exp(z) Mapping Strips

Complex Analysis · Axiom Academy

Using the conformal map w = e^ z to send a horizontal strip onto a half-plane Show that w = e^ z maps the horizontal strip onto the upper half-plane, and identify the image exactly. We will track the two boundary lines and the interior of the strip under the map, writing z = x + iy throughout. Lower boundary → the positive real axis (the ray ). Interior line → the ray at angle — each horizontal line y = c maps to the ray . Upper boundary → the negative real axis (the ray ). Nice work. You showed that w = e^ z carries the strip conformally onto the upper half-plane by tracking what happens to |w| and . The mechanism: |w| = e^ x sweeps every value in as x runs over , while is fixed by the height of the line. A horizontal line y = c becomes the open ray . Conformality: e^ z is analytic with derivative everywhere, so it preserves angles — the perpendicular grid of the strip maps to an orthogonal net of rays and circles. The height sets the wedge: a strip of height h opens into a wedge of angle h . Scale first with , then exponentiate — the general recipe is below. Half-planes, quadrants, and wedges are the standard targets in conformal mapping — once a region is on the upper half-plane, the whole Möbius toolkit is available to push it anywhere else.

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