Where does pole collide?

In electromagnetism a magnetic pole is a region of a magnet where the field lines are densest and appear to enter or leave the body, named north and south by convention. Griffiths sets this out in Introduction to electrodynamics, where the poles are not separable objects: cut a bar magnet in half and you get two smaller magnets, each with its own north and south, never a single pole on its own. The referent is a place on a physical body where a real magnetic field concentrates, measured in tesla. Dirac showed in 1931 that an isolated magnetic pole, a monopole, is allowed by quantum theory only under a strict condition, and none has ever been found. This pole is a region of space with a field in it, not a point on the Earth and not a singularity of a function.[1][2]

Unit

A region of concentrated magnetic field on a magnet

Recovery means

Not applicable

Measured by

Field strength and direction near the surface, in tesla, mapped to find where the lines converge

In complex analysis a pole is a kind of isolated singularity of a function: a point where the function grows without bound, but in a controlled way, so that multiplying by a finite power of the distance to the point makes it analytic again. Ahlfors treats this in Complex analysis, where the order of the pole is that power and the residue captures the leading term of the blow-up. The referent is a point in the complex plane attached to a particular function, not a place in physical space and not anything on the Earth. Near the pole the function tends to infinity, and the whole apparatus of contour integration turns on collecting residues at these points. It shares the word, and the underlying mathematics, with the control-theory pole, but here the object is any meromorphic function.[3]

Unit

An isolated point where a function tends to infinity

Recovery means

Not applicable

Measured by

Found as a zero of the denominator or from a Laurent expansion; described by its order and residue

In control theory a pole is a root of the denominator of a system's transfer function, and where it sits in the complex plane governs how the system behaves. Ogata sets this out in Modern control engineering: poles in the left half-plane give a response that settles, poles in the right half-plane one that runs away, and poles near the imaginary axis a response that rings for a long time. The referent is a specific number tied to a real linear system's dynamics, carrying a claim about stability. It is a special case of the complex-analysis pole, since the transfer function is a rational function whose denominator roots are its poles, but the two are worth keeping apart: one is a property of an abstract function, the other a statement about whether a physical system holds together.[4]

Unit

A root of a transfer function's denominator

Recovery means

Not applicable

Measured by

Computed as the roots of the characteristic denominator polynomial; read off the s-plane