for each \(i\le n\), \(\phi^{(i)}(x_0)=0\))
Greater0: \(o((x-x_0)^n)\)
Page: 224
Chapter: 5
Section: 3
Variables: \(E\) a closed interval,
\(x_0\) an endpoint,
\(n\in\mathbb N\),
\(\phi:E\to\mathbb R\)
Index: 2
Kind: Lem
Prefix Greater: \(\phi(x)=_{E\ni x\to x_0}\)
Assuming: \(\phi\) is \(n\)-th derivable on \(E\) interior,

Name: Taylor's formula with the Lagrange form of the remainder
Definition: \(f(x)=\sum_{i=0}^n\frac{f^{(i)}(x_0)}{i!}(x-x_0)^i+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}\) for some \(\xi\in E\).
Page: 225
Chapter: 5
Section: 3
Variables: \(E\) a closed interval,
\(x_0\) an endpoint,
\(n\in\mathbb N\),
\(f:E\to\mathbb R\)
Assuming: \(f\) has a derivable of order \(n+1\) on the open interval

Smaller0: \(f\) is nonincreasing
Greater0: \(f'\le 0\)
Page: 234
Chapter: 5
Section: 4.1
Variables: \(E\) an open interval,
\(f:E\to\mathbb R\)
Smaller to greater: iff
Assuming: \(f\) differentiable on \(]a,b[\)

Smaller0: \(f\) is nondecreasing
Greater0: \(f'\ge 0\)
Page: 234
Chapter: 5
Section: 4.1
Variables: \(E\) an open interval,
\(f:E\to\mathbb R\)
Index: 1
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) differentiable on \(]a,b[\)

Smaller0: \(f\) is constant
Greater0: \(f'\equiv 0\)
Page: 234, 455
Chapter: 5, 8
Section: 4.1, 4.1
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
Index: 1
Kind: Prop, corr
Smaller to greater: iff
Assuming: \(f\) differentiable on \(]a,b[\)

Smaller0: \(f'>0\)
Greater0: \(f\) is increasing
Page: 234
Chapter: 5
Section: 4.1
Variables: \(E\) an open interval,
\(f:E\to\mathbb R\)
Index: 1
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) differentiable on \(]a,b[\)

Smaller0: \(f'<0\)
Greater0: \(f\) is decreasing
Page: 234
Chapter: 5
Section: 4.1
Variables: \(E\) an open interval,
\(f:E\to\mathbb R\)
Index: 1
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) differentiable on \(]a,b[\)

Smaller0: \(x_0\) is an extremum
Greater0: \(f'(x_0)\) is 0 or not defined
Page: 235
Chapter: 5
Section: 4.2
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x_0\),
\(f:U\to\mathbb R\)
Index: 2
Kind: Prop

Smaller0: \(f^{(i)}(x_0)=0\) for all \(\mathring U(x_0)\), \(i<n\)
Greater0: \(x_0\) is a strict local maximum
Page: 237
Chapter: 5
Section: 4.2
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x_0\),
\(f:U\to\mathbb R\),
\(n\in\mathbb N\)
Index: 4
Kind: Prop
Assuming: \(f\) is \(n\)-th order derivable on \(U\)
Smaller1: \(f^{(n)}<0\) and \(n\) even

Smaller0: \(f^{(i)}(x_0)=0\) for all \(\mathring U(x_0)\), \(i<n\), \(f^{(n)}>0\) and \(n\) odd
Greater0: \(x_0\) is not an extremum
Page: 236
Chapter: 5
Section: 4.2
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x_0\),
\(f:U\to\mathbb R\),
\(n\in\mathbb N\)
Index: 4
Kind: Prop
Assuming: \(f\) is \(n\)-th order derivable on \(U\)

Smaller0: \(f^{(i)}(x_0)=0\) for all \(\mathring U(x_0)\), \(i<n\)
Greater0: \(x_0\) is not an extremum
Page: 236
Chapter: 5
Section: 4.2
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x_0\),
\(f:U\to\mathbb R\),
\(n\in\mathbb N\)
Index: 4
Kind: Prop
Hide middle: yes
Assuming: \(f\) is \(n\)-th order derivable on \(U\)
Smaller1: \(f^{(n)}<0\)
Smaller2: \(n\) odd

