FURTHER RESULTS ON THE NEUTRIX COMPOSITION OF THE DELTA FUNCTION

Abstract

Let F be a distribution in D and let f be a locally summable function. 

The composition F(f(x)) of F and f is said to exist and be equal to the

distribution h(x) if the neutrix limit of the sequence {Fn(f(x))} is equal

to h(x), where Fn(x) = F(x)∗δn(x) for n = 1,2,... and {δn(x)} is a

certain regular sequence converging to the Dirac delta. It is proved that

Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand e-mail: yong.33@hotmail.com

∗∗

Let F be a distribution in D and let f be a locally summable function.

(s) r 1/r
[ln (1 + x+ )] exists and is given by

the neutrix composition δ

s kr+r−1
kr + r − 1

i

k=0 i=0

for s = 0,1,2,... and r = 1,2,.... Further results are also proved.