Branches of the Logarithm

Complex Analysis · Axiom Academy

LESSON Branches of the Logarithm Making single-valued — and what it costs you: a discontinuous jump of across a branch cut. Write . Then , and the trouble is the angle: is only fixed up to whole turns. Walk the point once around the origin and its angle has grown by — so the value of has climbed by , even though you came back to the same z . Every integer k gives a value — infinitely many 2. The Principal Branch and Its Cut To pin down one value we restrict the angle. The principal branch uses the principal argument . That choice is continuous everywhere — except along the negative real axis , where the angle is forced to snap from down to just above . That ray is the branch cut . Approaching the negative real axis from above, . Approaching from below, — a gap of . The real part doesn't jump at all; only the imaginary part does. The cut must start at the origin (and run to ) — you can't encircle z=0 without a jump. At z = -2 : from above, ; from below, . Same for the real part, but the imaginary part jumps by across the cut. 3. Stacked Branches: the k th Sheet The principal branch is just k=0 . Add to the angle and you get the k th branch — a parallel, equally valid single-valued logarithm. Picture them as stacked sheets : every sheet is a full copy of the plane, neighbours separated by in the imaginary direction, joined along the cut. Crossing the branch cut moves you to the next sheet — which is exactly the jump from Step 2, seen as a hop between branches.

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