Smaller0: \(\langle\vec x,\vec y\rangle\)
Greater0: [$]\norm{\vec x}_p\norm{\vec y}_q [/$]
Page: 239
Chapter: 5
Section: 4.2a
Variables: \(x_1,y_1,\dots,x_n,y_n\ge 0\),
\(p\not\in\{0,1\}\),
\(q\) such that \(\frac 1p+\frac 1q=1\)
Smaller to greater: \(\le^{(p>1)}\)
Assuming: either \(p>0\) or the \(x_i>0\)
Name: Hölder's inequality
Greater to smaller: \(\ge^{(p>1)}\)

Smaller0: \(a=b\)
Greater0: \(\frac ap+\frac bq\)
Page: 239
Chapter: 5
Section: 4.2a
Variables: \(a,b>0\),
\(p\not\in\{0,1\}\),
\(q\) such that \(\frac 1p+\frac 1q=1\)
Smaller to greater: iff
Name: Young's equality
Greater1: \(\sqrt[p]a\sqrt[q]b\)
Connect Greater0: =
Construction: Both are equal to \(a\)
\[\frac ap+\frac bq=\frac ap+\frac{a}{\frac1{1-\frac1p}}=\frac ap+\frac{a}{\frac1{\frac{p-1}p}}=\frac ap+\frac{a}{\frac p{p-1}}=\frac ap+\frac{a(p-1)}{p}=\frac {ap}p=a=a^{1/p}a^{1-\frac1p}=\sqrt[p]{a}\sqrt[\frac1{1-\frac1p}]a=\sqrt[p]a\sqrt[q]b\]

Smaller0: [$]\norm{\vec x+\vec y}_p[/$]
Greater0: [$]\norm{\vec x}_p+\norm{\vec y}_p[/$]
Page: 240
Chapter: 5
Section: 4.2c
Variables: \(x_1,y_1,\dots,x_n,y_n\ge 0\), \(p\ne 0\)
Smaller to greater: \(\le^{(p>1)}\)
Name: Minkowski's inequality
Greater to smaller: \(\ge^{(p>1)}\)
Construction: [latex]\begin{eqnarray*}&
\left(\sum_{i=1}^n(x_i+y_i)^p\right)^{\frac{1}p}\left(\sum_{i=1}^n(x_i+y_i)^p\right)^{\frac{p-1}p}
\\=&\sum_{i=1}^n(x_i+y_i)^p
\\=&\sum_{i=1}^n x_i(x_i+y_i)^{p-1} + \sum_{i=1}^n y_i(x_i+y_i)^{p-1}
\\\le^{(p>1)}&\norm{\vec x}_p \left(\sum_{i=1}^n(x_i+y_i)^{p-1}\right)^{\frac{p-1}p} + \norm{\vec y}_p \left(\sum_{i=1}^n(x_i+y_i)^{p-1}\right)^{\frac{p-1}p}\end{eqnarray*}[/latex]
using Hölder's inequality, thus [$]\norm{\vec x+\vec y}_p\le^{(p>1)}\norm{\vec x}_p+\norm{\vec y}_p[/$]

Smaller0: the \(x\)'s and the \(y\)'s vector are colinear
Greater0: [$]\norm{\vec x}_p+\norm{\vec y}_p[/$]
Page: 240
Chapter: 5
Section: 4.2a
Variables: \(x_1,y_1,\dots,x_n,y_n\ge 0\), \(p\ne 0\)
Greater1: [$]\norm{\vec x+\vec y}_p[/$]
Connect Greater0: =

Smaller0: \(f'\) is nondcreasing(increasing)
Greater0: \(f\) is (strictly) convex
Page: 243
Chapter: 5
Section: 4.3
Variables: \(f:]a,b[\to\mathbb R\)
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) is differentiable

Name: A point of inflexion of \(f\)
Definition: A point \(x\in]a,b[\) such that \(f\) is convex on \(U^-(x)\) and concave on \(U^+(x)\) or reciprocally.
Page: 245
Chapter: 5
Section: 4.3
Variables: \(f:]a,b[\to\mathbb R\)
Index: 3
Kind: Def

