Delta Function - Gauge-institute.org

More documents

Recommendations

Info

Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic DannonThe Lower Integral is the Integration Sum where f ( x ) is replacedby its lowest value on each interval2.2∑x∈[ a, b]⎛⎜⎝dx dx2 2[ x − , x + ]⎞inf f ( t)dx⎠⎟x− ≤t≤ x+dx dx2 2The Upper Integral is the Integration Sum where f ( x ) is replacedby its largest value on each interval2.3∑x∈[ a, b]⎛⎜⎝dx dx2 2[ x − , x + ]⎞ sup f ( t)dx⎠⎟x− ≤t≤ x+dx dx2 2If the integral is a finite hyper-real, we have2.4 A hyper-real function has a finite integral if and only if itsupper integral and its lower integral are finite, and differ by aninfinitesimal.12
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon3.Delta FunctionIn [Dan5], we have defined the Delta Function, and established itsproperties1. The Delta Function is a hyper-real function defined from thehyper-real line into the set of two hyper-reals⎧⎪ 1 ⎫⎨0, ⎪⎬⎪⎩dx⎭⎪ . Thehyper-real0 is the sequence 0, 0, 0,... . The infinite hyperreal1dxdepends on our choice of dx .2. We will usually choose the family of infinitesimals that isspanned by the sequences1n , 12n,1n3,… It is asemigroup with respect to vector addition, and includes allthe scalar multiples of the generating sequences that arenon-zero. That is, the family includes infinitesimals withnegative sign. Therefore,1dxwill mean the sequence n .Alternatively, we may choose the family spanned by thesequences12 n ,13 n ,14 n ,… Then, 1dxwill mean the13
Page 4: Gauge Institute Journal, Volume 8,
Page 9 and 10: Gauge Institute Journal, Volume 8,

Delta Function - Gauge-institute.org

Create successful ePaper yourself

Delete template?

Save as template?