Elementary functions: analysis and complex properties

Elementary Functions in Complex Analysis: A Rigorous Axiomatic Construction

The exponential function is defined via a power series with an infinite radius of convergence:

\exp(z) = \sum_{k=0}^\infty \frac{z^k}{k!}, \quad z = t + i\tau \in \mathbb{C}

It is an entire function, mapping a complex argument to a complex value. Direct verification confirms it satisfies the Cauchy problem:

\frac{dw}{dz} = w, \quad w(0) = 1, \quad w(z) = \exp(z)

The group property of ODE solutions yields multiplicativity:

\exp(z_1 + z_2) = \exp(z_1) \exp(z_2)

The exponential never vanishes at any point in \mathbb{C}, otherwise it would be identically zero. On the real axis, \exp(t) > 0 and is monotonically increasing.

Define e := \exp(1) > 1. From multiplicativity, it follows:

\exp(m/n) = \sqrt[n]{e^m}, \quad m \in \mathbb{Z}, \quad n \in \mathbb{N}

Hence the limits: \lim_{t \to \infty} \exp(t) = \infty, \lim_{t \to -\infty} \exp(t) = 0. This justifies the notation e^z := \exp(z).

Logarithm and Generalizations

The natural logarithm is the inverse of the exponential:

\ln: (0, \infty) \to \mathbb{R}, \quad \ln \exp t = t, \quad \exp \ln \xi = \xi, \quad \xi > 0

General logarithm and power:

\log_a \xi := \frac{\ln \xi}{\ln a}, \quad a^t := \exp(t \ln a), \quad a > 0, \ a \ne 1

These definitions preserve algebraic properties in the complex plane when using the principal branch.

Complex Trigonometric Functions

Cosine and sine are defined via the exponential:

\cos z := \frac{e^{iz} + e^{-iz}}{2}, \quad \sin z := \frac{e^{iz} - e^{-iz}}{2i}

This yields the Pythagorean identity and derivatives:

\cos^2 z + \sin^2 z = 1, \quad \frac{d}{dz} \cos z = -\sin z, \quad \frac{d}{dz} \sin z = \cos z

Euler's formula:

e^{iz} = \cos z + i \sin z

Geometric Interpretation on the Unit Circle

A parametric curve in \mathbb{R}^2:

x(t) = \cos t, \quad y(t) = \sin t

Satisfies:

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 (x(t), y(t)) give a parametric equation of the unit circle centered at the origin. The parameter t is natural: speed \sqrt{\dot{x}^2 + \dot{y}^2} = 1. Increasing t corresponds to counterclockwise traversal.

Periodicity: \cos(t + 2\pi) = \cos t, \sin(t + 2\pi) = \sin t, where \pi is half the circumference.

Natural Parameter and Arc Length

For a smooth curve \gamma(t) = (x(t), y(t)), x, y \in C^1[t_1, t_2], \dot{\gamma} \ne 0:

\ell = \int_{t_1}^{t_2} \sqrt{\dot{x}^2(t) + \dot{y}^2(t)} \, dt

The parameter is natural if the radical expression equals 1. Then t is proportional to arc length, as in the case of the unit circle.

Key properties for analysis:

• The exponential is the only entire solution to the Cauchy problem dw/dz = w, w(0)=1.
• Multiplicativity ensures zero values are impossible.
• Trigonometry reduces to the exponential, preserving differential properties.
• Circle parameterization demonstrates the link to Euclidean geometry.
• The natural parameter unifies the concept of length in differential geometry.

Key Takeaways

• The exponential is defined by a Taylor series with infinite radius of convergence, an entire function.
• Multiplicativity \exp(z_1 + z_2) = \exp(z_1)\exp(z_2) follows from the ODE.
• Trigonometric functions are defined via the exponential and satisfy classical identities.
• The parametric curve (\cos t, \sin t) is the unit circle with a natural parameter.
• Arc length is computed via the integral of speed; the natural parameter simplifies calculations.

— Editorial Team