Title: Theory / Important / Math / Analysis / Zorich{ / 19 Asymptotic Expansions

Name: An asymptotic problem
Definition: Problem of finding a description of a problem when a parameter becomes aribtrarily small or big
Page: 588
Chapter: 19
Extra: The problem of finding a clear, convenient, and essentially correct description of a phenomenon being studied using the specifics of the situation that arises when some parameter (or combination of parameters) that characterizes the phenomenon is small (tends to zero) or, contrariwise, large (tends to infinity).

Name: An asymptotic method
Definition: A method for solving asymptotic problem
Page: 589
Chapter: 19

Name: Asymptotic estimates of \(f\)
Definition: An equation of the form \(f=O(g)\) or \(f=o(g)\)
Page: 590
Chapter: 19
Section: 1.1
Variables: \(f\) a function

Name: Asymptotic equivalence of \(f\) over \(\mathcal B\)
Definition: An equation of the form \(f\sim_{\mathcal B} g\)
Page: 590
Chapter: 19
Section: 1.1
Variables: \(f\) a funciton
\(\mathcal B\) a base
Name2: Asymptotic equality of \(f\) over \(\mathcal B\)

Name: Asymptotic formulas
Definition: Asymptotic estimates
Page: 590
Chapter: 19
Section: 1.1
Variables: \(f\) a funciton
\(\mathcal B\) a base
Definition2: asymptotic equivalence
Conjdef: Both

Question: \(\ln n!\) can be approximated by
Answer: \(n\ln n+O(n)\)
Page: 591
Chapter: 19
Section: 1.1
Index: 6
Kind: Ex

Question: [$]\int_1^x\frac{e^t}{t^n}\dd t[/$] is asymptotically equivalent to
Answer: \(x^{-n}e^x\)
Page: 591
Chapter: 19
Section: 1.1
Index: 7
Kind: Ex

Question: [$]\int_1^x\frac{e^t}{t}\dd t[/$] is approximated by
Answer: \(e^x\sum_{k=1}^n\frac{(k-1)!}{x^k}+O\left(\frac{e^x}{x^{n+1}}\right)\)
Page: 592
Chapter: 19
Section: 1
Index: 8
Kind: Ex

Definition: For each integer \(k\), \(f(x)=_{\mathcal B}\sum_{i=0}^{k-1}\psi_i(x)+o(\psi_{k}(x))\)
Page: 593
Chapter: 19
Section: 1
Variables: \(f,(\psi_i)\) functions with same domain and codomain
\(\mathcal B\) a basis
Notation: \(f(x)\simeq_{\mathcal B}\sum_{k=0}^\infty\psi_k(x)\)
Index: 2
Kind: Def

Name: An asymptotic expension of \(f\) in the base \(\mathcal B\)
Definition: A sequence \((\psi_i)\) such that \(f(x)\simeq_{\mathcal B}\sum_{k=0}^\infty\psi_k(x)\)
Page: 593
Chapter: 19
Section: 1
Variables: \(f\) functions with same domain and codomain
\(\mathcal B\) a basis
Index: 2
Kind: Def
Extra: \(f(x)\simeq_{\mathcal B}\sum_{k=0}^\infty\psi_k(x)\) means that:
\(f(x)\simeq_{\mathcal B}\sum_{k=0}^{n-1}\psi_k(x)+o(\psi_n(x))\) for all \(n\) over the basis \(\mathcal B\)
Intuition: Generalization of Taylor expansion, such that the sum does not necessarily converge and the function does not have to be derivable

Smaller0: \((\psi_i)\) is an asymptotic expansion
Greater0: \((\psi_i)\) is an asymptotic sequence
Page: 593
Chapter: 19
Section: 1
Variables: \(\mathcal B\) a basis
A sequence \((\psi_i)\)
Index: 2
Kind: Def
Counterexample: \(\sum_{i=0}^{+\infty}x^i\) is a valid asymptotic sequenc without being an expansion of a function

Name: An asymptotic sequence over the base \(\mathcal B\)
Definition: \(\psi_{i+1}=o(\psi_{i})\)
Page: 593
Chapter: 19
Section: 1
Variables: \(\mathcal B\) a basis
Index: 3
Kind: Def
Definition2: no \(\psi_i\) is identically 0 on any element of \(\mathcal B\)
Example: \((x^{n_i})\) as \(x\to 0\) for \((n_i)\) strictly increasing
Conjdef: both
Example2: \((x^{n_i})\) as \(x\to\infty\) for \((n_i)\) strictly decreasing
DefType: A sequence \((\psi_i)\)

Name: The set of asymptotic sequences over the base \(\mathcal B\)
Page: 594
Section: 1
Chapter: 19
Index: 9
Variables: \(\mathcal B\) a basis
Kind: Ex
Under: multiplication of all term by the same function

Name: The set of pair of functions such that \(f=O(g)\)
Page: 596
Section: 1.2b
Chapter: 19
Index: 3
Variables: \(g\) nonnegative
Kind: Prop
Under: linear combination
Under2: integration

Name: The set of pair with a function and its asymptotic expansion
Page: 596
Section: 1.2b
Chapter: 19
Index: 2
Variables: \(\mathcal B\) a basis
\((\phi_i)\) an asymptotic sequence over the base \(\mathcal B\)
Kind: Prop
Under: linear combination
Condition: finite

Name: The set of pair of functions such that \(f\sim o(g)\)
Page: 596
Section: 1.2b
Chapter: 19
Index: 3
Variables: \(g\) nonnegative
Kind: Prop
Under: linear combination
Condition: finite
Under2: integration
Condition2: [$]\int_a^xg(t)\dd t[/$] diverges
Closure3: transitive

Name: The set of pair of functions such that \(f=o(g)\)
Page: 596
Section: 1.2b
Chapter: 19
Index: 3
Variables: \(g\) nonnegative
Kind: Prop
Extra: difference avec sim ?
Under: linear combination
Condition: finite
Under2: integration
Condition2: [$]\int_a^xg(t)\dd t[/$] diverges

Name: An asymptotic expansion of \(f\) with respect to the asymptotic sequence \(\{\phi_i\}\) over \(\mathcal B\)
Definition: An asymptotic expension of the form \(f(x)\simeq\sum_{i=0}^\infty c_i\phi_i(x)\)
Page: 594, 595
Chapter: 19
Section: 1
Variables: \(\mathcal B\) a basis
\((\phi_i)\) an asymptotic sequence over the base \(\mathcal B\)
\(f\) a function
Index: 4, 1
Kind: Def, Prop
Extra: checked
Cardinal: unique

Name: An asymptotic zero with respect to \((\phi_i)\)
Definition: A function \(f\) such that \(f(x)=_{\mathcal B}o(\phi_n(x))\) for each \(n\)
Page: 595
Chapter: 19
Section: 1.2a
Variables: \(\mathcal B\) a basis
\((\phi_i)\) an asymptotic sequence over the base \(\mathcal B\)
Context: Asymptotic sequences
Index: 5
Kind: Def
Example: \(e^{-x}\) is asymptotically zero with respect to \((\frac 1{x^n})\)

Page: 595
Definition: Functions \((f,g)\) such that \((f-g)\) is asymptotically zero
Chapter: 19
Section: 1.2a
Variables: \(\mathcal B\) a basis
\((\phi_i)\) an asymptotic sequence over the base \(\mathcal B\)
Name: Asymptotically equal functions over \(\mathcal B\)
Kind: Def, Prop
Index: 6, 1
Definition2: Functions \((f,g)\) having the same asymptotical extpansion over \(\mathcal B\)
Equivalence: yes