\[ \newcommand{\ci}{\mathrm{i}} \newcommand{\e}{\mathrm{e}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\eps}{\varepsilon} \newcommand{\dsum}{\displaystyle\sum} \]
Abstract
This article develops the exponential and trigonometric functions from first principles using infinite series, avoiding reliance on geometric intuition or pre-existing notions of angle. Motivated by foundational concerns raised in early twentieth-century analysis, the construction is carried out directly on the complex plane. After establishing the algebraic structure and completeness of \(\C\) as a normed vector space, basic results on complex series are proved, including absolute convergence and a Fubini-type theorem. The exponential function is then defined by its power series, shown to be well-defined on \(\C\), and its fundamental properties—such as the exponent law, positivity on \(\R\), and monotonicity—are derived. Trigonometric functions are introduced via the complex exponential, leading naturally to Euler’s formula, addition formulas, and the Pythagorean identity. This approach demonstrates that the classical properties of exponential and trigonometric functions arise purely from analytic and algebraic considerations, without geometric assumptions.