Name: Jensen's inequality
Definition: \(f(\sum \alpha_ix_i)\le \sum \alpha_if(x_i) \).
Page: 246
Chapter: 5
Section: 4.3
Variables: \(n\in\mathbb N\),
\(f:]a,b[\to\mathbb R\),
\(\alpha_1,\dots,\alpha_n>0\),
\(x_1,\dots,x_n\in]a,b[\)
Index: 7
Kind: Prop
Assuming: \(f\) convex, \(\sum \alpha_i=1\)

Name: Imaginary unit
Page: 263
Chapter: 5
Notation: \(i\)

Name: Their sum of \((x_j+iy_j)\)
Definition: \((x_1+x_2)+i(y_1+y_2)\).
Page: 263
Chapter: 5
Variables: \((x_j+iy_j)\in\mathbb C\)

Name: The product of the \(x_j+iy_j\)
Definition: \((x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)\).
Page: 263
Chapter: 5
Variables: \((x_j+iy_j)\in\mathbb C\)

Definition: \(r+i0\).
Page: 264
Chapter: 5
Variables: \(r\in\mathbb R\)
Context: The complex number
Notation: \(r\)

Name: Its inverse
Definition: [$]\frac{x-iy}{x^2+y^2}=\frac{\overline z}{\norm{z}^2}[/$].
Page: 264
Chapter: 5
Variables: \((x+iy)\in\mathbb C\)

Name: The norm of \(x+iy\)
Definition: [$]\defFun{\mathbb C}{\mathbb R^{\ge0}}{x+iy}{\sqrt{x^2+y^2}}[/$].
Page: 264
Chapter: 5
Section: 5.1 b
Variables: \((x+iy)\in\mathbb C\)
Notation: [$]\abs{x+iy}[/$]

Name: The argument of \(z\)
Definition: An equivalence class of real \(\phi\) modulo \(2\pi\mathbb Z\) such that [$](x+iy)=\abs{x+iy}(\cos \phi+i\sin\phi)[/$].
Page: 264
Chapter: 5
Section: 5.1 b
Variables: \((x+iy)\in\mathbb C\) not null
Abbreviation: \(\text{Arg }z\)

Name: Its polar coordinate
Definition: The pair composed of the modulus and of the argument.
Page: 265
Chapter: 5
Section: 5.1 b
Variables: \((x+iy)\in\mathbb C\) not null

Smaller0: \(\left(e^r(\cos\phi+i\sin\phi)\right)^n\)
Greater0: \(e^{nr}(\cos n\phi+i\sin n\phi)\)
Page: 266
Chapter: 5
Section: 5.1 b
Variables: \((r,\phi)\in\mathbb C\) the polar representation of a complex number
\(n\in\mathbb N\)
Smaller to greater: =
Name: de Moivre formula

Q: \(\sum_i z^i\) converges [...] for [converges].
A: absolutly; \(z\in\mathbb C\) of norm less than 1
Page: 266
Chapter: 5
Section: 5.2
Index: 4
Kind: Ex

Question: This series converges on
Answer: the open disk of center \(z_0\) and of radius [$]\frac1{\overline{\lim}_{n\to\infty}\sqrt[n]{\abs{c_n}} }[/$], and potentially on a part of the circle.
Page: 269
Chapter: 5
Section: 5.2
Variables: \(\sum c_i(z-z_0)^i\) a power series
Index: 3
Kind: Prop
Name: The Cauchy-Hadamard formula

Name: A serie of radius of convergence 1, diverging on the circle of radius 1
Page: 269
Chapter: 5
Section: 5.2
Kind: Ex
Example: \(\sum z^n\)

Name: A serie of radius of convergence 1, diverging for 1, converging for \(-1\)
Page: 269
Chapter: 5
Section: 5.2
Kind: Ex
Example: \(\sum\frac{z^n}{n}\)

Name: A serie of radius of convergence 1, converging on the disk of radius 1
Page: 269
Chapter: 5
Section: 5.2
Kind: Ex
Example: \(\sum\frac{z^n}{n^2}\)

