Sectorial asymptotic
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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.
- An Entire Function with Logarithmic Asymptotic mimics the logarithm in wide sectors while remaining holomorphic everywhere.
- Superfunctions such as SuTra exhibit sectorial asymptotics derived from their underlying transfer function.
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
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References
Keywords
«Asymptotic», «Sectorial asymptotic», «Domain of validity», «Complex analysis», «Special function»