Examples of sectorial and strip asymptotics

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 functions.

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

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Table

Function f(z)	Asymptotic form g(z)	Type	Sector / Strip	Limiting angles (t₁,t₂)	Reason for restriction
log z (principal branch)	log z	Sectorial	ℂ \ (-∞,0]	t₁ = -π, t₂ = π	Branch cut along negative real axis
Φ(z) = −SuTra(−z)	log z	Sectorial	Almost all plane	t₁ = −π+ε, t₂ = π−ε	Logarithmic behavior except near inherited cut
√(1+z) (principal branch)	√z	Sectorial	ℂ \ (−∞,−1]	t₁ = −π, t₂ = π	Branch cut starting at z = −1
√(1+z²)	+z	Sectorial	|arg z| < π/2 − ε	t₁ ≈ −π/2, t₂ ≈ π/2	Branch points at z = ±i
√(1+z²)	−z	Sectorial	π/2 + ε < |arg z| < π − ε	t₁ ≈ π/2, t₂ ≈ 3π/2	Change of dominant branch
Γ(z) (Gamma)	Stirling formula	Sectorial	|arg z| < π − ε	t₁ = −π+ε, t₂ = π−ε	Essential singularity at infinity
Natural arctetration	log z + const	Strip	|Im z| < Im L	—	Fixed point of logarithm
Tania function	Various exponentials	Sectorial + Strip	3 sectors + strip	Multiple (case-dependent)	Multiple dominant dynamics

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Keywords

Asymptotic,