Integral representations and integral transforms of some families of ...

More documents

Recommendations

Info

$Fractional and operational calculus with generalized fractional ...$

$Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...$

10 N. Elezović et al. 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 and F c (S μ+1 (r))(x) = √ π · 2 1/2−μ ∫ ∞ t μ+1/2 cos(xr)S μ+1 (r)dr = 0 Ɣ(μ + 1) x e t − 1 c(μ; x,t)dt (x > 0; μ>0), (47) ∫ ∞ where (1) s (μ; x,t), (2) s (μ; x,t) and c (μ; x,t) are defined by (44), (45) and (46), respectively. In a similar manner, we find from the integral representation (11) that and F s ( S (α,0) μ (r;{k 2/α }) ) (x) = ∫ ∞ 0 sin(xr)S μ (α,0) (r;{k 2/α })dr ∫ ∞ t μ−1/2 (∫ ∞ ) r 1/2−μ J Ɣ(μ) 0 e t μ−1/2 (rt) sin(xr)dr dt − 1 0 ( ∫ x t μ−1/2 Ɣ(μ) 0 e t − 1 (1) s (μ; x,t)dt t μ−1/2 ) (μ; x,t)dt √ π · 2 3/2−μ = √ π · 2 3/2−μ = + ∫ ∞ x e t − 1 (2) s ) ( x>0; μ> 1 2 F c ( S (α,0) μ (r;{k 2/α }) ) (x) = ∫ ∞ 0 cos(xr)S μ (α,0) (r;{k 2/α })dr √ π · 2 3/2−μ = Ɣ(μ) ∫ ∞ t μ−1/2 x ( x>0; μ> 1 2 (48) e t − 1 c(μ; x,t)dt ) , (49) where (1) s (μ; x,t), (2) s (μ; x,t) and c (μ; x,t) are defined, as before, by (44), (45) and (46), respectively. III. In the evaluation of the Fourier sine and the Fourier cosine transforms of the general Mathieu Series S μ (α,β) (r;{k 2/α }), we shall make use of the following integral formulas proven earlier by Miller and Srivastava [9, pp. 225 and 226]: ⎧ ∫ ∞ ⎨ (1) sin(2ax) 1 F 2 (α; β,γ;−b 2 x 2 s (α, β, γ ; a,b) (0
Integral Transforms and Special Functions 11 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 where, and in what follows, (1) s (α, β, γ ; a,b), (2) s (α, β, γ ; a,b) and c (α, β, γ ; a,b) are given by (1) s (α, β, γ ; a,b) := 1 ( ) 1 b2 2a 3 F 2 , 1,α; β,γ; 2 a 2 ( 0
Page 1 and 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Page 3 and 4: 101 102 103 104 105 106 107 108 109
Page 5 and 6: Integral Transforms and Special Fun
Page 9: Integral Transforms and Special Fun
Page 15: Integral Transforms and Special Fun

Integral representations and integral transforms of some families of ...

Create successful ePaper yourself

Delete template?

Save as template?