choosing branches consistently

Once you accept that log z, z^a, and the inverse trig functions are multi-valued, the everyday skill is not just picking a branch but picking branches that fit together, so that a formula built from several of them stays continuous and gives the value you actually want. Inconsistent choices are the source of most sign and 2 pi i errors.

The guiding rule is: track the argument continuously as your point moves, and never let a value jump unless you intend it to. Suppose you need sqrt(z) along a path. Choose a starting value, then as z moves let arg z change continuously; if the path swings the argument past your cut, the principal branch will jump, so either reroute the path, move the cut, or knowingly add the 2 pi i (or sign flip) yourself. When a formula combines functions, like sqrt((z-1)(z+1)) or log(f(z)), decide the branch of each ingredient and check that their cuts are compatible: sometimes two square-root cuts can be arranged to cancel, leaving the product single-valued on a larger region than either factor alone. A common method is to pick a reference point where every value is obvious, fix all branches there, and then extend by continuity.

Why it matters: contour integrals, the argument principle, and branch-cut integration all depend on a single coherent choice of argument along the whole contour; an accidental branch jump silently changes the answer by a multiple of 2 pi i or flips a sign. Software has its own default cuts, so a closed-form answer can disagree with a numerical one purely because the branches were chosen differently. The discipline is simple to state and easy to forget: choose your branches up front, state your cuts, keep arg continuous along every path, and account for any crossing on purpose.