Cmjv04i01p0075

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J. Comp. & Math. Sci. Vol.4 (1), 75-79 (2013)

Time-Domain Synthesis Problem Involving I-Function of Two Variables S. S. SRIVASTAVA1 and ANSHU SINGH2 1,2

Department of Mathematics, Govt. P. G. College, Shahdol, M. P., INDIA. (Received on: February 17, 2013) ABSTRACT The object of this paper is to evaluate an integral involving Bessel polynomial and I-function of two variables and employ it to obtain a particular solution of the general solution derived in this paper of the classical problem known as the, "time-domain synthesis problem", occurring in electrical network theory. Keywords:.

introduced by Sharma & Mishra13, will be defined and represented as follows:

1. INTRODUCTION The I–function of two variables 0, n : m1, n1 :m2, n2 x I [ y ] =Ip, q: r :p , q : p , q i i i´ i´ r´ : i´´ i´: r´´

[

[(aj; αj, Aj)1, n], [(aji; αji, Aji) n + 1, pi]

x | y

[(bji; βji, Bji)1, qi]

: [(cj; γj)1, n1], [(cji´; γji´) n1 + 1, pi´ ]; [(ej; Ej)1, n2], [(eji´´; Eji´´) n2 + 1, pi´´] : [(dj; δj)1, m1], [(dji´; δji´) m1 + 1, qi´ ]; [(fj; Fj)1, m2], [(fji´´; Fji´´) m2 + 1, qi´´] =

1 . 2 (2πω)

∫

∫

L1 L2

]

φ(ξ, η) θ2(ξ) θ3(η)xξ yη dξ dη,

where

φ1 (ξ, η) =

n Π Γ ( 1 − aj + αjξ + Ajη) j=1 r

pi

qi

Σ [ Π Γ(aji - αjiξ − Αjiη) Π Γ (1 − bji + βjiξ + Bjiη) i=1

j = n+ 1

j=1

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

(1)

76

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.4 (1), 75-79 (2013)

θ2 (ξ) =

m1 n1 Π Γ (dj − δjξ) Π Γ (1 − cj + γjξ) j=1 j=1 r´

qi´

Σ [Π i´ = 1

θ3 (η) =

pi´

Γ (1 − dji´ + δji´ξ) Π

j=m +1 1

j=n +1 1

Γ (cji´ − γji´ξ) ]

m2 n2 Π Γ (fj − Fjη) Π Γ (1 − ej + Ejη) j=1 j=1 r ´´

qi´´

Σ [Π i´´ = 1

pi´´

Γ (1 − fji´´ + Fji´´η) Π

j=m +1 2

Γ (eji´´ − Eji´´η) ]

j=n +1 2

x and y are not equal to zero, and an empty product is interpreted as unity pi, pi´, pi´´, qi, qi´, qi´´, n, n1, n2, nj and mk are non negative integers such that pi ≥ n ≥ 0, pi´ ≥ n1 ≥ 0, pi´´ ≥ n2 ≥ 0, qi >0, qi´ >0, qi´´ > 0, (i = 1, …, r; i´ = 1, …, r´; i´´ = 1, …, r´´; k = 1, 2) also all the A’s, α’s, B’s, β’s, γ’s, δ’s, E’s and F’s are assumed to be positive quantities for standardization purpose; the definition of Ifunction of two variables given above will however, have a meaning even if some of these quantities are zero. The contour L1 is in the ξ−plane and runs from – ω∞ to + ω∞,

with loops, if necessary, to ensure that the poles of Γ(djδjξ) (j = 1, ..........., m1) lie to the right, and the poles of Γ (1 − cj  γjξ) (j = 1, ..., n1), Γ (1− aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour. The contour L2 is in the η−plane and runs from – ω∞ to + ω∞, with loops, if necessary, to ensure that the poles of Γ(fj Fjη) (j=1,....., n2) lie to the right, and the poles of Γ (1− ej  Ejη) (j = 1, ..., m2), Γ ( 1− aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour.

qi´´ pi´´ qi pi R = Σ αji + Σ γji´ − Σ βji − Σ δji´ < 0, j=1 j =1 j=1 j=1 pi qi qi´´ pi´´ S = Σ Aji + Σ Eji´´ − Σ Bji − Σ Fji´´ < 0, j=1 j=1 j =1 j=1 pi´ pi qi n1 m1 qi´ U = Σ αji − Σ βji + Σ δj − Σ δji´ + Σ γj − Σ γji´ > 0, j = n +

j=1

j =1

j = m1 + 1

j =1

j = n1 + 1

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

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S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.4 (1), 75-79 (2013)

pi

V=− ÎŁ

j=n+1

and

q qi m2 i´´ Aji − ÎŁ Bji − ÎŁ Fj − ÎŁ

j=1

j =1

pi´´ n2 Fji´´ + ÎŁ Ej − ÎŁ

j = m2 + 1

, ; = 2F0

x/b . ;

(5)

Orthogonality property of Bessel polynomials2 భ ๎

e y x; a, 1 dx

๤ ! Ď€ δ , