scholarly journals On the Composition of Distributionsx−sln|x|and|x|μ

2007 ◽  
Vol 2007 ◽  
pp. 1-9
Author(s):  
Biljana Jolevska-Tuneska ◽  
Emin Özça¯g
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Dirac Delta ◽  
Neutrix Limit ◽  
Differentiable Functions ◽  
Infinitely Differentiable Functions

LetFbe a distribution and letfbe a locally summable function. The distributionF(f)is defined as the neutrix limit of the sequence{Fn(f)}, whereFn(x)=F(x)*δn(x)and{δn(x)}is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). The composition of the distributionsx−sIn|x|and|x|μis evaluated fors=1,2,…,μ>0andμs≠1,2,….

On the neutrix composition of x−−slnmx− and ln(1 + x+)

2018 ◽  
Vol 11 (06) ◽  
pp. 1850086
Author(s):  
Mongkolsery Lin
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Dirac Delta ◽  
Neutrix Limit ◽  
Differentiable Functions ◽  
Composition Formula ◽  
Infinitely Differentiable Functions

The neutrix composition [Formula: see text], [Formula: see text] is a distribution and [Formula: see text] is a locally summable function, is defined as the neutrix limit of the sequence [Formula: see text], where [Formula: see text] and [Formula: see text] is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function [Formula: see text]. The neutrix composition of the distributions [Formula: see text] and [Formula: see text] is evaluated for [Formula: see text] Further related results are also deduced.

scholarly journals Further results on the neutrix composition of distributions involving the delta function and the function cosh+-1(x1/r+1)$\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)$

2019 ◽  
Vol 52 (1) ◽  
pp. 249-255
Author(s):  
Brian Fisher ◽  
Kenan Tas
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Dirac Delta ◽  
Neutrix Limit ◽  
Differentiable Functions ◽  
Infinitely Differentiable Functions ◽  
Appl Math

AbstractThe neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) 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) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The function $\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by$$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition ${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$] exists and$${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$for r, s = 1, 2, . . . . Further results are also proved.Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh−1+(x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].

scholarly journals On the composition of the distributions x-s+ lnmx+ and xμ+

2009 ◽  
Vol 3 (2) ◽  
pp. 212-223 ◽  
Author(s):  
Brian Fisher
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Dirac Delta ◽  
Neutrix Limit ◽  
Differentiable Functions ◽  
Infinitely Differentiable Functions

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {Fn(f)}, where Fn(x) = F(x)*?n(x) and {?n(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function ?(x). The composition of the distributions x-s + lnm x+ and x? + is proved to exist and be equal to ?mx-s? + lnm x+ for ? > 0 and s,m = 1, 2,....

scholarly journals On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Brian Fisher ◽  
Adem Kılıçman
Keyword(s):  
Delta Function ◽  
Regular Sequence ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Integer Part ◽  
Hyperbolic Tangent ◽  
Dirac Delta ◽  
Inverse Functions

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix compositionδ(rs-1)((tanhx+)1/r)exists andδ(rs-1)((tanhx+)1/r)=∑k=0s-1∑i=0Kk((-1)kcs-2i-1,k(rs)!/2sk!)δ(k)(x)forr,s=1,2,…, whereKkis the integer part of(s-k-1)/2and the constantscj,kare defined by the expansion(tanh-1x)k={∑i=0∞(x2i+1/(2i+1))}k=∑j=k∞cj,kxj, fork=0,1,2,….Further results are also proved.

scholarly journals On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Brian Fisher ◽  
Adem Kılıçman
Keyword(s):  
Delta Function ◽  
Regular Sequence ◽  
Dirac Delta Function ◽  
Summable Function ◽  
Ordinary Sense ◽  
Dirac Delta ◽  
Hyperbolic Sine ◽  
Sine Functions ◽  
Inverse Hyperbolic Sine

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the compositionδ(s)[(sinh⁡-1x+)r]does not exists. In this study, it is proved that the neutrix compositionδ(s)[(sinh⁡-1x+)r]exists and is given byδ(s)[(sinh⁡-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), fors=0,1,2,…andr=1,2,…, wherecs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.

scholarly journals On the Noncommutative Neutrix Product of Distributions

2007 ◽  
Vol 2007 ◽  
pp. 1-10
Author(s):  
Emin Özçaḡ ◽  
İnci Ege ◽  
Haşmet Gürçay ◽  
Biljana Jolevska-Tuneska
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Test Functions ◽  
Dirac Delta ◽  
Neutrix Limit ◽  
Neutrix Product

Letfandgbe distributions and letgn=(g*δn)(x), whereδn(x)is a certain sequence converging to the Dirac-delta functionδ(x). The noncommutative neutrix productf∘goffandgis defined to be the neutrix limit of the sequence{fgn}, provided the limithexists in the sense thatN‐limn→∞〈f(x)gn(x),φ(x)〉=〈h(x),φ(x)〉, for all test functions in𝒟. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix productsx+rlnx+∘x−−r−1lnx−andx−−r−1lnx−∘x+rlnx+are proved to exist and are evaluated forr=1,2,…. It is consequently seen that these two products are in fact equal.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

scholarly journals Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

2001 ◽  
Vol 694 ◽  
Author(s):  
Fredy R Zypman ◽  
Gabriel Cwilich
Keyword(s):  
Probability Distribution ◽  
Disordered Systems ◽  
Electromagnetic Waves ◽  
Delta Function ◽  
Rayleigh Distribution ◽  
Dirac Delta Function ◽  
Universal Form ◽  
Dirac Delta ◽  
Diffusive Regime ◽  
Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

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