The Essential Singularity at z = 0

Complex Analysis · Axiom Academy

EXAMPLE The e^ 1/z Singularity Classifying the quintessential essential singularity, and finding its residue The function below has an isolated singularity at z_0 = 0 . Classify it (removable, pole, or essential) using its Laurent series, then compute its residue there. Brightness shows the magnitude . Hug the origin from the right and it blows up (bright); from the left it vanishes (dark) — extreme values pressed against the very same point. You classified e^ 1/z at z=0 straight from its Laurent series and pulled the residue right out of it. Laurent series: — substitute w = 1/z into e^ w . Type: essential singularity — the principal part has infinitely many negative powers (not finitely many, as a pole would). Residue: , the coefficient of 1/z — residues come from the Laurent series even when no derivative formula applies. Picard's theorem: near an essential singularity f hits every complex value infinitely often, with at most one exception. Here the omitted value is 0 , since e^ w is never zero. The count of negative-power terms is the whole story: removable, finitely many a pole, infinitely many essential.

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