20.5. Sequences and Series in the Complex Field#

The extension of sequences and series from the real line \(\mathbb{R}\) to the complex plane \(\mathbb{C} \cong \mathbb{R}^2\) preserves many fundamental operational properties while introducing deeper geometric and analytic structures. Most notably, the equivalence between complex differentiability (holomorphy) and power series convergence (analyticity) forms the cornerstone of complex analysis.

20.5.1. Complex Numerical Sequences and Series#

20.5.1.1. Complex Numerical Sequences#

A complex sequence \(\{z_n\}_{n \in \mathbb{N}} \subset \mathbb{C}\) can be decomposed into its real and imaginary components:

\[z_n = x_n + i y_n, \quad \text{where } x_n, y_n \in \mathbb{R}\]

20.5.1.1.1. Convergence#

Definition: Convergence in \(\mathbb{C}\)

A sequence \(\{z_n\}\) converges to \(L = u + i v \in \mathbb{C}\) (written \(\lim_{n \to \infty} z_n = L\)) if:

\[\lim_{n \to \infty} |z_n - L| = 0 \ .\]

Or, with the \(\espilon\)-definition: given \(\varepsilon > 0\), \(\exists N\) s.t. \(| z_n - L | < \varepsilon\) for \(\forall n > N\).

Equivalently, convergence in \(\mathbb{C}\) under the standard Euclidean metric can be written component-wise, with \(z_n = x_n + i y_n\),

\[\lim_{n \to \infty} z_n = L \iff \left( \lim_{n \to \infty} x_n = u \quad \text{and} \quad \lim_{n \to \infty} y_n = v \right)\]

20.5.1.2. Complex Numerical Series#

A complex infinite series is defined as the formal sum:

\[\sum_{n=0}^{+\infty} z_n, \quad z_n \in \mathbb{C}\]

It converges to \(S \in \mathbb{C}\) if the sequence of partial sums \(S_N = \sum_{n=0}^{N} z_n\) converges to \(S\).

20.5.1.2.1. Convergence#

Convergence. Using the convergence of a sequence, a series is convergent if the sequence of the partial sums \(S_N = \sum_{n=0}^{N} z_n\) converges.

Absolute Convergence: The series \(\sum z_n\) converges absolutely if the real-valued series of moduli converges:

\[\sum_{n=0}^{+\infty} |z_n| < +\infty\]

Property: Absolute convergence in \(\mathbb{C}\) implies standard convergence.

Conditional Convergence: The series converges, but \(\sum |z_n| = +\infty\). As an example:

\[1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ...\]

Tests for absolute convergence

Test

Statement

Ratio Test

If \(\lim_{n \to \infty} \left|\frac{z_{n+1}}{z_n}\right| = L\), then:
• \(L < 1 \implies\) Absolutely Convergent
• \(L > 1 \implies\) Divergent

Root Test

If \(\limsup_{n \to \infty} \sqrt[n]{|z_n|} = L\), then:
• \(L < 1 \implies\) Absolutely Convergent
• \(L > 1 \implies\) Divergent

20.5.2. Complex Sequences and Series of Functions#

Let \(\Omega \subseteq \mathbb{C}\) be an open set and let \(\{f_n\}_{n \in \mathbb{N}}\) be a sequence of complex-valued functions \(f_n: \Omega \to \mathbb{C}\).

20.5.2.1. Convergence types of Sequences#

  1. Pointwise Convergence: \(f_n \to f\) pointwise on \(\Omega\) if for every \(z \in \Omega\):

    \[\lim_{n \to \infty} |f_n(z) - f(z)| = 0\]

    (or for \(\forall z \in \Omega\), \(\forall \varepsilon > 0\), \(\exists N\) so that \(| f_n(z) - f(z) | < \varepsilon\) for \(\forall n > N\).)

  2. Uniform Convergence: \(f_n \rightrightarrows f\) uniformly on \(E \subseteq \Omega\) if:

    \[\lim_{n \to \infty} \sup_{z \in E} |f_n(z) - f(z)| = 0\]

    (or for \(\forall \varepsilon > 0\), \(\exists N\) so that \(| f_n(z) - f(z) | < \varepsilon\) for \(\forall n > N\), \(\forall z \in \Omega\).)

  3. Locally Uniform (Compact) Convergence: \(f_n \to f\) uniformly on every compact subset \(K \subset \Omega\).

20.5.2.2. Weierstrass theorem#

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