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Delta Sequence

A delta sequence is a sequence of strongly peaked functions for which

lim_(n->infty)int_(-infty)^inftydelta_n(x)f(x)dx=f(0)
(1)

so that in the limit as n->infty, the sequences become delta functions.

Examples include

delta_n(x)={0 for x<-1/(2n); n for -1/(2n)<x<1/(2n); 0 for x>1/(2n)
(2)
=n/(sqrt(pi))e^(-n^2x^2)
(3)
=n/pisinc(nx)
(4)
=1/(pix)(e^(inx)-e^(-inx))/(2i)
(5)
=1/(2piix)[e^(ixt)]_(-n)^n
(6)
=1/(2pi)int_(-n)^ne^(ixt)dt
(7)
=1/(2pi)(sin[(n+1/2)x])/(sin(1/2x))
(8)

(Arfken 1985, pp. 482 and 488-489).

See also

Delta Function

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.

Referenced on Wolfram|Alpha

Delta Sequence

Cite this as:

Weisstein, Eric W. "Delta Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeltaSequence.html

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