Regular Point

regular point

[¦reg·yə·lər ′pȯint]
(mathematics)
Any point or a surface that is not a singular point.

Regular Point

a mathematical term used in different senses.

A regular point of a function f(z) of the complex variable z is a point z0 = x0 + iy0 such that in some neighborhood ǀz - z0ǀ < ρ of it the function is single-valued and can be represented in the form of the series

where the Cn are constants.

In the analytic theory of differential equations, a singular point is said to be regular for the equation

if the point is a pole of order at most k for the coefficients pk, k = 1,2.

In Russian usage, the point x0 is called a regular point of discontinuity of the function f(x) if

f(x0)= ½{f(x0 + 0) + f(x0 - -0)}

where f(x0 - 0) and f(x0 + 0) are the limits on the left and on the right, respectively, of the function. This concept is made use of in the theory of Fourier series. In English, such an x0 is usually called a removable discontinuity.

REFERENCES

Smirnov, V. I. Kurs vysshei matematiki, 8th ed., vol. 3, part 2. Moscow, 1969.
Markushevich, A. I. Kratkii kurs teorii analiticheskikh funklsii, 3rd ed. Moscow, 1966.
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