the exponential and logarithm as conformal maps

Beyond Mobius maps, the most useful conformal mappings come from the elementary functions you already know — and the two most versatile are the exponential e^z and its inverse, the logarithm log z. What makes them so handy is the simple way they trade rectangles for wedges and strips for sectors. They are the standard tools for handling regions bounded by rays and parallel lines, which Mobius maps alone cannot reshape.

Watch the exponential at work. Write z = x + i y; then e^z = e^x * e^(i y), so the modulus of the image is e^x and its argument is y. This means a vertical line x = constant maps to a circle of radius e^x, and a horizontal line y = constant maps to a ray from the origin at angle y. Consequently a horizontal strip, say 0 < y < pi (all points between two horizontal lines), maps conformally onto the upper half-plane: as y sweeps from 0 to pi the argument of e^z sweeps from 0 to pi, filling exactly the upper half. Narrow the strip and you get a wedge of smaller angle. The derivative of e^z is e^z, which is never zero, so the map is conformal everywhere — angles between the strip's bounding lines (a right angle where a vertical crosses a horizontal) are faithfully preserved as the right angle between a ray and a circle.

The logarithm log z runs this in reverse: it unrolls a wedge or sector back into a strip. A sector 0 < arg z < alpha (a pie-slice) maps under log z to the horizontal strip 0 < y < alpha, turning a curved angular region into a flat-sided one where problems are far simpler. This is exactly why log is the go-to map for corner and wedge domains. The essential caveat is multivaluedness: log z is many-valued (arg z is only defined up to multiples of 2 pi), so to use it as a single-valued conformal map you must fix a branch and a branch cut — typically slitting the plane along a ray so that arg z stays inside one interval of length 2 pi. Cross the cut and the imaginary part of log jumps by 2 pi.