Laurent Series and Classifying Isolated Singularities
Allow negative powers to expand a function on an annulus, and sort singularities by how the negative part behaves.
What a learner can do afterwards
- Find the Laurent expansion of a function on a stated annulus
- Classify a singularity as removable, a pole of stated order, or essential
- Give the behaviour near an essential singularity that separates it from a pole
1 · Read
Laurent series extend Taylor series by allowing negative powers, so they can describe functions like 1 divided by z that blow up at a point. On a ring between two singularities, those negative powers capture blowup at the inner edge while positive powers handle the smooth outer part. Coefficients still come from loop integrals, with negative indices joining in.
Singularities come in three flavours. Removable ones are missing points you can fill with the limiting value. Poles blow up like a finite negative run, with the order counting its depth: 1 over z is order 1. Essential points need infinitely many negative powers, and nearby the function edges arbitrarily close to almost any value you like. The negative run, the principal part, fingerprints the case.
Near an essential point functions go wild. Every punctured neighbourhood lands arbitrarily close to any number you name, and Picard sharpens this to hitting every value with at most two exceptions. Poles never do this: near a pole the function is uniformly large, marching to infinity from every direction.
Read the negative powers first and ignore the rest. None refillable means removable, a deepest minus n means a pole of order n, and no deepest means essential. Never judge by the value at the point itself, since the point is the hole.
Negative powers describe the hole, and their run tells you which beast it is.
2 · Watch
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.