Some Indefinite Integrals in the Light of Hypergeometric Function

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1 Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 3 Issue 6 Version.0 Year 03 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: Some Indefinite Integrals in the Light of Hypergeometric Function By Salahuddin & Intazar Husain P.D.M College of Engineering, India Abstract - In this paper we have evaluated some indefinite integrals associated to Hypergeometric function. The results represent here are assume to be new. Keywords : pochhammer symbol gaussian hypergeometric function kampé de fériet double hypergeometric function and srivastava s triple hypergeometric function. GJSFR-F Classification : MSC NO: 33C05,33C45,33C5,33D50,33D60 Some Indefinite Integrals in the Light of Hypergeometric Function Strictly as per the compliance and regulations of : 03. Salahuddin & Intazar Husain. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Some Indefinite Integrals in the Light of Hypergeometric Function 03 Salahuddin α & Intazar Husain σ Abstract - In this paper we have evaluated some indefinite integrals associated to Hypergeometric function. The results represent here are assume to be new. Keywords : pochhammer symbol gaussian hypergeometric function kampé de Fériet double hypergeometric function and srivastava s triple hypergeometric function. The Pochhammer s symbol or Appell s symbol or shifted factorial or rising factorial or generalized factorial function is defined by Γ(b + k b(b + (b + (b + k if k,, 3, (b, k (b k if k 0 Γ(b k! if b, k,, 3, where b is neither zero nor negative integer and the notation Γ stands for Gamma function. a Generalized Gaussian Hypergeometric Function Generalized ordinary hypergeometric function of one variable is defined by a, a,, a A AF B z (a k (a k (a A k z k (b b, b,, b B k (b k (b B k k! k0 or AF B (a A (b B z A F B (a j A j (b j B j z k0 ((a A k z k ((b B k k! where denominator parameters b, b,, b B are neither zero nor negative integers and A, B are non-negative integers. b Kampé de F eriet s General Double Hypergeometric Function In 9, Appell s four double hypergeometric functions F, F, F 3, F 4 and their confluent forms Φ, Φ, Φ 3, Ψ, Ψ, Ξ, Ξ were unified and generalized by Kampé de Fériet. (. Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year 9 Author α : P.D.M College of Engineering, Bahadurgarh, Haryana, India. Author σ : Delhi Institute of Technology, Management and Research, Ballabhgarh, Haryana, India. vsludn@gmail.com 03 Global Journals Inc. (US

3 We recall the definition of general double hypergeometric function of Kampé de Fériet in slightly modified notation of H.M.Srivastava and R.Panda: (a A :(b B (d D F A:BD ((a E:GH x, y A m+n ((b B m ((d D n x m y n (. ((e (e E :(g G (h H m,n0 E m+n ((g G m ((h H n n! 03 where for convergence (i A + B < E + G +, A + D < E + H + x <, y <, or (ii A + B E + G +, A + D E + H +, and Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year 0 x (A E + y (A E < max { x, y } < c Srivastava s General Triple Hypergeometric Function, if E < A, if E A In 967, H. M. Srivastava defined a general triple hypergeometric function F (3 in the following form i,j,k0 F (3 (a A :: (b B (d D (e E : (g G (h H (l L (m M :: (n N (p P (q Q : (r R (s S (t T x, y, z ((a A i+j+k ((b B i+j ((d D j+k ((e E k+i ((g G i ((h H j ((l L k x i y j z k ((m M i+j+k ((n N i+j ((p P j+k ((q Q k+i ((r R i ((s S j ((t T k i! j! k! d Wright s Generalized Hypergeometric Function (α, A,, (α p, A p pψ q x Γ(α + ma Γ(α + ma Γ(α p + ma p x m Γ(λ (λ, B,, (λ q, B q m0 + mb Γ(λ + mb Γ(λ q + ma q (.4 (α, A,, (α p, A p pψ (α q x ma (α ma (α p map x m (.5 (λ (λ, B,, (λ q, B q m0 mb (λ mb (λ q mbq [ ( +x sin3 y cosy sin 3m+ y (sin y 3m F, 3m 0 3 +x, cos y (.3 + Constant (. 03 Global Journals Inc. (US

