Singular Points

Complex Analysis · Axiom Academy

Where an analytic function breaks down — and the one number near each break that runs all of complex analysis. A function can be perfectly smooth — analytic — across the whole plane except at a handful of points where it simply stops behaving. Those isolated breakdowns, called singularities , decide almost everything about the function: its domain, the value of its integrals, the way it grows. Before naming the three kinds, watch the most famous breakdown happen. Here is the simplest singular function, f(z)= z . Press play and watch a point z slide in toward the origin. The height bar tracks the size of the output, |f(z)| = |z| : as z closes on z_0=0 , the output races off to infinity. That runaway is a pole . As z 0 the value of 1/z leaves every bound behind. That is what it means for a point to be a singularity — the function refuses to settle there. Three kinds — read off the principal part Not every singularity blows up. Around an isolated singularity a function has a Laurent series — a power series allowed to carry negative powers of z . Those negative-power terms are the principal part , and counting them sorts every isolated singularity into exactly three kinds. Pick a function and watch its principal part assemble. Zero negative terms → removable . Finitely many, down to z^ -n → a pole of order n . Infinitely many → essential . The principal part is the classification. One number per break: the residue

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