3.1 Limits and continuity

Complex functions build on real-valued functions, extending concepts like limits and continuity to the complex plane. These ideas are crucial for understanding analytic functions, which are smooth and well-behaved in complex analysis.

Limits and continuity in complex analysis mirror their real counterparts but with some key differences. The main distinction is that we're now dealing with two-dimensional complex numbers, which adds depth to these fundamental concepts.

Limits of Complex Functions

Definition of Limit at a Point and Infinity

• The limit of a complex function $f(z)$ as $z$ approaches a point $z_0$ is a complex number $L$ if, for any positive real number $\varepsilon$, there exists a positive real number $\delta$ such that $|f(z) - L| < \varepsilon$ whenever $0 < |z - z_0| < \delta$
  • If the limit $L$ exists, we write $\lim_{z \to z_0} f(z) = L$
  • If the limit does not exist, we say the limit of $f(z)$ as $z$ approaches $z_0$ does not exist or is undefined
• The limit of a complex function $f(z)$ as $z$ approaches infinity is a complex number $L$ if, for any positive real number $\varepsilon$, there exists a positive real number $M$ such that $|f(z) - L| < \varepsilon$ whenever $|z| > M$
  • If the limit $L$ exists as $z$ approaches infinity, we write $\lim_{z \to \infty} f(z) = L$
  • If the limit does not exist, we say the limit of $f(z)$ as $z$ approaches infinity does not exist or is undefined

Evaluating Limits Using Real and Imaginary Parts

• The limit of a complex function $f(z)$ as $z$ approaches a point $z_0$ or infinity can be evaluated by considering the limits of the real and imaginary parts of $f(z)$ separately, provided both limits exist
  • Let $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$
  • $\lim_{z \to z_0} f(z) = \lim_{(x, y) \to (x_0, y_0)} u(x, y) + i \lim_{(x, y) \to (x_0, y_0)} v(x, y)$
  • If both limits exist, then the limit of $f(z)$ exists and is equal to the sum of the limits of the real and imaginary parts
  • Example: $\lim_{z \to 1+i} (z^2 + 2z) = \lim_{(x, y) \to (1, 1)} (x^2 - y^2 + 2x) + i \lim_{(x, y) \to (1, 1)} (2xy + 2y) = 3 + 4i$

Continuity of Complex Functions

Conditions for Continuity

• A complex function $f(z)$ is continuous at a point $z_0$ if and only if the following three conditions are satisfied:
  1. $f(z_0)$ is defined (i.e., $z_0$ is in the domain of $f$)
  2. $\lim_{z \to z_0} f(z)$ exists
  3. $\lim_{z \to z_0} f(z) = f(z_0)$
• A complex function $f(z)$ is continuous on a domain $D$ if it is continuous at every point in $D$

Properties of Continuous Functions

• The sum, difference, product, and quotient of continuous complex functions are also continuous, provided the quotient is not divided by zero
  • If $f(z)$ and $g(z)$ are continuous at $z_0$, then $f(z) \pm g(z)$, $f(z) \cdot g(z)$, and $f(z) / g(z)$ (provided $g(z_0) \neq 0$) are also continuous at $z_0$
• Polynomial functions, exponential functions, and trigonometric functions are continuous on their entire domains
  • Example: $f(z) = z^3 + 2z + 1$ is continuous on the entire complex plane
• Rational functions are continuous on their domains, except at points where the denominator is zero
  • Example: $f(z) = \frac{1}{z - 1}$ is continuous on the complex plane except at $z = 1$

Evaluating Limits of Complex Functions

Properties of Limits

• The limit of a constant function is the constant itself: $\lim_{z \to z_0} c = c$, where $c$ is a complex constant
• The limit of the sum (or difference) of two complex functions is the sum (or difference) of their limits: $\lim_{z \to z_0} [f(z) \pm g(z)] = \lim_{z \to z_0} f(z) \pm \lim_{z \to z_0} g(z)$
• The limit of the product of two complex functions is the product of their limits: $\lim_{z \to z_0} [f(z) \cdot g(z)] = \lim_{z \to z_0} f(z) \cdot \lim_{z \to z_0} g(z)$
• The limit of the quotient of two complex functions is the quotient of their limits, provided the limit of the denominator is not zero: $\lim_{z \to z_0} [f(z) / g(z)] = \lim_{z \to z_0} f(z) / \lim_{z \to z_0} g(z)$, where $\lim_{z \to z_0} g(z) \neq 0$

Squeeze Theorem

• If $f(z) \leq g(z) \leq h(z)$ for all $z$ in a neighborhood of $z_0$ (except possibly at $z_0$) and $\lim_{z \to z_0} f(z) = \lim_{z \to z_0} h(z) = L$, then $\lim_{z \to z_0} g(z) = L$
  • The squeeze theorem is useful for evaluating limits of functions that are "sandwiched" between two other functions with known limits
  • Example: If $0 \leq |f(z)| \leq |z|$ for all $z$ in a neighborhood of 0 and $\lim_{z \to 0} |z| = 0$, then $\lim_{z \to 0} f(z) = 0$

Proving Continuity of Complex Functions

Using the Definition of Continuity

• To prove that a complex function $f(z)$ is continuous at a point $z_0$ using the definition of continuity, one must show that for any $\varepsilon > 0$, there exists a $\delta > 0$ such that $|f(z) - f(z_0)| < \varepsilon$ whenever $|z - z_0| < \delta$
  • The proof typically involves algebraic manipulation of the inequality $|f(z) - f(z_0)| < \varepsilon$ to isolate $|z - z_0|$ and determine a suitable value for $\delta$ in terms of $\varepsilon$
  • The value of $\delta$ may depend on $z_0$ and the specific form of the function $f(z)$
  • Once a suitable $\delta$ is found, the proof is complete, demonstrating that $f(z)$ is continuous at $z_0$

Proving Continuity on a Domain

• To prove continuity on a domain, one must show that the function is continuous at every point in the domain, which may require considering different cases or using properties of continuous functions
  • For example, to prove that a polynomial function is continuous on the entire complex plane, one can use the properties of continuous functions (sum, product, and composition of continuous functions are continuous)
  • For a rational function, one might need to consider the continuity at points where the denominator is zero separately from the rest of the domain
  • In some cases, it may be more efficient to prove continuity on a domain by showing that the real and imaginary parts of the function are continuous on the corresponding real domain