Delta Function - Gauge-institute.org

More documents

Recommendations

Info

Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon2.Integral of a Hyper-real FunctionIn [Dan3], we defined the integral of a Hyper-real Function.Let f () x be a hyper-real function on the interval [ ab] , .The interval may not be bounded.f () x may take infinite hyper-real values, and need not bebounded.At eacha≤x≤b,there is a rectangle with basedx dx[ x − , x + 2], height f () x , and area2f ( xdx. )We form the Integration Sum of all the areas for the x ’s thatstart at x = a, and end at x = b,∑ f ( xdx ) .x∈[ a, b]If for any infinitesimal dx , the Integration Sum has the samehyper-real value, then f () x is integrable over the interval [ ab] , .Then, we call the Integration Sum the integral of f () x from x = a,to x= b, and denote it by10
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonx=b∫ f ( xdx ) .x=aIf the hyper-real is infinite, then it is the integral over [, ab] ,If the hyper-real is finite,x=b∫ fxdx ( ) = real part of the hyper-real . x=a2.1 The countability of the Integration SumIn [Dan1], we established the equality of all positive infinities:We proved that the number of the Natural Numbers,Card , equals the number of Real Numbers,2 Card Card = , andwe have2 Card2Card Card = ( Card) = .... = 2 = 2 = ... ≡ ∞.In particular, we demonstrated that the real numbers may bewell-ordered.Consequently, there are countably many real numbers in theinterval [ ab] , , and the Integration Sum has countably many terms.While we do not sequence the real numbers in the interval, thesummation takes place over countably many f ( xdx. )11
Page 4: Gauge Institute Journal, Volume 8,
Page 13 and 14: Gauge Institute Journal, Volume 8,

Delta Function - Gauge-institute.org

Create successful ePaper yourself

Delete template?

Save as template?