\[ \newcommand{\ci}{\mathrm{i}} \newcommand{\e}{\mathrm{e}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\eps}{\varepsilon} \newcommand{\dsum}{\displaystyle\sum} \]
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.
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 \(\C\), 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 \(\R\) (it’s a complete field with an Archimedean total order), and will build everything else from scratch.
Complex Series
Complex Numbers
We define
\[ \C = \{(a, b) : a, b \in \R\} \]
With addition
\[ \begin{array}{rrl} \cdot + \cdot : & \C \times \C & \to \C \\ & (a, b), (c, d) & \mapsto (a+c, b+d) \end{array} \]
And multiplication
\[ \begin{array}{rrl} \cdot \cdot \cdot : & \C \times \C & \to \C \\ & (a, b), (c, d) & \mapsto (ac-bd, ad+bc) \end{array} \]
We denote \(1 = (1, 0), \ci = (0, 1)\), then \((a, b)\) will be denoted as \(a + \ci b\).
We define
\[ \begin{array}{rrl} \Re : & \C & \to \R \\ & a + \ci b & \mapsto a \end{array} \]
\[ \begin{array}{rrl} \Im : & \C & \to \R \\ & a + \ci b & \mapsto b \end{array} \]
Then \(\forall z \in \C, z = \Re(z) + \ci \Im(z)\).
Completeness of \(\C\)
We see \(\C\) as a vector space on \(\R\) and define the norm on \(\C\) as
\[ \begin{array}{rrl} | \cdot | : & \C & \to \R \\ & a+\ci b & \mapsto \sqrt{a^2+b^2} \end{array} \]
Theorem (Completeness of \(\C\)) \(\C\) is complete under this norm.
Proof. For any Cauchy sequence on \(\C\), that is, \(\{z_k = x_k + \ci y_k\} \subset \C\) that satisfies
\[ \forall \eps \in \R, \exists N \in \N, \forall n, m > N, |z_n - z_m| < \eps \]
Since
\[ |x_n - x_m|^2 + |y_n - y_m|^2 = |z_n - z_m|^2 \]
We have
\[ |x_n - x_m| < |z_n - z_m| < \eps \]
And \(\{x_k\}\) constructs a Cauchy sequence on \(\R\). Say
\[ x = \lim_{k \to \infty} x_k \]
Similarly, we assume that
\[ y = \lim_{k \to \infty} y_k \]
Then, \(\forall \eps \in \R\), choose \(N \in \N\) such that
\[ \forall k > N, |x_k - x| < \frac{\eps}{2} \]
Choose \(M \in \N\) such that
\[ \forall k > M, |y_k - y| < \frac{\eps}{2} \]
Let \(z = x + \ci y\). Then
\[ \forall k > \max\{N, M\}, |z_k - z| = \sqrt{|x_k - x|^2 + |y_k - y|^2} < \sqrt{\left(\frac{\eps}{2}\right)^2 + \left(\frac{\eps}{2}\right)^2} < \eps \]
Thus \(\{z_k\}\) converges to \(z\).
Since \(\C\) is a complete normed vector space (Banach space) over \(\R\), we naturally have
Theorem (Absolute Convergence of Complex Series) If a complex series converges absolutely, it converges.
Proof. Let \(\dsum_{k=0}^{\infty} |z_k|\) converge. Then, since it’s a Cauchy sequence, \(\forall \eps > 0, \exists N \in \N, \forall n, m > N\),
\[ \dsum_{k=n}^{m} |z_k| < \eps \]
For sequence \(\{z_k\}\), we have
\[ \left|\dsum_{k=n}^{m} z_k\right| \le \dsum_{i=n}^{m} |z_k| < \eps \]
Thus \(\{z_k\}\) is a Cauchy sequence. Then it converges.