4 [ ( +x cos3 y sin y cosec y cos 3m+ y 3m + F 0, 3m+ 3m+3 +x, cos y + Constant (. [ ( +x tan3 y tan3m+ y (3m + F 0, 3m+ 3m+3 +x, tan y +Constant (.3 03 [ ( +x cot3 y cot3m+ y (3m + F 0 sin(y cos (y 3m+ sec 3m+ (y F 0 cosy(sin (y 3m+ cosec 3m+ y F 0 Derivation of integral (. ( m( +x m0 m0 m, 3m+ 3m+3 [ ( +x sec3 y, +3m 3 dx ( ( +x cosech3 y, +3m 3 +x, cot y +x, sin y +x, cos y [ ( + x sin 3 y [ ( +x sin3 y ( m( +x m sin 3m y m0 ( m( +x m [ ( cosy sin 3m+ y(sin y 3m F cosy sin 3m+ y (sin y 3m F 0, 3m 3, 3m 3 +x, cos y + Constant (.4 + Constant (.5 + Constant (.6 sin 3m y cos y + Constant + Constant (3. Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year 03 Global Journals Inc. (US

5 03 Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year Derivation of integral (. [ ( +x cos3 y m0 m0 ( m( +x m sin y cosec y cos 3m+ y 3m + Derivation of integral (.3 m0 m0 ( m( +x m cos 3m y [ m0 ( + x sin y cos 3m+ [ y cosec y F (3m + ( m( +x m F 0 [ ( +x tan3 y ( m( +x m tan 3m y tan3m+ y (3m + F 0 Derivation of integral (.4 m0 m0 tan 3m+ [ y (3m + F, 3m+ 3m+3 [, 3m+ 3m+3 [ ( +x cot3 y ( m( +x m cot 3m y ( m( +x m cot3m+ y (3m + F 0 cot 3m+ [ y (3m + F m0 cos 3 y ( m( +x m ( + x, 3m+ 3m+3, 3m+ 3m+3 [, 3m+ 3m+3 m0 +x, cos y tan 3 y ( m( +x m +x, tan y ( + x cos 3m y cos y + Constant + Constant (3. tan 3m y tan y + Constant cot 3 y ( m( +x m, 3m+ 3m+3 +x, cot y + Constant (3.3 cot 3m y cot y + Constant + Constant ( Global Journals Inc. (US

6 Derivation of integral (.5 ( ( +x sec3 y m0 m0 ( m( +x m ( m( +x m sin(y cos (y 3m+ sec 3m+ (y F 0 Derivation of integral (.6 m0 m0 ( m( +x m sec 3m y m0 ( m( +x m [ siny cos (y 3m+ sec 3m+ y F, 3m+ ( ( +x cosec3 y ( m( +x m cosec 3m y, +3m 3 [ m0 3 sec 3m y +x, sin y ( + x cosec 3 y ( m( +x m ( cosy (sin (y 3m+ cosec 3m+ y F [ cosy(sin (y 3m+ cosec 3m+ y F 0 [, +3m 3 ( + x sec 3 y sin y + Constant + Constant (3.5 cosec 3m y, 3m+ 3 +x, cos y cos y + Constant + Constant (3.6 In our work we have established hypergeometric form of some indefinite integrals. We can only expect that the development presented in this work will stimulate further interest and research in this important area of classical special functions. Just as the mathematical properties of the Gauss hypergeometric function are alrea of immense and significant utility in mathematical sciences and numerous other areas of pure and applied mathematics, the elucidation and discovery of the formula of hypergeometric functions considered herein should certainly eventually prove useful to further developments in the broad areas alluded to above. 03 Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year 3. Agarwal, R. P. Resonance of Ramanujan s Mathematics. Vol. I, New Age International (P Ltd., New Delhi, Agarwal, R. P. Resonance of Ramanujan s Mathematics. Vol. II, New Age International (P Ltd., New Delhi, Global Journals Inc. (US

7 3. Srivastava, H.M. and Daoust, M.C. A note on the convergence of Kampé de Fériet double hypergeometric series.math.nachr.53( Qureshi,M.I., Salahuddin.,Chaudhary, M. P. and Quraishi, K.A. Evaluation of Certain Elliptic Type Single, Double Integrals of Ramanujan and Erdélyi, J. Mathematics Research., (00, Global Journal of Science Frontier Research F Volume XIII Issue VI V ersion I Year 4 03 Global Journals Inc. (US