Strip asymptotic

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In complex analysis, a **strip asymptotic** describes the asymptotic behavior of a function in an **unbounded horizontal or vertical strip** of the complex plane, rather than in an angular sector.

In TORI, by default, the branch cuts are horizontal half-lines directed toward the negative of the Real axis. In the current version, the Strip asymptotic is defined only for this case.

Definition

Let \(f\) be a complex-valued function. We say that \(f\) has a strip asymptotic \[ f(z)\sim g(z) \] in a strip if there exist real numbers \(a<b\) and \(R>0\) such that \[ \lim_{\Re z\to \pm\infty} \big(f(z)-g(z)\big)=0 \quad\text{uniformly for } a<\Im z<b . \]

The strip \[ \{z\in\mathbb C : a<\Im z<b\} \] is called the domain of validity of the strip asymptotic.

Remarks

• A strip asymptotic is **not** a special case of a sectorial asymptotic.
• In contrast to sectorial asymptotics, the strip asymptotics are not tied to a fixed angular direction.
• Strip asymptotics naturally arise in problems involving iteration, transfer equations, and quasiperiodic behavior.

Examples

Tania function

The Tania function exhibits a strip asymptotic for \[ |\Im z|<\pi,\quad \Re z<0. \]

It is not the same as for the similar function WrightOmega

Natural arctetration

Natural arctetration admits the strip asymptotic in \[ |\Im z|<\Im(L), \] where \(L\approx 0.3+1.3\,\mathrm i\) is the fixed point of the natural Logarithm.

Rerences

No definition of term Strip asymptotic is found in dictionaries. The closest topic is found at Wikipedia:

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», «Sectorial asymptotic», «Strip», «Strip asymptotic», «Natural arctetration», «Tania function»