Theorem (Fubini, for real series) If \(\{a_{m, n}\}_{m, n \in \N} \subset \R\) is a double-indexed sequence of real numbers and \(\dsum_{(m, n) \in \N \times \N} a_{m, n}\) is absolutely convergent, then
\[ \dsum_{(m, n) \in \N \times \N} a_{m, n} = \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \]
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
\[ b_{m, n} = \max\{a_{m, n}, 0\}, c_{m, n} = \max\{-a_{m, n}, 0\} \]
Then we guarantee that \(b_{m, n}\) and \(c_{m, n}\) are convergent, non-negative sequences that satisfy
\[ \forall m, n \in \N, a_{m, n} = b_{m, n} - c_{m, n} \]
Then
\[ \begin{aligned} \dsum_{(m, n) \in \N \times \N} a_{m, n} =& \dsum_{(m, n) \in \N \times \N} b_{m, n} - \dsum_{(m, n) \in \N \times \N} c_{m, n} \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} b_{m, n} - \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} c_{m, n} \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \end{aligned} \]
Theorem (Fubini, for complex series) If \(\{a_{m, n}\}_{m, n \in \N} \subset \C\) is a double-indexed sequence of complex numbers and \(\dsum_{(m, n) \in \N \times \N} a_{m, n}\) is absolutely convergent, then
\[ \dsum_{(m, n) \in \N \times \N} a_{m, n} = \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \]
Proof. Since \(\{\Re(a_{m, n})\}\) and \(\{\Im(a_{m, n})\}\) are also absolutely convergent, we have
\[ \begin{aligned} \dsum_{(m, n) \in \N \times \N} a_{m, n} =& \dsum_{(m, n) \in \N \times \N} \Re(a_{m, n}) + \ci \dsum_{(m, n) \in \N \times \N} \Im(a_{m, n}) \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} \Re(a_{m, n}) + \ci \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} \Im(a_{m, n}) \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \\ \end{aligned} \]
Exponential Function
Definition (Exponential Function)
\[ \begin{array}{rrl} \exp : & \C & \to \C \\ & z & \mapsto \e^z = \dsum_{k=0}^{\infty} \frac{z^k}{k!} \end{array} \]
Theorem (Well-definition of \(\exp\) on \(\R_{\ge 0}\)) \(\forall x \ge 0, \exp x\) converges.
Proof. Let \(a_k = \frac{x^k}{k!}\). Choose \(M \in \N, M > 2x\). Then,
\[ \forall k \ge M, \frac{a_{k+1}}{a_k} = \frac{x}{k+1} < \frac{x}{M} < \frac{1}{2} \]
By induction, we have
\[ \forall k \ge M, a_k < \frac{a_M}{2^{k-M}} \]
Then
\[ \dsum_{k=M}^{\infty} \frac{x^k}{k!} < \dsum_{k=M}^{\infty} \frac{x^M}{M! 2^{k-M}} = \frac{2x^M}{M!} < +\infty \]
Thus \(\dsum_{k=0}^{\infty} \frac{x^k}{k!}\) converges.
Theorem (Well-definition of \(\exp\)) \(\forall z \in \C, \exp z\) converges.
Proof. The absolute series of \(\exp z\)
\[ \dsum_{k=0}^{\infty} \left| \frac{z^k}{k!} \right| = \dsum_{k=0}^{\infty} \frac{|z|^k}{k!} = \exp |z| < +\infty \]
Converges. Therefore \(\exp z\) converges.
Theorem (Exponent Law)
\[ \forall z_1, z_2 \in C, \e^{z_1 + z_2} = \e^{z_1} \cdot \e^{z_2} \]
Specifically, \(\forall z \in \C\), \(\e^z \neq 0\); \(\forall x \in \R\), \(\e^x \in \R_{>0}\).
Proof.
\[ \begin{aligned} \e^{z_1} \cdot \e^{z_2} =& \dsum_{k=0}^{\infty} \frac{z_1^k}{k!} \dsum_{k=0}^{\infty} \frac{z_2^k}{k!} \\ =& \dsum_{i=0}^{\infty} \dsum_{j=0}^{\infty} \frac{z_1^i z_2^j}{i! j!} \\ =& \dsum_{k=0}^{\infty} \dsum_{i+j=k} \frac{z_1^i z_2^j}{i! j!} \\ =& \dsum_{k=0}^{\infty} \frac{1}{k!} \dsum_{i+j=k} \begin{pmatrix}k \\ i\end{pmatrix} z_1^i z_2^j \\ =& \dsum_{k=0}^{\infty} \frac{1}{k!} (z_1 + z_2)^k \\ =& \e^{z_1 + z_2} \\ \end{aligned} \]
By the definition of \(\exp\), \(\e^0 = 1\).
Since \(\forall z \in C, \e^z \e^{-z} = 1\), \(\e^z \neq 0\).
By the definition of \(\exp\),
\[ \forall x \in \R_{>0}, \e^x = \dsum_{k=0}^{\infty} \frac{x^k}{k!} > \frac{x^0}{0!} = 1 \]
Moreover, \(\forall x < 0, \e^x = (\e^{-x})^{-1} \in (0, 1)\).
