Poles

Complex Analysis · Axiom Academy

The well-behaved infinity of complex analysis — a point where a function blows up like , no worse. 1. A Pole Is Controlled Infinity Take near its singular point z_0 . Push z toward z_0 from any direction and the magnitude |f(z)| shoots up without bound. That is the defining behavior of a pole: The size of f runs off to infinity at the pole Watch a point slide toward the pole at z_0 = 1 for . As the gap |z - z_0| shrinks, the magnitude tower on the right races upward — halving the distance quadruples the height (order m = 2 ). 2. The Order Counts the Negative Powers Expand f in a Laurent series centered at z_0 . The terms with negative powers form the principal part . For a pole this part is finite — it stops at some deepest term with . That bottom rung m is the order . Equivalently, z_0 is a pole of order m exactly when this limit is finite and nonzero — multiplying by (z-z_0)^m cancels the blow-up perfectly: Just one negative term, . Example: at z=2 . Deepest term is (z-z_0)^ -2 . Example: at z=0 . Principal part runs from (z-z_0)^ -m up to (z-z_0)^ -1 — at most m negative terms. For in lowest terms, the order at a root of Q equals that root's multiplicity . For , write with g(z)=e^z analytic and . The deepest term is (z)^ -3 , so z=0 is a pole of order 3 . 3. Pole, Removable, or Essential Every isolated singularity falls into exactly one of three boxes, and the principal part of the Laurent series is what sorts them. A pole is the middle case — the finite one:

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