Complex Analysis: Standing Notation

(i) Base. This setting extends The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, whose chain includes Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, fixing the modulus $|z|$, the real and imaginary parts (Real and Imaginary Parts of a Complex Number (i)) and the metric space $(\mathbb{C},d_{\mathbb{C}})$: their conventions and notation are in force.

(ii) Conjugates. For $z\in\mathbb{C}$, $\overline{z}$ is the complex conjugate of $z$, as in The Complex Conjugate of a Complex Number (i).

(iii) Real series. Convergent sequences in $\mathbb{R}$ are as in Convergent Sequences of Real Numbers (ii). For a sequence $(x_{n})$ in $\mathbb{R}$, partial sums, convergence of the series $\sum_{k=1}^{\infty}x_{k}$ and its sum, also written $\sum_{k=1}^{\infty}x_{k}$, and absolute convergence are as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (i), Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii) and Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii).

(iv) The real exponential. $\exp_{\mathbb{R}}:\mathbb{R}\to\mathbb{R}$ is the exponential function of The Real Exponential Function (ii), and $e=\exp_{\mathbb{R}}(1)$ is the number of The Real Exponential Function (iii).