Download Index Transforms by Semyon B Yakubovich PDF

By Semyon B Yakubovich

This ebook bargains with the speculation and a few purposes of critical transforms that contain integration with recognize to an index or parameter of a different functionality of hypergeometric style because the kernel (index transforms). the elemental index transforms are thought of, akin to the Kontorovich-Lebedev remodel, the Mehler-Fock remodel, the Olevskii remodel and the Lebedev-Skalskaya transforms. The Lp concept of index transforms is mentioned, and new index transforms and convolution buildings are established. For the 1st time, the primarily multidimensional Kontorovich-Lebedev remodel is introduced. normal index remodel formulae are received. the relationship among the multidimensional index kernels and G and H capabilities of numerous variables is gifted. The publication is self-contained, and contains a checklist of symbols with definitions, writer and topic indices, and an updated bibliography.This paintings may be of curiosity to researchers and graudate scholars within the mathematical and actual sciences whose paintings contains crucial transforms and exact capabilities.

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Example text

191) is the most important one we may consider separately the Mellin transform, which can be obtained from the Fourier transform by an exponential replacement and by rotating the complex plane through a right angle /*(*) = M{f(t)\ s} = [ F > / 2 ? / ( C » ) ] ( - M ) = / + °° f{t)t*-ldt. 203) Jo0 In fact, its inversion is given by the formula 1 1 f{x) = /(x) = M-'U'is);x} ^"Mr(»); *} == — irZlTl ri/+t'oo ri/+t'oo JU—too J7TZ Jv—ioo f'{s)x-'ds, r(>)*-*>, v» == »a, *«. 204) and undoubtedly, we can observe its connection with the class of hypergeometric type functions as it was described in detail in Marichev [1].

DStX>0f. ^ * 1 " ' * , x > 0. 17. Let f*{v + it) G LP(R), 1 < p < 2, v € R, /i(x) € I 1 _ I / f P (R+). 203) of the function h(x). 214). 190) and usually we shall define it as [Lf](x) = [°°e-xtf(t)dt, x > 0. 32) allows us to conclude that T(v + it) € LP(R), 1 < p < 2, v € R+. 214) as 1 [Lf](x) = — rv+ioo T(s)fm(1 - s)z- 5 0. 214) is slightly different from the Mellin convolution V*9)(*) = J~f(j)9(t)j (1-217) and we can definitely call the Laplace transform of positive variable x as the Mellin convolution type integral transform as noted in Widder [1], Hirschman and Widder [1], Marichev [1], Brychkov et al.

13) does not coincide with the space X r ( R + ) . 12). But choosing different weights one can easily verify that there exists some function, which belongs to Z r (R+), but does not belong to the space Z r (R + ;/o) and vice versa. 1). 230). It was introduced in slightly different form in Yakubovich and Luchko [2], Chapter 6. 14) JO where e £ (0,7r). 91) of the order 1. Proof. 16) JO provided that we choose Sx + 82 + e > IT. 3). 1 and interchange the order of integration in obtained iterated integral.

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