branch cut
A multivalued function like the square root or the logarithm cannot be drawn as one clean surface — it has overlapping layers, and walking around a branch point shuffles you between them. A branch cut is the curtain you hang to keep that from happening: a curve in the plane that you declare off-limits to your paths, so that within the remaining region the function stops being multivalued and settles into a single, continuous branch.
Practically, you choose a curve connecting branch points (for the logarithm, a ray from the origin to infinity; for sqrt((z-1)(z+1)), the segment from -1 to +1) and forbid any path from crossing it. Inside the slit plane the function is now single-valued and analytic, but it pays a price: it jumps discontinuously as you step across the cut. The function approaches different limiting values from the two sides, and the size of that jump is information, not error — it is exactly what you integrate when a cut runs through your contour. The cut's location is a free choice; only the branch points are fixed.
Branch cuts are indispensable in evaluating real integrals by contour methods. To integrate something like x^{a} / (1 + x^2) from 0 to infinity, you place a cut along the positive real axis and run a keyhole contour around it; the function's jump across the cut reproduces a multiple of the original real integral, and residues at the poles finish the job. The same idea — a cut as a controlled discontinuity carrying physical meaning — appears in dispersion relations, where the jump across a cut encodes absorption or particle production.
For Log z with the principal branch, the cut runs along the negative real axis. Just above it at z = -1 the imaginary part is +i pi; just below it the imaginary part is -i pi. The function jumps by 2 pi i as you cross. Move the cut to the positive real axis instead and the function is perfectly fine on the negative axis — the cut was your choice, the branch point at z = 0 was not.
The cut is movable, the jump across it is real information, and only the branch points are forced.
A branch cut is a choice of where to put the discontinuity, not a property of the function itself. Two people may place the cut differently and both be right, as long as it joins the branch points; comparing answers requires agreeing on the convention first.