Convergence of Generalized Sum of Complex Numbers
Theorem
Let $\family {z_j}_{j \mathop \in I}$ be an $I$-indexed family of complex numbers.
That is, let $z_j \in \C$ for all $j \in I$.
Let $\map \Re {z_j}$ and $\map \Im {z_j}$ denote the families of real and imaginary parts of the family $z_j$.
Then the following are equivalent:
- $(1): \quad \ds \sum_{j \mathop \in I} z_j$ converges to $z \in \C$
- $(2): \quad \ds \sum_{j \mathop \in I} \map \Re {z_j}, \sum_{j \mathop \in I} \map \Im {z_j}$ converge to $\map \Re z, \map \Im z \in \R$, respectively
Corollary
Suppose that $\ds \sum_{j \mathop \in I} z_j$ converges to $z \in \C$.
Then $\ds \sum_{j \mathop \in I} \overline {z_j}$ converges to $\overline z$, where $\overline z$ denotes the complex conjugate of $z$.
Proof
$(2)$ implies $(1)$
By Generalized Sum is Linear, the stated convergences lead to:
| \(\ds z\) | \(=\) | \(\ds \map \Re z + i \map \Im z\) | Definition of Real Part and Definition of Imaginary Part | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} \map \Re {z_j} + i \sum_{j \mathop \in I} \map \Im {z_j}\) | Statement $(2)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} \paren {\map \Re {z_j} + i \map \Im {z_j} }\) | Generalized Sum is Linear | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in I} z_j\) | Definition of Real Part and Definition of Imaginary Part |
$\Box$
$(1)$ implies $(2)$
Statement $(1)$, according to the definition of convergence, amounts to the following:
For every $\epsilon > 0$, there exists a finite $G \subseteq I$ such that:
- For every finite $F \subseteq I$ with $G \subseteq F$:
- $\ds \cmod {z - \sum_{j \mathop \in F} z_j} < \epsilon$
Now suppose that for $\epsilon > 0$, $G$ and $F$ are as above. Then observe that:
| \(\ds \epsilon^2\) | \(>\) | \(\ds \cmod {z - \sum_{j \mathop \in F} z_j}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\map \Re z - \sum_{j \mathop \in F} \map \Re {z_j} }^2 + \paren {\map \Im z - \sum_{j \mathop \in F} \map \Im {z_j} }^2\) | Definition of Modulus of Complex Number |
Hence, by Square of Real Number is Non-Negative, both of the terms on the right hand side are smaller than $\epsilon^2$.
It follows that, taking square roots, $G$ satisfies, for any finite $F \supseteq G$:
- $\ds \size {\map \Re z - \sum_{j \mathop \in F} \map \Re {z_j} } < \epsilon$
- $\ds \size {\map \Im z - \sum_{j \mathop \in F} \map \Im {z_j} } < \epsilon$
As $\epsilon > 0$ was arbitrary, using the definition of convergence, this implies precisely that:
- $\ds \sum_{j \mathop \in I} \map \Re {z_j}, \sum_{j \mathop \in I} \map \Im {z_j}$ converge to $\map \Re z, \map \Im z \in \R$, respectively.
Hence, $(1)$ is shown to imply $(2)$.
$\blacksquare$
Also see
- Generalized Sum is Linear, of which this is a partial converse.