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Home > Pole (complex analysis)

Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U − {a} → C is a holomorphic function. If there exists a holomorphic function g : U → C and a natural number n such that

for all z in U − {a}, then a is called a pole of f. If n is chosen as small as possible, then n is called the order of the pole.

The number a is a pole of order n of f if and only if the Laurent series expansion of f around a has only finitely many negative degree terms, starting with (z - a)⁻ⁿ.

A pole of order 0 is a removable singularity. In this case the limit lim_z→a f(z) exists as a complex number. If the order is bigger than 0, then lim_z→a f(z) = ∞.

The first derivative of f has a pole of order 1 (a "simple" pole) at a iff a is a branch point of f.

A non-removable singularity that is not a pole or a branch point is called an essential singularity.

A holomorphic function whose only singularities are poles is called meromorphic.

See also

• Zero (complex analysis)
• Electronic filterAn electronic filter eliminates unwanted frequencies from an electronic signal. There are several kinds of electronic filters: A low-pass filter passes low frequencies. A high-pass filter passes high frequencies. A band-pass filter passes a limited range

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Topics: Pole (complex Analysis) Order Complex Function Singularity...

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Pole vaultPole vaulting is an athletics event where competitors use a long, flexible pole as an aid to leap over a bar, similar to the high jump, but at much greater heights. Pole jumping competitions were known to the ancient Greeks, as well as the Cretans and Cel Pole weaponA pole weapon or polearm is a close combat weapon with the main fighting part of the weapon placed on the end of a long shaft, typically of wood. The use of pole weapons is very old, and the first spears date to the stone age. The purpose of using pole we PoleThe noun pole and adjective polar may refer to multiple things. A Pole is a member of a Slavic ethnic group from Poland. A pole is a long and straight stick, usually vertical or intended to be used vertically. See barber, pole vault, pole (object). A pole Pole (complex analysis)In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f ''z 1 z n at z 0; a pole of a function f is a point a such that f ''z approaches infinity as z approaches a''. Formally, suppose U is Pole vault technologyCompetitive pole vaulting began with iron bars. Bamboo was then used. Today's pole vaulters benefit from wrapped sheets of fiberglass. As in the high jump, the landing area was originally a heap of sawdust where athletes landed on their feet. As technolog Pole PositionPole Position is a video game that was first released in 1982 by Namco, also the creator of Pac-Man''. In Pole Position the player had to complete a lap in a certain amount of time in order to qualify for an F1 race at the Fuji Racetrack. After qualifying Pole bending Polemarch Poles Polecat Poleszuk Polesie Polewiki Polemic Poletown