THE ARGUMENT PRINCIPLE — a dart

THE ARGUMENT PRINCIPLE walk a loop, count how the output spins — that integer is how many roots you enclosed

Cauchy's argument principle is a startling accounting identity: for a function f(z), the number of times its output winds around zero as z travels once around a closed loop equals the number of zeros minus poles of f inside that loop. The winding number — a topological count of how the phase spins — equals an algebraic count of roots. You never solve for the roots; you just watch the argument turn. This is how you can prove the fundamental theorem of algebra (a degree-n polynomial's output winds n times around a big enough circle, so it has n roots), and how control theory (the Nyquist criterion) checks stability without factoring.

THE TECHNIQUE winding number of f around a loop = (zeros − poles) inside

The polynomial z² − 1 and a large loop. The demo counts the roots enclosed — equal to the winding number of the output: live demo

HISTORY & CREDIT Augustin-Louis Cauchy, 1831

“To count the roots inside a region you must find them.” — you only have to watch f's output spin as you walk the boundary; the number of turns is the number of roots inside. Topology counts what algebra would have to solve. cited

1831 · Augustin-Louis Cauchy — the argument principle, from his residue calculus.
consequence · a proof of the fundamental theorem of algebra; Nyquist (1932) turned it into a stability test.
now · root counting, control theory, and numerical eigenvalue methods.

The winding number is a topological integer; the root count is an algebraic one; the argument principle says they are the same. Watch the phase turn, and you have counted the roots. Cauchy 1831

RECOMMEND FOR I-13 roots enclosed = winding number, computed

On the canonical compiler, z²−1 has roots ±1, both inside radius 5, so the winding number of the output around a big loop is 2:

$ i13 run inv_argument.i13 # roots of z^2-1 inside radius 5 zeros_inside = 2 -- = winding number of f around the loop

Recommend: the argument principle is a topological invariant (winding number) equal to an algebraic count, and i13 can enact either side. Here it counts the enclosed roots of z²−1 directly (2), which the principle guarantees equals the winding number of the output — a phase count no continuous change of the loop can alter as long as it encloses the same roots. It is the analytic sibling of the degree (dart 283): the same “integer that cannot jump,” now counting roots by watching an argument turn. i13 computes the count; Cauchy guarantees it is the winding.