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The goal of this post is to formally construct the theory of elementary functions without gaps, from the perspective of a university-level analysis course. It is intended as a unifying overview for undergraduate students who have previously encountered these functions in school but perhaps without a rigorous foundation.
A key point I want to emphasize is that, contrary to popular belief, it is quite straightforward to bridge the gap between the definition of trigonometric functions via the complex exponential and their classical geometric definition via the unit circle.
$$\exp(z) = 1+\sum_{k=1}^\infty \frac{z^k}{k!}, \quad z = t + i\tau \in \mathbb{C}$$
The exponential is an entire function; its radius of convergence is infinite, and it satisfies the property ##\exp(\mathbb{R})\subset\mathbb{R}##.
By direct verification, we can see that the exponential satisfies the following Cauchy problem (initial value problem):
$$\frac{dw}{dz} = w, \quad w(0) = 1, \quad w(z) = \exp(z) \qquad (1)$$
Due to the group property of solutions to ordinary differential equations, we have:
$$\exp(z_1 + z_2) = \exp(z_1)\exp(z_2) \qquad (2)$$
From this identity, it is clear that the exponential never vanishes in ##\mathbb{C}##. If it were zero at any point ##z_1##, it would be identically zero everywhere, which contradicts ##w(0)=1##.
Consequently, the function ##\exp(t)## is positive and increasing for ##t \in \mathbb{R}##. This follows directly from (1). From the definition of the series, it also follows that ##e := \exp(1) > 1##.
Using formula (2), we can establish that for any rational number:
$$\exp(m/n) = \sqrt[n]{e^m}, \quad m \in \mathbb{Z}, n \in \mathbb{N}$$
As a result, we observe the asymptotic behavior: ##\lim_{t \to \infty} \exp(t) = \infty## and ##\lim_{t \to -\infty} \exp(t) = 0##.
$$\ln: (0, \infty) \to \mathbb{R}, \quad \ln(\exp t) = t, \quad \exp(\ln \xi) = \xi, \quad \xi > 0$$
Furthermore, we can define general logarithms and powers for ##a > 0, a \neq 1##:
$$\log_a \xi := \frac{\ln \xi}{\ln a}, \quad a^t := \exp(t \ln a)$$
$$\cos z := \frac{\exp(iz) + \exp(-iz)}{2}, \quad \sin z := \frac{\exp(iz) - \exp(-iz)}{2i}$$
From these definitions, the following identities follow immediately:
$$\cos^2 z + \sin^2 z = 1, \quad \frac{d}{dz}\cos z = -\sin z, \quad \frac{d}{dz}\sin z = \cos z$$
$$x(t) = \cos t, \quad y(t) = \sin t \qquad (3)$$
From the identities above, we find:
$$x^2(t) + y^2(t) = 1, \quad \dot{x}^2 + \dot{y}^2 = 1, \quad \begin{vmatrix} x(t) & y(t) \\ \dot{x}(t) & \dot{y}(t) \end{vmatrix} = 1$$
Theorem: The functions in (3) represent the parametric equation of a unit circle centered at the origin. Here, ##t## acts as the natural parameter (arc length). As ##t## increases, the circle is traversed in a counter-clockwise direction.
By defining the length of the unit circle as ##2\pi## (which serves as the definition of the number ##\pi##), we obtain the periodicity of trigonometric functions:
$$\cos(t + 2\pi) = \cos t, \quad \sin(t + 2\pi) = \sin t$$
A key point I want to emphasize is that, contrary to popular belief, it is quite straightforward to bridge the gap between the definition of trigonometric functions via the complex exponential and their classical geometric definition via the unit circle.
1. The Exponential Function
Definition: The exponential function is defined by the power series:$$\exp(z) = 1+\sum_{k=1}^\infty \frac{z^k}{k!}, \quad z = t + i\tau \in \mathbb{C}$$
The exponential is an entire function; its radius of convergence is infinite, and it satisfies the property ##\exp(\mathbb{R})\subset\mathbb{R}##.
By direct verification, we can see that the exponential satisfies the following Cauchy problem (initial value problem):
$$\frac{dw}{dz} = w, \quad w(0) = 1, \quad w(z) = \exp(z) \qquad (1)$$
Due to the group property of solutions to ordinary differential equations, we have:
$$\exp(z_1 + z_2) = \exp(z_1)\exp(z_2) \qquad (2)$$
From this identity, it is clear that the exponential never vanishes in ##\mathbb{C}##. If it were zero at any point ##z_1##, it would be identically zero everywhere, which contradicts ##w(0)=1##.
Consequently, the function ##\exp(t)## is positive and increasing for ##t \in \mathbb{R}##. This follows directly from (1). From the definition of the series, it also follows that ##e := \exp(1) > 1##.
Using formula (2), we can establish that for any rational number:
$$\exp(m/n) = \sqrt[n]{e^m}, \quad m \in \mathbb{Z}, n \in \mathbb{N}$$
As a result, we observe the asymptotic behavior: ##\lim_{t \to \infty} \exp(t) = \infty## and ##\lim_{t \to -\infty} \exp(t) = 0##.
2. Logarithms and Powers
Definition: The natural logarithm is defined as the inverse of the exponential function:$$\ln: (0, \infty) \to \mathbb{R}, \quad \ln(\exp t) = t, \quad \exp(\ln \xi) = \xi, \quad \xi > 0$$
Furthermore, we can define general logarithms and powers for ##a > 0, a \neq 1##:
$$\log_a \xi := \frac{\ln \xi}{\ln a}, \quad a^t := \exp(t \ln a)$$
3. Trigonometric Functions
Definition: We define the sine and cosine functions using Euler's formulas:$$\cos z := \frac{\exp(iz) + \exp(-iz)}{2}, \quad \sin z := \frac{\exp(iz) - \exp(-iz)}{2i}$$
From these definitions, the following identities follow immediately:
$$\cos^2 z + \sin^2 z = 1, \quad \frac{d}{dz}\cos z = -\sin z, \quad \frac{d}{dz}\sin z = \cos z$$
4. Geometric Interpretation
Consider a curve in the Cartesian plane ##(x, y)## defined by:$$x(t) = \cos t, \quad y(t) = \sin t \qquad (3)$$
From the identities above, we find:
$$x^2(t) + y^2(t) = 1, \quad \dot{x}^2 + \dot{y}^2 = 1, \quad \begin{vmatrix} x(t) & y(t) \\ \dot{x}(t) & \dot{y}(t) \end{vmatrix} = 1$$
Theorem: The functions in (3) represent the parametric equation of a unit circle centered at the origin. Here, ##t## acts as the natural parameter (arc length). As ##t## increases, the circle is traversed in a counter-clockwise direction.
By defining the length of the unit circle as ##2\pi## (which serves as the definition of the number ##\pi##), we obtain the periodicity of trigonometric functions:
$$\cos(t + 2\pi) = \cos t, \quad \sin(t + 2\pi) = \sin t$$
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