Paths in the Complex Plane

Complex Analysis · Axiom Academy

A curve is something you can walk, one instant at a time On the real line, an integral can only run one way: left to right. The complex plane is two-dimensional, so a path from one point to another can bend, loop, and double back — and soon you'll integrate along those paths. The first move is humbler than it sounds: describe a curve as the trail of a single moving point. A parametrization is a function (t)=x(t)+i\,y(t) that, as the real parameter t ticks from a to b , names a point in at every instant. Press play and watch one point z(t) leave a trail from z_0 to z_1 : the trail is the path . The path isn't a picture you draw all at once — it's the record of a point in motion. That motion is exactly what a parametrization captures. The simplest path: a straight line between two points The shortest path from z_0 to z_1 is the line segment (t)=z_0+t\,(z_1-z_0) for t [0,1] — at t=0 you're at z_0 , at t=1 you're at z_1 , and you slide steadily between. Drag the slider to set t and watch the point walk the segment; the readout shows the exact z(t) you're standing on. Same recipe as the animation, now under your thumb: feed in a real t , read out a complex point z(t) . That map from a real interval into is all a path ever is. Around and back: the circle, and which way you go

This is the written version of the interactive lesson above. See the full Complex Analysis course.