Understanding Complex Exponentiation

Complex exponentiation is the generalization of raising numbers to powers to the case where both the base and the exponent can be complex numbers. It is a core concept in complex analysis with profound and sometimes surprising results.

The fundamental challenge—and beauty—of the subject is that while \( a^b \) is straightforward for positive real \(a\) and real \(b\), it becomes inherently multi-valued when extended to the complex plane. The definition is built upon two foundational pillars: Euler's formula and the complex logarithm.

1. The Foundation: Euler's Formula

Everything begins with Euler's formula, which connects the complex exponential function to trigonometry:

\[ e^{i\theta} = \cos\theta + i\sin\theta \]

This identity is the most important relationship in complex analysis.

2. The Complex Exponential Function \(e^z\)

For any complex number \( z = x + iy \) (with \(x, y \in \mathbb{R}\)), we define the exponential function as:

\[ e^z = e^{x+iy} = e^x \cdot e^{iy} = e^x(\cos y + i\sin y) \]

This function is single-valued, analytic (holomorphic) everywhere in the complex plane, and reduces to the standard real exponential function when \(y=0\).

3. The General Case \(w^z\)

To define exponentiation with an arbitrary complex base \(w\) and exponent \(z\), we preserve the fundamental property from real analysis:

\[ a^b = e^{b \ln a} \]

For a complex base \(w\), we must use the complex logarithm, denoted \(\ln(w)\).

The Multi-Valued Nature of the Complex Logarithm

Because the complex exponential is periodic with period \(2\pi i\) (i.e., \( e^{i\theta} = e^{i(\theta + 2\pi k)} \) for any integer \(k\)), its inverse, the logarithm, has infinitely many values.

Express \(w\) in polar form \( w = re^{i\theta} \) (where \(r > 0\) and \(\theta = \arg(w)\)). The complex logarithm is defined as:

\[ \ln(w) = \ln(r) + i(\theta + 2\pi k), \quad k \in \mathbb{Z} \]

Here, \(\ln(r)\) is the ordinary real logarithm of the positive number \(r\). The term \(i(\theta + 2\pi k)\) accounts for the infinite possible angles (arguments) of \(w\).

The principal value, denoted \(\operatorname{Log}(w)\) (with a capital L), is the single value obtained by restricting the argument to the principal branch, typically \(-\pi < \theta \leq \pi\), corresponding to \(k=0\).

4. Formal Definition of Complex Exponentiation

\[ w^z := e^{\,z \, \ln(w)} = \exp\left(\,z \left[ \ln|w| + i(\arg(w) + 2\pi k) \right] \,\right), \quad k \in \mathbb{Z} \]

This definition leads directly to the central insight:

The expression \(w^z\) is, in general, multi-valued. Each integer \(k\) gives a potentially distinct result.

• It yields infinitely many distinct values if \(z\) is not a rational number.
• It yields a finite set of values if \(z\) is a rational number. For example, \(w^{1/2}\) gives two distinct square roots, and \(w^{1/n}\) gives \(n\) distinct \(n\)-th roots.
• It yields a single unique value if \(z\) is an integer.

5. A Famous Example: \(i^{\,i}\)

This calculation perfectly illustrates the process and its surprising consequences.

Step 1: Write the base in polar form. The principal polar form is \( i = e^{i\pi/2} \).

Step 2: Apply the definition \( i^i = e^{i \cdot \ln(i)} \).

Step 3: Compute the logarithm: \(\ln(i) = \ln|i| + i(\arg(i) + 2\pi k) = \ln(1) + i\left(\frac{\pi}{2} + 2\pi k\right) = i\left(\frac{\pi}{2} + 2\pi k\right)\).

Step 4: Substitute back: \[ i^i = e^{i \cdot \left[ i\left(\frac{\pi}{2} + 2\pi k\right) \right]} = e^{-\left(\frac{\pi}{2} + 2\pi k\right)}. \]

Conclusion: An imaginary base raised to an imaginary power produces infinitely many real numbers.

The principal value (with \(k=0\)) is \( e^{-\pi/2} \approx 0.20788 \).
Other values include \( e^{-5\pi/2} \) (for \(k=1\)), \( e^{3\pi/2} \) (for \(k=-1\)), and so on.

6. Important Properties and Caveats

Familiar exponent rules often fail. Identities like \((a^b)^c = a^{bc}\) and \(a^b \cdot a^c = a^{b+c}\), which hold for positive real bases, are not generally true in the complex realm due to multi-valuedness. Applying them without care can lead to contradictions.

Branch cuts are necessary. To create a single-valued, analytic function from \(w^z\), one must choose a specific branch of the complex logarithm (e.g., the principal branch) and consistently restrict the argument.

Analyticity: For a fixed branch, functions like \(f(z) = a^z\) (with \(a > 0\)) or \(f(z) = e^z\) are analytic (holomorphic) everywhere. Functions like \(f(z) = z^c\) are analytic on domains that exclude a branch cut.

7. Applications and Significance

Complex exponentiation is far more than a mathematical curiosity; it is a fundamental tool.

In Pure Mathematics: It is central to complex analysis, number theory (e.g., the Riemann zeta function), and solving differential equations.

In Physics and Engineering:

• Quantum Mechanics: Wave functions are inherently complex, with time evolution often expressed as \(e^{-iEt/\hbar}\).
• Electrical Engineering: AC circuit analysis uses phasors, represented as complex exponentials \(e^{i\omega t}\), to simplify calculations with sinusoidal voltages and currents.
• Signal Processing: The Fourier transform relies on complex exponentials to decompose signals into frequencies.
• Fluid Dynamics and Electromagnetism: Complex potentials provide elegant solutions to Laplace's equation.

It provides a profound unification, showing that exponential growth/decay and rotational oscillation are two facets of the same fundamental operation in the complex plane.

Summary

Complex exponentiation is defined via the core identity:

\[ w^z = e^{z \ln w} \]

Its defining and essential characteristic is multi-valuedness, which arises from the periodic nature of the complex exponential function. While this requires careful and rigorous handling, it unlocks a powerful and elegant generalization of one of algebra's most basic operations. It reveals a deep geometric structure where imaginary exponents can yield real numbers, and where algebra, geometry, and analysis beautifully intertwine.