Difference between revisions of "Sectorial asymptotic"

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(Created page with "{{top}} This article is under construction. In complex analysis, the term Sectorial asymptotic refers to the asymptotic behavior of a complex-valued function along rays c...")
 
 
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This article is under construction.
  
This article is under construction.
   
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In complex analysis, the term [[Sectorial asymptotic]] refers to the asymptotic behavior of a complex-valued function along rays contained in a sector of the complex plane.
 +
A [[Sectorial asymptotic]] is a special case of an [[Asymptotic]] whose [[domain of validity]] is restricted to a [[sector of the complex plane]].
   
  +
This type of asymptotic is widely used in [[complex analysis]] and in the theory of [[special function]]s, especially for describing the behavior of functions at infinity.
−
The concept is used when global asymptotic behavior does not exist or is not meaningful, but a well-defined asymptotic expansion holds in restricted angular regions.
  
   
 
==Definition==
  
==Definition==
   
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Let \(f\) and \(g\) be complex-valued functions defined on a domain containing all sufficiently large points of a sector
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Let \(f\) be a complex-valued function and let \(C\) be a complex number or \( \infty \).
  +
  +
An [[asymptotic]] \(A\) of \(f\) at \(C\) is called a **sectorial asymptotic** if the limit
 
\[
  
\[
  +
\lim_{z\to C} (f(z)-A(z))\,B(z)=0
−
S = \{\, z = r e^{\mathrm i t} \mid r>R,\ t_1 < t < t_2 \,\},
  
 
\]
  
\]
  +
is taken only over values of (z) belonging to a fixed sector of the complex plane.
−
where \(R>0\) and \(-\pi < t_1 < t_2 < \pi\).
  
  +
  +
Function \(B\) qualifies the quality of the asumptotic \(A\).
  +
Both \(A\) and \(B\) are supposed to be easier to evaluate than the initial function \(f\).
  +
Usually, \(B\) is just a [[power function]] or its combination with a [[Logarithm]].
  +
  ⚫
== Case 1: Sectorial asymptotic at infinity ==
   
  +
For \(C=\infty\), the allowed values of \(z\) are assumed to satisfy
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The function \(f\) is said to have a [[sectorial asymptotic]] \(g\) in the sector \(S\) if, for every fixed real \(t\) with \(t_1 < t < t_2\),
  
 
\[
  
\[
−
\lim_{r\to+\infty} \bigl( f(r e^{\mathrm i t}) - g(r e^{\mathrm i t}) \bigr) = 0.
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z = r\,\exp(\mathrm i t),
  +
\qquad r \to +\infty\,
  +
\qquad t_{\min} < t < t_{\max},
 
\]
  
\]
  +
where (t_{\min}) and (t_{\max}) are fixed real numbers with
  ⚫
\[
  +
-\pi < t_{\min} < t_{\max} < \pi.
  ⚫
\]
  +
  +
The angle interval \((t_{\min},t_{\max})\) is fixed and
  +
does not depend on \(r\).
  +
  +
In this case, the asymptotic behavior of \(f(z)\) may differ for different sectors, even when \( |z| \to \infty \).
  +
  +
== Case 2: Sectorial asymptotic at a finite point ==
   
  +
For a finite point \(C\), a sectorial asymptotic refers to the approach
−
This is denoted by
  
 
\[
  
\[
−
f(z) \sim g(z)
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z \to C
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\quad \text{as } |z|\to\infty,\ z\in S.
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\quad\text{with}\quad
  ⚫
z-C = r\,\exp(\mathrm i t),
  +
\qquad r \to 0^+\,
  +
\qquad t_{\min} < t < t_{\max}.
 
\]
  
\]
  +
  +
Such asymptotics describe directional behavior near branch points or other singularities.
   
 
==Remarks==
  
==Remarks==
   
  +
* Sectorial asymptotics are typically **not uniform** across the whole complex plane.
−
* The limit is taken along rays with fixed argument; no uniformity in \(t\) is required unless explicitly stated.
  
  ⚫
* Different sectors may admit **different asymptotic expansions** for the same function.
−
* Sectorial asymptotic behavior is weaker than global asymptotic behavior, but stronger than pointwise convergence along isolated directions.
  
  +
* Transitions between sectors are often associated with rapid changes of behavior; in asymptotic analysis, these boundaries are sometimes related to [[Stokes lines]].
⚫
* Different sectors may admit different asymptotic expansions for the same function.
  
−
* Sectorial asymptotics are common in the theory of asymptotic expansions, summability theory, and the study of functional equations.
  
   
 
==Examples==
  
==Examples==
   
  +
* The [[logarithm]] has sectorial asymptotics at infinity, valid in any sector avoiding its [[branch cut]].
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* The principal branch of the [[Logarithm]] satisfies
  
⚫
\[
  
⚫
\log z \sim \log(r e^{\mathrm i t})
  
−
\quad \text{as } r\to\infty
  
⚫
\]
  
−
in any sector avoiding the negative real axis.
  
−
* Certain [[entire function]]s may exhibit logarithmic or power-like sectorial asymptotics while remaining single-valued and holomorphic on the whole complex plane.
  
