The exponential is loyal; its inverse is not
Recall from Volume I that the real logarithm is the inverse function of e^x: each positive number has exactly one logarithm. In the complex plane that clean one-to-one story breaks. The complex exponential e^z is periodic — Euler's formula gives e^{i theta} = cos theta + i sin theta, and adding 2 pi to theta lands you on the same point. So e^z and e^{z + 2 pi i} are equal. Any function that tries to undo e^z therefore cannot pin down a single answer.
Write a nonzero complex number in polar form, z = r e^{i theta}, where r = |z| is its modulus and theta is its argument (its angle). Then the complex logarithm is log z = ln r + i theta, where ln r is just the ordinary real logarithm of the positive number r. The trouble is theta: the angle of z is only defined up to adding any whole multiple of 2 pi. So the honest formula is log z = ln|z| + i(theta + 2 pi k) for every integer k.
Walk a loop and you climb a staircase
Picture yourself standing at z = 1, where one natural choice gives log 1 = 0. Now walk slowly counterclockwise on the unit circle, keeping the angle theta increasing continuously. When you have gone all the way around and arrived back at z = 1, your angle has risen from 0 to 2 pi — so the logarithm has climbed from 0 to 2 pi i. You returned to the same point in the plane but to a different value of the function. Loop again and you climb another 2 pi i.
The pivot of all this winding is the origin. A point you must circle to accumulate angle is called a branch point — for the logarithm it is z = 0. (Infinity is a branch point too, in a precise sense.) A branch point is the place where the multivaluedness is born: encircle it and the function does not come home to its old value. Encircle a point that is not a branch point and nothing changes.
Cut the plane to choose a branch
To get an honest single-valued function we make a deal: forbid the loops that cause trouble. Draw a curve from the branch point out to infinity and declare it off-limits — you may not cross it. This forbidden curve is a branch cut, and a branch cut for the logarithm is most often the negative real axis. With that barrier in place you can never circle the origin, so the angle theta can no longer drift; pinning it to one interval, say -pi < theta <= pi, selects one consistent value everywhere.
This particular choice — modulus times the angle in (-pi, pi] — is called the principal branch, written Log z (capital L) or principal log. On it, Log z is a perfectly good analytic function off the cut: its derivative is the familiar d/dz Log z = 1/z, exactly as in Volume I, and it satisfies the Cauchy-Riemann equations everywhere except on the cut. The cut is not a feature of the logarithm itself — it is a feature of the SLICE you chose. Put the cut along the positive imaginary axis instead and you get an equally valid (different) single-valued branch.
What does the cut cost you? Continuity across it. Approach a point on the negative real axis from just above and Log z has imaginary part near +pi; approach from just below and it is near -pi. The function jumps by 2 pi i as you step across — a discontinuity of exactly one full turn of the staircase. That jump is real and you must respect it: never let a path of integration wander across a branch cut as if nothing were there.
Complex powers and what i^i really means
Once you have a logarithm, you have all powers, because every power is defined through it: z^a = e^{a log z} for any complex exponent a. Since log z is multivalued, so in general is z^a. This is why complex powers also carry branch points and need a branch cut. Choosing the principal logarithm gives the principal value of the power — the one a calculator returns — but it is one choice among many.
Now the famous puzzle. Take z = i, which has modulus 1 and angle pi/2, so log i = i(pi/2 + 2 pi k). Then i^i = e^{i log i} = e^{i times i(pi/2 + 2 pi k)} = e^{-(pi/2 + 2 pi k)}. Every value is a positive REAL number! The principal one (k = 0) is e^{-pi/2}, about 0.2079. So i to the power i is not imaginary, not mysterious — it is a real number, and the surprise is fully explained by the multivaluedness of the log buried inside the definition.
z = r e^{i theta}
log z = ln r + i (theta + 2 pi k), k = 0, +-1, +-2, ...
Log z = ln r + i theta, -pi < theta <= pi (principal)
z^a = e^{a log z}
i^i = e^{i log i} = e^{-(pi/2 + 2 pi k)} -> principal e^{-pi/2} ~ 0.2079
sqrt(z) = z^{1/2}: two values, opposite signs (branch point at 0)A clean sanity check: the square root sqrt(z) = z^{1/2} has exactly two values (e^{i pi k} for k = 0, 1 gives the two opposite signs), matching the two roots you already know from Volume I. Its branch point is again z = 0, and a cut along the negative real axis makes it single-valued. A rational exponent p/q gives q values; an irrational or genuinely complex exponent like i gives infinitely many.
Putting branches to work
Branch cuts are not a nuisance to apologize for — they are a tool. The whole power of complex methods, which the earlier guide on the residue theorem built up, rests on integrating analytic functions along carefully chosen contours; but those functions are only analytic on a branch you have fixed. So a contour must be routed to respect the cuts. The standard technique is the keyhole contour: a path that comes in along one side of the cut, loops around the branch point on a tiny circle, and returns along the other side.
Here is why it pays off. Because the function jumps by a known amount across the cut, the integral along the top of the cut and the integral along the bottom do NOT cancel — they combine into a clean multiple of the real integral you actually want. Encircling the enclosed poles and applying the residue theorem then evaluates the whole loop, and a little algebra delivers a real integral that no elementary real method could touch — things like the definite integral of x^{a-1}/(1+x) from 0 to infinity, which works out to pi / sin(pi a).
- Identify the branch points of the integrand (where a log or non-integer power sits) and choose a cut that keeps your region simply connected.
- Build a contour that hugs both sides of the cut and skirts the branch point on a small circle — a keyhole.
- Use the known jump across the cut to relate the top and bottom pieces to your target real integral.
- Sum the residues at the enclosed poles, check that the small and large arcs vanish, and solve for the real integral.
One honest caution. Routing a real problem through the complex plane and across a branch cut is a fully legitimate method, not a trick or a cheat — every step is justified by analyticity and the geometry of the chosen branch. But it is unforgiving of bookkeeping errors: pick the wrong cut, forget the 2 pi i jump, or let an arc fail to vanish, and the answer silently goes wrong. The reward for that discipline is the same one contour integration promised throughout this rung — and it scales onward to Laurent series, the argument principle, and conformal mapping of multi-sheeted regions.