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Pole


The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a complex function. In inversive geometry, the inversion pole is related to inverse points with respect to an inversion circle. The term "pole" is also used to denote the degenerate points phi=0 and phi=pi in spherical coordinates, corresponding to the north pole and south pole respectively. "All-poles method" is an alternate term for the maximum entropy method used in deconvolution. In triangle geometry, an orthopole is the point of concurrence certain perpendiculars with respect to a triangle of a given line, and a Simson line pole is similarly defined based on the Simson line of a point with respect to a triangle. In projective geometry, the perspector is sometimes known as the perspective pole.

In complex analysis, an analytic function f is said to have a pole of order n at a point z=z_0 if, in the Laurent series, a_m=0 for m<-n and a_(-n)!=0. Equivalently, f has a pole of order n at z_0 if n is the smallest positive integer for which (z-z_0)^nf(z) is holomorphic at z_0. A analytic function f has a pole at infinity if

 lim_(z->infty)f(z)=infty.

A nonconstant polynomial P(z) has a pole at infinity of order degP, i.e., the polynomial degree of P.

PoleContoursPoleReIm

The basic example of a pole is f(z)=1/z^n, which has a single pole of order n at z=0. Plots of 1/z and 1/z^2 are shown above in the complex plane.

For a rational function, the poles are simply given by the roots of the denominator, where a root of multiplicity n corresponds to a pole of order n.

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

Renteln and Dundes (2005) give the following (bad) mathematical jokes about poles:

Q: What's the value of a contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe.

Q: Why did the mathematician name his dog "Cauchy?" A: Because he left a residue at every pole.


See also

Argument Principle, Complex Residue, Essential Singularity, Flat Pole, Holomorphic Function, Inversion Pole, Isopole, Laurent Series, Magnetic Pole Differential Equation, Meromorphic Function, Orthopole, Removable Singularity, Simple Pole, Singular Point, Sphere Pole Explore this topic in the MathWorld classroom

Portions of this entry contributed by Todd Rowland

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References

Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005.

Referenced on Wolfram|Alpha

Pole

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Pole." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pole.html

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