Sectorial asymptotic

This article is under construction.

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.

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

Let \(f\) and \(g\) be complex-valued functions defined on a domain containing all sufficiently large points of a sector \[ S = \{\, z = r e^{\mathrm i t} \mid r>R,\ t_1 < t < t_2 \,\}, \] where \(R>0\) and \(-\pi < t_1 < t_2 < \pi\).

The function \(f\) is said to have a sectorial asymptotic \(g\) in the sector \(S\) if, for every fixed real \(t\) with \(t_1 < t < t_2\), \[ \lim_{r\to+\infty} \bigl( f(r e^{\mathrm i t}) - g(r e^{\mathrm i t}) \bigr) = 0. \]

This is denoted by \[ f(z) \sim g(z) \quad \text{as } |z|\to\infty,\ z\in S. \]

Remarks

• The limit is taken along rays with fixed argument; no uniformity in \(t\) is required unless explicitly stated.
• Sectorial asymptotic behavior is weaker than global asymptotic behavior, but stronger than pointwise convergence along isolated directions.
• 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

• 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 functions may exhibit logarithmic or power-like sectorial asymptotics while remaining single-valued and holomorphic on the whole complex plane.

Relation to other notions

Sectorial asymptotic behavior is closely related to:

• asymptotic expansions in sectors,
• Stokes phenomena,
• Phragmén–Lindelöf type principles.

References

https://en.wikipedia.org/wiki/Asymptotic_analysis In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

Keywords

«Asymptotic behavior», «Complex analysis», «Entire function», «Logarithm», «Sector», «Sectorial asymptotic»

«Entire Function with Logarithmic Asymptotic», «Superfunction», «Natural tetration», «Tania function», «WrightOmega»,