View source for Examples of sectorial and strip asymptotics

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Terms related to [[asymptotic]] behavior of functions of a complex variable caused confusions.

On the other hand, the calculus of the asymptotic behavior is important tool
for implementation of [[special function]]s.

In particular, the asymptotic expansions are used for evaluation of functions defined in book "[[Superfunctions]]".

The description of [[asymptotic]] analysis at Wikipedia <ref>
https://en.wikipedia.org/wiki/Asymptotic_analysis
In mathematical analysis, asymptotic analysis, also known as [[asymptotic]]s, is a method of describing limiting behavior. ..
</ref>
seem to be not sufficient. 

Editor and [[ChatGPT]] find no appropriate definitions in other dictionaries.

The table below is suggested to elaborate the better definitions of the related terms.
==Warning==
This article is generated by [[ChatGPT]] and may require serious revision.

==Table==

<table border="1" cellpadding="1" cellspacing="0" border-collapse="collapse">
<tr>
<th>Function f(z)</th>
<th>Asymptotic form g(z)</th>
<th>Type</th>
<th>Sector / Strip</th>
<th>Limiting angles (t₁,t₂)</th>
<th>Reason for restriction</th>
</tr>

<tr>
<td><b>log z</b> (principal branch)</td>
<td>log z</td>
<td>Sectorial</td>
<td>ℂ \ (-∞,0]</td>
<td>t₁ = -π, t₂ = π</td>
<td>Branch cut along negative real axis</td>
</tr>

<tr>
<td><b>Φ(z) = −SuTra(−z)</b></td>
<td>log z</td>
<td>Sectorial</td>
<td>Almost all plane</td>
<td>t₁ = −π+ε, t₂ = π−ε</td>
<td>Logarithmic behavior except near inherited cut</td>
</tr>

<tr>
<td><b>√(1+z)</b> (principal branch)</td>
<td>√z</td>
<td>Sectorial</td>
<td>ℂ \ (−∞,−1]</td>
<td>t₁ = −π, t₂ = π</td>
<td>Branch cut starting at z = −1</td>
</tr>

<tr>
<td><b>√(1+z²)</b></td>
<td>+z</td>
<td>Sectorial</td>
<td>|arg z| &lt; π/2 − ε</td>
<td>t₁ ≈ −π/2, t₂ ≈ π/2</td>
<td>Branch points at z = ±i</td>
</tr>

<tr>
<td><b>√(1+z²)</b></td>
<td>−z</td>
<td>Sectorial</td>
<td>π/2 + ε &lt; |arg z| &lt; π − ε</td>
<td>t₁ ≈ π/2, t₂ ≈ 3π/2</td>
<td>Change of dominant branch</td>
</tr>

<tr>
<td><b>Γ(z)</b> (Gamma)</td>
<td>Stirling formula</td>
<td>Sectorial</td>
<td>|arg z| &lt; π − ε</td>
<td>t₁ = −π+ε, t₂ = π−ε</td>
<td>Essential singularity at infinity</td>
</tr>

<tr>
<td><b>Natural arctetration</b></td>
<td>log z + const</td>
<td>Strip</td>
<td>|Im z| &lt; Im L</td>
<td>—</td>
<td>Fixed point of logarithm</td>
</tr>

<tr>
<td><b>Tania function</b></td>
<td>Various exponentials</td>
<td>Sectorial + Strip</td>
<td>3 sectors + strip</td>
<td>Multiple (case-dependent)</td>
<td>Multiple dominant dynamics</td>
</tr>
</table>

==Description==

==References==
{{ref}}

{{fer}}

==Keywords==

[[Asymptotic]],

[[Category:Asymptotic]]