Publication Date:
1992-01-01
Description:
Four selfreciprocal integral transformations of Hankel type are defined through(ℋi,μf)(y)=Fi(y)=∫0∞αi(x)ℊi,μ(xy)f(x)dx, ℋi,μ−1=ℋi,μ,wherei=1,2,3,4;μ≥0;α1(x)=x1+2μ,ℊ1,μ(x)=x−μJμ(x),Jμ(x)being the Bessel function of the first kind of orderμ;α2(x)=x1−2μ,ℊ2,μ(x)=(−1)μx2μℊ1,μ(x);α3(x)=x−1−2μ,ℊ3,μ(x)=x1+2μℊ1,μ(x), andα4(x)=x−1+2μ,ℊ4,μ(x)=(−1)μxℊ1,μ(x). The simultaneous use of transformationsℋ1,μ, andℋ2,μ, (which are denoted byℋμ) allows us to solve many problems of Mathematical Physics involving the differential operatorΔμ=D2+(1+2μ)x−1D, whereas the pair of transformationsℋ3,μandℋ4,μ, (which we express byℋμ*) permits us to tackle those problems containing its adjoint operatorΔμ*=D2−(1+2μ)x−1D+(1+2μ)x−2, no matter what the real value ofμbe. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation∫0∞f(x)g(x)dx=∫0∞(ℋμf)(y)(ℋμ*g)(y)dy,which is now valid for all realμ.
Print ISSN:
0161-1712
Electronic ISSN:
1687-0425
Topics:
Mathematics
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