“Wilbraham-Gibbs Constant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Wilbraham-Gibbs Constant

释义

Wilbraham-Gibbs Constant

N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.

Let a piecewise smooth function with only finitely many discontinuities (which are all jumps) be defined on with Fourier Series

			(1)
			(2)

(3)

Let a discontinuity be at

, with

(4)

(5)

Define

(6)

and let

be the first local minimum and

the first local maximum of

on either side of

. Then

(7)

(8)

where

(9)

Here,

is the Sinc Function. The Fourier Series of

therefore does notconverge to

and

at the ends, but to

and

. This phenomenon was observed by Wilbraham (1848) andGibbs (1899). Although Wilbraham was the first to note the phenomenon, the constant

is frequently (and unfairly)credited to Gibbs and known as the Gibbs Constant. A related constant sometimes also called the GibbsConstant is

(10)

(Le Lionnais 1983).

References

Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed. New York: Dover, 1930.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/gibbs/gibbs.html

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36 and 43, 1983.

Zygmund, A. G. Trigonometric Series 1, 2nd ed. Cambridge, England: Cambridge University Press, 1959.

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