The Argument Principle

Complex Analysis · Axiom Academy

One contour integral counts every zero and pole inside it — by watching how many times the image winds around the origin. 1. Counting Roots by Counting Loops Let f be meromorphic inside and on a simple closed contour , with no zero or pole sitting on . As z travels once around , watch the image point w = f(z) . The argument principle says the integral of f'/f equals how many net times that image winds around the origin. zeros minus poles inside = winding number of about 0 2. Zeros Add a Turn, Poles Subtract One Why does the count come out as Z - P ? Near a zero of order m , f'/f has residue +m ; near a pole of order n , residue -n . Summing the residues (the residue theorem) makes each zero push the argument +1 time around and each pole pull it back -1 . The running total settles on Z - P . f'/f has residue +m . The image sweeps m extra turns counter-clockwise — winding goes up . f'/f has residue -n . The image unwinds n turns clockwise — winding goes down . A triple zero contributes +3 , a double pole -2 . Order is weight. Everything analytic and nonzero integrates to 0 ; only zeros and poles leave a mark. f(z) = z^3 - z on |z| = 2 : zeros at 0, 1, -1 , no poles, so Z - P = 3 - 0 = 3 . And on |z| = 3 : zeros , one pole at 2 , so Z - P = 2 - 1 = 1 — no integration by hand required. 3. Rouché's Theorem: the Dominant Term Wins

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