Complex Powers and Roots

The one definition that runs the whole show

We have climbed a long way for this. The exponential gave us e^z and its 2 pi i periodicity; sine and cosine grew unbounded; the logarithm splintered into infinitely many values; and last guide we learned to tame that splintering with a branch cut and a principal branch. Now everything cashes out in a single, daring question: what should z^c mean when c is itself a complex number — even something like i? There is no "multiply z by itself i times" to fall back on. We need a genuinely new idea.

The idea is to lean on the two functions we trust most. For real positive numbers, a^c already equals e^(c ln a) — exponentiation is secretly "e to the log." We simply promote that identity to a definition for the complex world. A complex power is z^c = e^(c log z), where log z is the complex logarithm. Read it slowly: take the log of the base, multiply by the exponent, exponentiate. Every complex power is built from exactly the two elementary functions this rung has been about.

z^c  =  e^(c log z)              (the definition)

log z = ln|z| + i (arg z + 2 pi k),   k = 0, +-1, +-2, ...

so       z^c = e^( c [ ln|z| + i arg z + i 2 pi k ] )
               \___________________________/   \_______/
                  one chosen value               the k-ambiguity

Multivalued by inheritance

Here is the honest consequence you must not skip past: because log z is multivalued, z^c is multivalued too. The logarithm offers a whole ladder of values differing by 2 pi i k, and each rung feeds a different answer into e^(c log z). So in general z^c is not one number but a set of numbers — and the set can be infinite. This is not a defect to apologize for; it is the true shape of complex exponentiation, and pretending otherwise is exactly the kind of falsehood-for-convenience we refuse to tell.

Let us actually compute the headline example, i^i, and feel the surprise. Write i = e^(i pi/2), so log i = i(pi/2 + 2 pi k). Then i^i = e^(i log i) = e^(i · i(pi/2 + 2 pi k)) = e^(-(pi/2 + 2 pi k)). Every one of these values is a positive REAL number — i raised to an imaginary power is real, which no real-variable intuition prepares you for. With k = 0 we get the principal value e^(-pi/2) ≈ 0.20788; other k give e^(-pi/2 - 2 pi k), a whole geometric family marching off toward 0 and toward infinity.

The principal power, and when the surprises stop

Choosing the principal logarithm everywhere gives the principal value of the power, z^c = e^(c Log z), and this is what a calculator returns. On the cut plane — the plane slit along the negative real axis, where Log z lives single-valued and holomorphic — the principal power is a perfectly respectable holomorphic function of z. But never forget what the choice cost: the same branch cut you placed for the logarithm is inherited wholesale by the power. The principal z^c is discontinuous as you cross the negative real axis, jumping by the factor e^(c · 2 pi i) that the 2 pi i log-jump produces.

So when does the multivaluedness quietly disappear? Exactly when the jump factor e^(c · 2 pi i) equals 1 — that is, when c is an integer. If c = n is an integer, then e^(c · 2 pi i k) = e^(2 pi i n k) = 1 for every k, so all the branch values collapse into a single number and z^n is honestly single-valued. This is the reassuring fine print: ordinary integer powers like z^2 or z^(-3) were never multivalued, and our grand definition agrees perfectly with the elementary one. The strangeness is real, but it is confined to non-integer exponents.

Roots: the same machine, run at c = 1/n

Now watch the most beautiful payoff. A root is just a power with a fractional exponent: the nth root of z is z^(1/n). Feed c = 1/n into our definition and the 2 pi k inside the logarithm no longer washes out — instead it sprays the values around. The factor e^((1/n) · 2 pi i k) = e^(2 pi i k/n) takes exactly n distinct values as k runs 0, 1, ..., n-1, and then repeats. So z^(1/n) yields precisely n distinct roots, no more and no fewer — the n nth roots of a complex number you first met back in the foundations, now derived from the single power law rather than asserted.

Geometrically the picture is gorgeous and simple: the n roots all share the modulus |z|^(1/n), and their arguments are spaced evenly by 2 pi/n. They sit at the vertices of a regular n-gon centred on the origin. Take the special case z = 1 and you recover the roots of unity — the points e^(2 pi i k/n) marching around the unit circle, the cleanest pattern in all of elementary complex analysis. This is also exactly what de Moivre's theorem was telling you in polar coordinates, now seen as one face of the complex power.

1. Write the base in polar form, z = r e^(i theta) with r = |z| and theta a value of arg z.
2. The common modulus of every root is r^(1/n) (the ordinary real nth root of the positive number r).
3. The n arguments are (theta + 2 pi k)/n for k = 0, 1, ..., n-1 — evenly spaced by 2 pi/n.
4. Assemble each root as r^(1/n) · e^(i (theta + 2 pi k)/n); together they form a regular n-gon, and beyond k = n-1 the values just repeat.

Branch points, and the rules that quietly break

Where do those roots get tangled? At the origin and at infinity — the branch points of z^(1/n). Encircle the origin once and the argument advances by 2 pi, so each root rotates by 2 pi/n and lands on the next vertex of the n-gon: the roots cyclically permute. That is the unmistakable fingerprint of a branch point, the same one the logarithm carries. To single out one continuous root function you again slit the plane along a branch cut from 0 to infinity and pick a branch — most often the principal one, built from Log z.

Finally, a warning that catches even careful people. Once exponents go non-integer, the schoolbook power laws are no longer guaranteed. The identity (z w)^c = z^c w^c can fail because Log(z w) need not equal Log z + Log w — the arguments may add up past pi and force a 2 pi correction. The classic trap: (-1)·(-1) = 1 gives sqrt(1) = 1, yet sqrt(-1)·sqrt(-1) = i·i = -1. Nothing is broken; you have merely landed on different branches. The cure is the discipline from last guide — fix your branch first, then compute — which is why choosing branches consistently is the habit this whole rung has been training.