Smaller0: the series converges on \(z^*\)
Greater0: the series converges absolutely on any value \(z\) satisfying [$]\abs{z-z_0}<\abs{z^*-z_0}[/$]
Page: 269
Chapter: 5
Section: 5.2
Variables: \(\sum c_i(z-z_0)^i\) a power series
Kind: Cor
Name: Abel's first theorem on power series

Question: Their product is
Answer: an aboslutely convergent series whose sum is the product of the sum of the two series.
Page: 270
Chapter: 5
Section: 5.2
Variables: \(\sum a_i,\sum b_j\) series
Kind: Prop
Assuming: Both series are absolutly convergent

Definition: [$]\defFun{\mathbb C}{\mathbb C}{z}{\sum\frac{z^i}{i!}}[/$]
Page: 271
Chapter: 5
Section: 5.3
Notation: \(e^z\)

Smaller0: \(e^{z_1+z_2} \)
Greater0: \(e^{z_1} e^{z_2} \)
Page: 273
Chapter: 5
Section: 5.3
Variables: \(z_1,z_2\in\mathbb C\)
Smaller to greater: \(=\)
Hide middle: yes

Question: To graphically represent \(f\):
Answer: represents a set \(S\) and \(f(S)\).
Page: 275
Chapter: 5
Section: 5.4
Variables: \(f\) a complex function

Question: the \(k\)-th derivative of the power series \(f(z)=\sum_{n=0} ^\infty c_n(z-z_0)^n\) is
Answer: \(\sum _{n=0}^\infty\frac{d^k}{dz^k}c_n(z-z_0)^n\)
Page: 179
Chapter: 5
Index: 1
Kind: Theo

Q: \([...]=[...] f^{(n)}([...]) \)
A: c_n; \frac1{n!}; z_0
Page: 280
Chapter: 5
Variables: \(f(z)=\sum _{n=0} ^\infty c_n(z-z_0)^n\),
\(k\in\mathbb N\)
Index: 1
Kind: Theo

Name: A function analytic at \(z_0\)
Definition: A function which can be representend in a neighborhood of \(z_0\) as \(\sum _{n=0} ^\infty c_n(z-z_0)^n\)
Page: 281
Chapter: 5
Variables: \(z_0\in\mathbb C\)

Smaller0: \(f\) is analytic at that \(z_0\)
Greater0: \(f\) is infinitely differentiable at \(z_0\)
Page: 281
Chapter: 5
Variables: \(z_0\in\mathbb C\)
\(f\) a complex function defined in a complex neighborhood of \(z_0\)
Smaller to greater: iff
Greater1: its Taylor series converges to it in a neighborhood of \(z_0\)

Smaller0: \(f\) is analytic at \(z_0\)
Greater0: the taylor series of \(f\) converges to \(f\) on a neighborhood of \(z_0\)
Page: 281
Chapter: 5
Variables: \(z_0\in\mathbb C\)
\(f\) a complex function defined in a neighborhood of \(z_0\)
Smaller to greater: iff
Assuming: \(f\) infinitely differentiable
Context: Complex analysis

Smaller0: \(P\) is not constant
Greater0: \(P\) has a root
Page: 283
Chapter: 5
Section: 5.5
Variables: \(P\) a complex polynomial
Index: 2
Kind: Theo

Question: \(P\) can be factorized as
Answer: \(c_n\Pi(z-z_i)\) where \(z_i\) are roots
Page: 284
Chapter: 5
Section: 5.5
Variables: \(P\) a complex polynomial
Index: 1
Kind: Cor

Question: \(P\) can be factorized as
Answer: \(c_n\Pi p_i\) with \(p_i\) monomial and quadratic polynomials
Page: 285
Chapter: 5
Section: 5.5
Variables: \(P\) a real polynomial
Index: 2
Kind: Cor

Name: The \(p\)-norm of \(\vec x\)
Definition: \(\left(\sum x_i^p\right)^{\sqrt p}\)
Page: 240
Chapter: 5
Section: 4.2a
Variables: \(x_1,\dots,x_n\ge 0\),
\(p\in\mathbb N^+\)
Notation: [$]\norm{\vec x}_p[/$]
Name2: \(L^p\) norm of \(\vec x\)
Extra: Not here in the book, but defining it heres simplifying recalling of Minkowski's inequality