Theorem (Monotonicity of \(\exp\) on \(\R\))
\[ \forall x_1, x_2 \in \R, x_1 < x_2 \Leftrightarrow \e^{x_1} < \e^{x_2} \]
Proof.
\[ x_1 < x_2 \Leftrightarrow x_2 - x_1 > 0 \Leftrightarrow \e^{x_2 - x_1} > 1 \Leftrightarrow \e^{x_2} > \e^{x_1} \]
Trigonometric Functions
Definition (Trigonometric Functions)
\[ \begin{array}{rrl} \cos : & \C & \to \C \\ & z & \mapsto \displaystyle\frac{\e^{\ci z} + \e^{-\ci z}}{2} = \dsum_{k=0}^{\infty} \frac{(-1)^k z^{2k}}{(2k)!} \end{array} \]
\[ \begin{array}{rrl} \sin : & \C & \to \C \\ & z & \mapsto \displaystyle\frac{\e^{\ci z} - \e^{-\ci z}}{2\ci} = \dsum_{k=0}^{\infty} \frac{(-1)^k z^{2k+1}}{(2k+1)!} \end{array} \]
The definitions indicate that \(\sin\) is an odd function and \(\cos\) is an even function.
Theorem (Euler’s Formula) \(\forall z \in \C\),
\[ \e^{\ci z} = \cos z + \ci \sin z \]
Proof.
\[ \cos z + \ci \sin z = \frac{\e^{\ci z} + \e^{-\ci z}}{2} + \ci \frac{\e^{\ci z} - \e^{-\ci z}}{2\ci} = \e^{\ci z} \]
Theorem (Sum Formulas) \(\forall z_1, z_2 \in \C\),
\[ \begin{cases} \cos(z_1 + z_2) = \cos z_1 \cos z_2 - \sin z_1 \sin z_2 \\ \sin(z_1 + z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2 \end{cases} \]
Proof.
\[ \begin{aligned} \cos(z_1 + z_2) =& \frac{\e^{\ci (z_1 + z_2)} + \e^{-\ci (z_1 + z_2)}}{2} \\ =& \frac{\e^{\ci z_1} \e^{\ci z_2} + \e^{-\ci z_1} \e^{-\ci z_2}}{2} \\ \end{aligned} \]
\[ \begin{aligned} \cos z_1 \cos z_2 - \sin z_1 \sin z_2 =& \frac{\e^{\ci z_1} + \e^{-\ci z_1}}{2} \frac{\e^{\ci z_2} + \e^{-\ci z_2}}{2} - \frac{\e^{\ci z_1} - \e^{-\ci z_1}}{2\ci} \frac{\e^{\ci z_2} - \e^{-\ci z_2}}{2\ci} \\ =& \frac{1}{4}((\e^{\ci z_1} + \e^{-\ci z_1})(\e^{\ci z_2} + \e^{-\ci z_2}) + (\e^{\ci z_1} - \e^{-\ci z_1})(\e^{\ci z_2} - \e^{-\ci z_2})) \\ =& \frac{1}{4}(2\e^{\ci z_1} \e^{\ci z_2} + 2\e^{-\ci z_1} \e^{-\ci z_2}) \\ =& \cos(z_1 + z_2) \end{aligned} \]
\[ \begin{aligned} \sin(z_1 + z_2) =& \frac{\e^{\ci (z_1 + z_2)} - \e^{-\ci (z_1 + z_2)}}{2\ci} \\ =& \frac{\e^{\ci z_1} \e^{\ci z_2} - \e^{-\ci z_1} \e^{-\ci z_2}}{2\ci} \\ \end{aligned} \]
\[ \begin{aligned} \sin z_1 \cos z_2 + \cos z_1 \sin z_2 =& \frac{\e^{\ci z_1} - \e^{-\ci z_1}}{2\ci} \frac{\e^{\ci z_2} + \e^{-\ci z_2}}{2} + \frac{\e^{\ci z_1} + \e^{-\ci z_1}}{2} \frac{\e^{\ci z_2} - \e^{-\ci z_2}}{2\ci} \\ =& \frac{1}{4\ci}((\e^{\ci z_1} - \e^{-\ci z_1})(\e^{\ci z_2} + \e^{-\ci z_2}) + (\e^{\ci z_1} + \e^{-\ci z_1})(\e^{\ci z_2} - \e^{-\ci z_2})) \\ =& \frac{1}{4\ci}(2\e^{\ci z_1} \e^{\ci z_2} - 2\e^{-\ci z_1} \e^{-\ci z_2}) \\ =& \sin(z_1 + z_2) \end{aligned} \]
Theorem (Pythagorean’s Identity) \(\forall z \in \C\),
\[ \cos^2 z + \sin^2 z = 1 \]
Proof. \(1 = \cos 0 = \cos(z-z) = \cos z \cos(-z) - \sin z \sin(-z) = \cos^2 z + \sin^2 z\)
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 \(\C\) 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.