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
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Laurent series. A function holomorphic on an annulus r < |z - a| < R expands as a sum over all integers n of c_n (z - a)^n, converging locally uniformly there. The centre and annulus are part of the specification; the same function expands differently on different annuli. A missing or problem point a is an isolated singularity when f is holomorphic on some punctured disk 0 < |z - a| < R. If f is already holomorphic at a, the point is regular, not a singularity.
Worked two-region expansion. For f(z) = 1/(z - 1) about 0, on |z| < 1 write -1/(1 - z) and expand as -sum_{n>=0} z^n, with no negative powers. On |z| > 1 write (1/z)/(1 - 1/z) and expand as sum_{n>=1} z^(-n), all negative powers. The ratio selection decides which powers appear; each expansion is valid only on its stated annulus.
Classifying isolated singularities. On a punctured disk 0 < |z - a| < R, read the principal part, the negative powers with nonzero coefficients after combining like powers. No negative powers means removable: the limit fills the point. Finitely many nonzero negative powers means a pole of positive order m, where the deepest nonzero term is (z - a)^(-m), not the term count: 7/(z - a)^5 + 2/(z - a) has order 5. Infinitely many nonzero negative powers means essential. The classification needs the expansion reaching arbitrarily close to a, so an outer annulus cannot classify a point. The coefficient c_-1 is the local residue.
Pole versus essential behaviour. Near a pole of order m, |f(z)| tends to infinity as z approaches a, uniformly in every direction. Near an essential singularity, the Casorati picture says the image of every punctured neighbourhood is dense in the complex plane: values land arbitrarily close to every finite complex number, without necessarily attaining them. e^(1/z) never equals 0, yet approaches 0 along the negative real axis and grows along the positive axis.
Worked essential expansion. The exponential series e^w = sum_{n>=0} w^n/n! holds for every complex w. Multiplying e^w by e^(-w) gives 1, so the exponential is never zero. Substitute w = 1/z: e^(1/z) = sum_{n>=0} 1/(n! z^n) = 1 + 1/z + 1/(2z²) + 1/(6z³) + ..., valid for z nonzero. The n = 0 term is the constant 1, and every later term is a negative power with nonzero coefficient, running without end. The infinitely many nonzero negative powers mark an essential singularity at 0.
Removable singularities. If f is holomorphic and bounded on a punctured disk 0 < |z - a| < R, the singularity is removable: the finite limit as z approaches a fills the point and extends f analytically. The constant coefficient c_0 of the Laurent expansion becomes the filled value. Boundedness alone without punctured holomorphicity does not give the extension, as conj(z)/z shows: modulus 1, but values 1 on the real axis and -1 on the imaginary axis.
Negative powers name the hole, the principal part fingerprints the type, and density of values separates essential from pole.
All nine extra tasks and three readings are original authorship created for this topic.
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