Name: The maximum-norm of \(\vec x\)
Definition: \(\max{x_i}\)
Page: 240
Chapter: 5
Section: 4.2a
Variables: \(x_1,\dots,x_n\ge 0\)
Notation: [$]\norm{\vec x}_\infty[/$]
Name2: The \(L^{\infty}\)-norm of \(\vec x\)
Name3: The uniform norm of \(\vec x\)

Question: Multiplicity of \(r\) as a root of \(P'\) is
Answer: \(k-1\)
Page: 285
Chapter: 5
Section: 5.5
Variables: \(P\) a complex polynomial,
\(k\in\mathbb N\)
\(r\) a root of multiplicity \(k\) of \(P\)
Index: 3
Kind: cor
Assuming: \(k>1\)

Name: A strictly convex/concave function
Definition: it is convex/concave function and the inequality relations are strict.
Page: 241
Chapter: 5
Section: 4.3
Index: 1
Kind: Def
Conjdef: A function \(f\) such that
Extra: Information theory Entropy Ch 2.6, Page 25,
Zorich 1. Ch 5.4.3, P 241

Smaller0: \(\langle\vec x,\vec y\rangle\)
Greater0: [$]\norm{\vec x}_p\norm{\vec y}_q[/$]
Page: 239
Chapter: 5
Section: 4.2a
Variables: \(x_1,y_1,\dots,x_n,y_n>0\), \(p\not\in\{0,1\}\), \(q\) such that \(\frac 1p+\frac 1q=1\)
Smaller to greater: =
Assuming: the \(x\)'s and the \(y\)'s vector are colinear

Question: [$]\frac{d \log_a\abs{x}}{dx^n}[/$]
Answer: equals \(\frac{(-1)^{n-1}(n-1)!x^{-n} }{\ln a}\).
Page: 207
Chapter: 5
Section: 2.6
Variables: \(n\in\mathbb N\)
Kind: Ex

Smaller0: \(\sqrt[p]a\sqrt[q]b\)
Greater0: \(\frac ap+\frac bq\)
Page: 239
Chapter: 5
Section: 4.2a
Variables: \(a,b>0\),
\(p\not\in\{0,1\}\),
\(q\) such that \(\frac 1p+\frac 1q=1\)
Here \(^{(p-1)}\) is the identity operator if \(p>1\), reverse operator if \(p<1\) and transform into equality for \(p=1\)
Smaller to greater: \(\le^{(p>1)}\)
Name: Young's inequality
Greater to smaller: \(\ge^{(p>1)}\)
Extra: \(q>1\) also.
\(q=\frac1{1-\frac1p}=\frac{1}{\frac{p-1}{p}}=\frac{p}{p-1}\)

Name: \(\cosh\)
Definition: [$]\defFun{\mathbb C}{\mathbb C}{x}{\frac{e^x+e^{-x} }{2}}[/$]
Page: 199, 274
Chapter: 5
Section: 2.3
Variables: \(x\in\mathbb C\)
Index: 14
Kind: Ex
Definition2: [$]\defFun{\mathbb C}{\mathbb C}{x}{\cos iz}[/$]
Intuition: mean between \(e^x\) and \(e^{-x}\)

Smaller0: \(f\) is continuous, differentiable on \(]a,b[\) and \(f(a)=f(b)\)
Greater0: \(f'(x)=0\) for some \(x\in]a,b[\)
Page: 213
Chapter: 5
Section: 3.1
Variables: \(a<b\in \mathbb R\)
\(f:]a,b[\to\mathbb R\)
Index: 1
Kind: Prop
Hide middle: yes
Name: Rolle's theorem

Smaller0: \(f^{(i)}(x_0)=0\) for all \(\mathring U(x_0)\), \(i<n\), \(f^{(n)}>0\) and \(n\) even
Greater0: \(x_0\) is a strict local minimal
Page: 237
Chapter: 5
Section: 4.2
Variables: \(x_0\in\mathbb R\),
\(U\) a neighborhood of \(x_0\),
\(f:U\to\mathbb R\),
\(n\in\mathbb N\)
Index: 4
Kind: Prop
Assuming: \(f\) is \(n\)-th order derivable on \(U\)
Extra: n>0 ?

