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Complex Exponentiation — Definition, Formula & Examples

Complex exponentiation is the operation of raising a complex number to a complex power, written zwz^w where both zz and ww can be complex. It is defined using the exponential function and the complex logarithm, and it is generally multi-valued.

For complex numbers z≠0z \neq 0 and ww, the complex power zwz^w is defined as zw=ewln⁡zz^w = e^{w \ln z}, where ln⁡z=ln⁡∣z∣+i(arg⁡z+2πk)\ln z = \ln|z| + i(\arg z + 2\pi k) for integer kk. Choosing a branch of the logarithm (typically the principal branch with −π<arg⁡z≤π-\pi < \arg z \le \pi) yields a single-valued principal value.

Key Formula

zw=ewln⁡z=ew(ln⁡∣z∣+iarg⁡z)z^w = e^{w \ln z} = e^{w(\ln|z| + i\arg z)}
Where:
  • zz = The base, a nonzero complex number
  • ww = The exponent, a complex number
  • ln⁡z\ln z = The complex (natural) logarithm of z
  • ∣z∣|z| = The modulus (absolute value) of z
  • arg⁡z\arg z = An argument of z (determined up to multiples of 2π)

How It Works

To compute zwz^w, you first express zz in polar form z=reiθz = re^{i\theta}, then take its logarithm ln⁡z=ln⁡r+iθ\ln z = \ln r + i\theta. You multiply this by ww and exponentiate the result using ewln⁡ze^{w \ln z}. Because arg⁡z\arg z is only determined up to multiples of 2π2\pi, different choices of branch yield different values — this is why complex exponentiation is multi-valued in general. When ww is an integer, all branches agree and you recover the familiar single-valued power.

Worked Example

Problem: Find the principal value of i^i.
Step 1: Write i in polar form. Its modulus is 1 and its principal argument is π/2.
i=eiπ/2i = e^{i\pi/2}
Step 2: Take the principal logarithm of i.
ln⁡i=ln⁡1+iπ2=iπ2\ln i = \ln 1 + i\frac{\pi}{2} = i\frac{\pi}{2}
Step 3: Multiply by the exponent i and exponentiate.
ii=ei⋅iπ/2=e−π/2i^i = e^{i \cdot i\pi/2} = e^{-\pi/2}
Answer: The principal value of iii^i is e−π/2≈0.2079e^{-\pi/2} \approx 0.2079, a real number.

Why It Matters

Complex exponentiation is central to complex analysis, Fourier theory, and quantum mechanics. Evaluating expressions like zsz^s for complex ss is the foundation of the Riemann zeta function and analytic number theory. Engineers encounter it in signal processing whenever phasors are raised to non-integer powers.

Common Mistakes

Mistake: Treating zwz^w as single-valued for non-integer w.
Correction: Because the complex logarithm is multi-valued (differing by 2πik2\pi i k), zwz^w generally has infinitely many values. Always specify which branch you are using, or state the principal value.