List of basic formulas and concepts related to the limits of functions of a complex variable:
- Definition of limit:
[
\lim_{z \to z_0} f(z) = L
]
If for any (\epsilon > 0) there exists (\delta > 0) such that if (0 < |z – z_0| < \delta), then (|f(z) – L| < \epsilon). - One-sided limits:
- Left-hand limit:
[
\lim_{z \to z_0^-} f(z) = L
] - Right-hand limit:
[
\lim_{z \to z_0^+} f(z) = L
]
- Limits at infinity:
[
\lim_{|z| \to \infty} f(z) = L
]
If for any (\epsilon > 0) there exists (R > 0) such that if (|z| > R), then (|f(z) – L| < \epsilon). - Properties of limits:
- Linearity:
[
\lim_{z \to z_0} (af(z) + bg(z)) = a \lim_{z \to z_0} f(z) + b \lim_{z \to z_0} g(z)
]
where (a) and (b) are constants. - Product:
[
\lim_{z \to z_0} (f(z)g(z)) = \lim_{z \to z_0} f(z) \cdot \lim_{z \to z_0} g(z)
]
provided that both limits exist. - Private:
[
\lim_{z \to z_0} \frac{f(z)}{g(z)} = \frac{\lim_{z \to z_0} f(z)}{\lim_{z \to z_0 } g(z)}
]
provided that (\lim_{z \to z_0} g(z) \neq 0).
- Limits of composite functions:
If (g(z)) is continuous at (z_0) and (\lim_{z \to z_0} f(z) = L), then:
[
\lim_{z \to z_0} g(f(z)) = g(L)
]