   
  +
* The function \( \sqrt{1+z} \) has a sectorial asymptotic \( \sqrt{z} \) for \( |z|\to\infty \) in sectors avoiding the negative real axis.
⚫
==Relation to other notions==
  
   
  +
* An [[Entire Function with Logarithmic Asymptotic]] mimics the logarithm in wide sectors while remaining holomorphic everywhere.
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Sectorial asymptotic behavior is closely related to:
  
  +
−
* asymptotic expansions in sectors,
  
  +
* Superfunctions such as [[SuTra]] exhibit sectorial asymptotics derived from their underlying [[transfer function]].
−
* Stokes phenomena,
  
  +
−
* Phragmén–Lindelöf type principles.
  
  ⚫
==Relation to other asymptotics==
  +
  +
* A [[Strip asymptotic]] restricts \(z\) by bounding \( \Im(z) \) instead of \( \arg(z) \).
  +
  +
* More general asymptotics may involve curved domains or piecewise-defined regions.
  +
  +
==Acknowledgement==
  +
[[ChatGPT]] helps to improve this article.
   
 
==References==
  
==References==
 
{{ref}}
  
{{ref}}
  +
−
https://en.wikipedia.org/wiki/Asymptotic_analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
 
  +
https://dlmf.nist.gov
  +
  +
https://en.wikipedia.org/wiki/Asymptotic_analysis
  +
 
{{fer}}
  
{{fer}}
   
 
==Keywords==
  
==Keywords==
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«[[Asymptotic behavior]]»,
 +
«[[Asymptotic]]»,
  +
«[[Sectorial asymptotic]]»,
  +
«[[Domain of validity]]»,
 
«[[Complex analysis]]»,
  
«[[Complex analysis]]»,
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«[[Entire function]]»,
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«[[Special function]]»
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«[[Logarithm]]»,
  
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«[[Sector]]»,
  
⚫
«[[Sectorial asymptotic]]»
  
−  
−
«[[Entire Function with Logarithmic Asymptotic]]»,
  
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«[[Superfunction]]»,
  
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«[[Natural tetration]]»,
  
−
«[[Tania function]]»,
  
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«[[WrightOmega]]»,
  
   
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[[Category:Complex analysis]]
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[[Category:Asymptotic]]
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[[Category:Asymptotic behavior]]
  
 
[[Category:Sectorial asymptotic]]
  
[[Category:Sectorial asymptotic]]

Latest revision as of 21:03, 12 January 2026


This article is under construction.

A Sectorial asymptotic is a special case of an Asymptotic whose domain of validity is restricted to a sector of the complex plane.

This type of asymptotic is widely used in complex analysis and in the theory of special functions, especially for describing the behavior of functions at infinity.

Definition

Let \(f\) be a complex-valued function and let \(C\) be a complex number or \( \infty \).

An asymptotic \(A\) of \(f\) at \(C\) is called a **sectorial asymptotic** if the limit \[ \lim_{z\to C} (f(z)-A(z))\,B(z)=0 \] is taken only over values of (z) belonging to a fixed sector of the complex plane.

Function \(B\) qualifies the quality of the asumptotic \(A\). Both \(A\) and \(B\) are supposed to be easier to evaluate than the initial function \(f\). Usually, \(B\) is just a power function or its combination with a Logarithm.

Case 1: Sectorial asymptotic at infinity

For \(C=\infty\), the allowed values of \(z\) are assumed to satisfy \[ z = r\,\exp(\mathrm i t), \qquad r \to +\infty\, \qquad t_{\min} < t < t_{\max}, \] where (t_{\min}) and (t_{\max}) are fixed real numbers with \[ -\pi < t_{\min} < t_{\max} < \pi. \]

The angle interval \((t_{\min},t_{\max})\) is fixed and does not depend on \(r\).

In this case, the asymptotic behavior of \(f(z)\) may differ for different sectors, even when \( |z| \to \infty \).

Case 2: Sectorial asymptotic at a finite point

For a finite point \(C\), a sectorial asymptotic refers to the approach \[ z \to C \quad\text{with}\quad z-C = r\,\exp(\mathrm i t), \qquad r \to 0^+\, \qquad t_{\min} < t < t_{\max}. \]

Such asymptotics describe directional behavior near branch points or other singularities.

Remarks

  • Sectorial asymptotics are typically **not uniform** across the whole complex plane.
  • Different sectors may admit **different asymptotic expansions** for the same function.
  • Transitions between sectors are often associated with rapid changes of behavior; in asymptotic analysis, these boundaries are sometimes related to Stokes lines.

Examples

  • The logarithm has sectorial asymptotics at infinity, valid in any sector avoiding its branch cut.
  • The function \( \sqrt{1+z} \) has a sectorial asymptotic \( \sqrt{z} \) for \( |z|\to\infty \) in sectors avoiding the negative real axis.

Relation to other asymptotics

  • A Strip asymptotic restricts \(z\) by bounding \( \Im(z) \) instead of \( \arg(z) \).
  • More general asymptotics may involve curved domains or piecewise-defined regions.

Acknowledgement

ChatGPT helps to improve this article.

References

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