The sin(z)/z Singularity

Complex Analysis · Axiom Academy

EXAMPLE The sin(z)/z Singularity Classify the singularity at the origin and read its residue straight off the Laurent series. The function is undefined at z_0 = 0 , where it gives the indeterminate form . Classify the singularity there (removable, pole, or essential) and find . The tool: residue = the a_ -1 Laurent coefficient Nice work. You classified a singularity and found its residue with one move: expand into a Laurent series and look at the coefficients. Laurent series tells you the type. No negative powers of z removable ; finitely many a pole (the order is the most negative power); infinitely many essential . Here the series is — all non-negative powers, so the singularity is removable. The residue is just one coefficient. , the coefficient of . There is no term here, so — as it is for every removable singularity. Three checks, one conclusion. The limit exists ( ) and f stays bounded near 0 — by Riemann's removable-singularity theorem, both independently confirm "removable." Defining f(0)=1 patches the hole and makes entire. Payoff for integration. Residue 0 means . Contrast or , each a simple pole with residue 1 . This is the prototype removable singularity: a that isn't really a singularity at all — the Laurent series exposes it as an ordinary Taylor series in disguise.

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