The Definition of Exponential Function and Trigonometric Functions and Their Properties

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 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 , and its fundamental properties—such as the exponent law, positivity on , 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.

Introduction

The definition of exponential function and trigonometric functions originates from some elementary perspectives at
first. Exponential function was defined as an extension of exponentiation with rational exponents, and trigonometric
arose from geometric connections between angles and lengths.

However, in modern maths, these intuitive definitions aren’t well enough. G. H. Hardy noted in his 1908 work A Course
of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory,
because it depends implicitly on a notion of angle that can be measured by a real number. Exponential function extended
from rational exponents are also hard to achieve some of the basic properties easily.

Therefore, modern definitions express exponential function and trigonometric functions as infinite series or as
solutions of differential equations. This paper will discuss about the former in detail.

This paper is going to define these functions directly on , so the first part is some properties of complex series,
followed by definitions and properties of exponential and trigonometric functions. We assume some of the basic
properties of (it’s a complete field with an Archimedean total order), and will build everything else from scratch.

Complex Series

Complex Numbers

We define

With addition

And multiplication

We denote , then will be denoted as .

We define

Then .

Completeness of

We see as a vector space on and define the norm on as

Theorem (Completeness of ) is complete under this norm.

Proof. For any Cauchy sequence on , that is, that satisfies

Since

We have

And constructs a Cauchy sequence on . Say

Similarly, we assume that

Then, , choose such that

Choose such that

Let . Then

Thus converges to .

Since is a complete normed vector space (Banach space) over , we naturally have

Theorem (Absolute Convergence of Complex Series) If a complex series converges absolutely, it converges.

Proof. Let converge. Then, since it’s a Cauchy sequence,
,

For sequence , we have

Thus is a Cauchy sequence. Then it converges.

Theorem (Fubini, for real series) If ${a_{m, n}}{m, n \in \N} \subset \R\dsum{(m, n) \in \N \times \N} a_{m, n}$ is absolutely convergent, then

Proof. By Riemann’s rearrangement theorem, the sum doesn’t change whatever the order of a non-negative sequence is,
which indicates Fubini’s theorem for non-negative series. For an arbitrary absolutely convergent sequence, let

Then we guarantee that and are convergent, non-negative sequences that satisfy

Then

Theorem (Fubini, for complex series) If ${a_{m, n}}{m, n \in \N} \subset \C\dsum{(m, n) \in \N \times \N} a_{m, n}$ is absolutely convergent, then

Proof. Since and are also absolutely convergent, we have

Exponential Function

Definition (Exponential Function)

Theorem (Well-definition of on ) converges.

Proof. Let . Choose . Then,

By induction, we have

Then

Thus converges.

Theorem (Well-definition of ) converges.

Proof. The absolute series of

Converges. Therefore converges.

Theorem (Exponent Law)

Specifically, , ; , .

Proof.

By the definition of , .

Since , .

By the definition of ,

Moreover, .

Theorem (Monotonicity of on )

Proof.

Trigonometric Functions

Definition (Trigonometric Functions)

The definitions indicate that is an odd function and is an even function.

Theorem (Euler’s Formula) ,

Proof.

Theorem (Sum Formulas) ,

Proof.

Theorem (Pythagorean’s Identity) ,

Proof.

Conclusion

By defining the exponential and trigonometric functions through power series on the complex numbers, this paper provides
a self-contained and conceptually rigorous foundation for these central objects of analysis. The completeness of
ensures the convergence of the relevant series and justifies algebraic manipulations such as termwise addition and
rearrangement. From these definitions, the familiar laws of exponentiation, Euler’s formula, and the fundamental
identities of trigonometry follow naturally and transparently. This series-based construction not only resolves the
foundational issues associated with geometric or rational-exponent definitions, but also highlights the unity of
exponential and trigonometric functions within complex analysis. As a result, the classical real-variable theory emerges
as a special case of a broader and more coherent complex-analytic framework.