By A.M. Mathai
The subject matters of specified H-function and fractional calculus are at the moment present process speedy alterations either in idea and alertness. taking into consideration the newest study effects, the authors delve into those themes as they relate to purposes to difficulties in data, physics, and engineering, quite in condensed subject physics, plasma physics, and astrophysics.
The booklet units forth the definitions, contours, lifestyles stipulations, and specific situations for the H-function, then explores the houses and relationships one of the Laplace, Fourier, Hankel, and different transforms. From right here, the H-functions are applied for functions in statistical distribution idea, buildings of random variables, generalized distributions, Mathai’s pathway types, and flexible integrals. services of matrix argument are brought with a spotlight on real-valued scalar features whilst the matrices are actual or Hermitian positive-definite. The textual content concludes with very important fresh functions to actual difficulties in response, diffusion, reaction-diffusion thought and information, and superstatistics. Generalized entropies in addition to purposes in astrophysics are dealt with.
Over the previous couple of years, fabric during this booklet has been additional to numerous classes and constructed to satisfy the desires of students on the PhD point. All workouts within the ebook were used to probe the data and skill of arithmetic, facts, and physics to scholars and researchers.
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6. 5 are satisfied. 5. 7. 5 be satisfied. Let either of the following conditions hold: < 1; jarg. ı/; arg. 5. 7 then we arrive at the following theorem given by Kilbas et al. (2006, p. 140). 8. 5 be satisfied, and let 2 R. 5. 140). The utility and importance of the generalized Wright function is realized in recent years due to its occurrence in certain problems of applied character. This function is in the proximity of the H -function so its utility is further increased. Nearly all the Mittag-Leffler functions and their generalizations can be expressed in terms of this function; in this connection one can refer to the paper by Kilbas et al.
0; . 15) the above line reduces to xˇ 1 1 X . ˇ/ > 0; <. 46). 1. ˇ/ > 0; and jas ˛ j < 1. ı/ < ˛ > 0I ; ˛; s C; > 0, satisfy the condition Ä <. bj / Bj > 0 for ˛ > 0 or ˛ D 0; Ä <. ı/C 1 2 " 1 0I and > 0; < 0. s/ > 0, there holds the formula ( L x min1Ä j 1. 2) and the well-known gamma function formula. 9 Inverse Laplace Transform of the H -Function Due to the importance and utility of inverse Laplace transforms of special functions in physical problems, we present the inverse Laplace transform of the H -function in this section.
Ap C Ap s/; n < p D . 1 bq Bq s/; m < q D s . b1 C B1 s/; m > 0 D s . 1 a1 A1 s/; n > 0 D s . ap C Ap s/; n < p D s . 3). s/ is given as (Buschman 1974a, p. 151). 10. kCa/ 1 X . kCa/ A Á; where A > 0. 11. 1;1/ a;A/ i DA 1 1 X . a k 1/ A Á; where A > 0. 12. 13. 1 s/ zk . 14. 15. Evaluate 1 ; z 2 C; z ¤ 0; < 0; <. s a ; 2 N0 . 16. Prove that the Mellin–Barnes integral (Paris and Kaminski 2001, p. 113) Z 1 Ci 1 2 i i1 . z/j < 2 and the contour C separates the poles at s D ; 2 N0 from the pole s D a (a is not a positive integer).