Analyzing f(z) = 1/z

Complex Analysis · Axiom Academy

Work out the inversion mapping: its domain, its real and imaginary parts, and what it does to length and angle. Analyze the function : find its domain and its component functions, describe what it does to the modulus and the argument, and evaluate it at z = 1 + i . z = 1 + i (blue) sits at distance — just outside the unit circle — at angle . Its image (orange) sits at distance — inside the circle — at angle . The length is inverted and the angle is reflected across the real axis. Nice work — you fully analyzed the inversion map f(z) = 1/z , from its domain all the way to a worked value. Domain: everything except the pole, — f is undefined at z = 0 . Components: multiplying by the conjugate gives . Geometry: and — an inversion (inside outside the unit circle) plus a reflection across the real axis. The unit circle is the dividing line: points on it stay on it, points inside swap with points outside, and every angle flips its sign — the cleanest non-trivial map in complex analysis.

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