Smaller0: \(f'\) is nonincreasing
Greater0: \(f\) is concave
Page: 243
Chapter: 5
Section: 4.3
Variables: \(f:]a,b[\to\mathbb R\)
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) is differentiable

Smaller0: \(f'\) is decreasing
Greater0: \(f\) is strictly concave
Page: 243
Chapter: 5
Section: 4.3
Variables: \(f:]a,b[\to\mathbb R\)
Kind: Prop
Smaller to greater: iff
Assuming: \(f\) is differentiable

Smaller0: \(x^\alpha-\alpha x+\alpha-1\)
Greater0: 0
Page: 238
Chapter: 5
Section: 4
Variables: \(x>0\)
\(\alpha\in\mathbb R\)
Index: 7
Kind: ex
Smaller to greater: \(\le^{(0<\alpha<1)}\)
Greater to smaller: \(\ge^{(0<\alpha<1)}\)
Hide sides: yes

Definition: \(\le\) if \(x\in I\),
\(=\) if \(x\) is a boundary of \(I\)
\(\ge\) otherwise
Page: 238
Chapter: 5
Section: 4
Variables: \(x\)
\(I\) an open interval
Notation: \(\le^{(x\in I)}\)
Index: 7
Kind: ex
Extra: Introduced by card's author for the sake of simplicity

Name: The set of compact subsets of \(\mathbb R^m\)
Page: 414
Section: 1.3
Chapter: 7
Index: 3
Kind: Prop
Under: taking a subset
Condition: closed subset
Under2: union
Condition2: finite
Under3: intersection
Condition3: arbitrary

Name: The scalar product of \(\vec x\) and \(\vec y\)
Definition: \(\sum x_iy_i\)
Page: 239
Chapter: 5
Section: 4.2a
Variables: \(n\in\mathbb N\)
\(\vec x,\vec y\in\mathbb R^n\)
Notation: \(\langle\vec x,\vec y\rangle\)
Name2: The dot product of \(\vec x\) and \(\vec y\)
Notation2: \(\vec x\cdot\vec y\)

Page: 236
Chapter: 5
Name: Complex numbers
Set: \(a+ib\) for \(a,b\in\mathbb R\)
Notation: \(\mathbb C\)
*: \((a+ib)(c+id)=(ac-bd)+i(ad+bc)\)
+: \((a+ib)+(c+id)=(a+c)+i(b+d)\)
0: 0+i0
1: 1+i0
÷: [$]\frac{\overline{a+ib}}{\norm{a+ib}^2}[/$]
-: \(-a+i(-b)\)
Commutative: y

Smaller0: \(f\) is differentiable on \((x,x+h)\)
Greater0: [$]\abs{f(x+h)-f(x)}\le\sup_{t\in]x,x+h[}\abs{f'(x)}\abs{h}[/$]
Page: 214, 454
Chapter: 5, 8
Section: 3.2, 4.1
Variables: \(m,n\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x,x+h\in G\)
\(f:G\to\mathbb R^n\)
Smaller1: \(f\) is continuous and defined on \([x,x+h]\)
Name: Lagrange finite-increment theorem
Extra: \(]x,x+h[\) is the segment from \(x\) to \(x+h\)

Question: Relation between \(\sin(z)\) and \(\cos(z)\)
Answer: \(\sin^2(z)+\cos^2(z)=1\)
Page: 273
Chapter: 5
Section: 5.3
Variables: \(z\in\mathbb C\)

Question: What can be tell of the convergence of \(\sum _{n=0}^\infty c_n(z-z_0)^n\)
Answer: it converges infinitely on its convergence disk
Page: 279
Chapter: 5
Index: 1
Kind: Theo

Question: Relationship between cosh and sinh
Answer: \(\cosh^2z-\sinh^2z=1\)
Page: 273
Chapter: 5
Section: 5.3
Variables: \(z\in\mathbb C\)
Index: 32
Kind: Ex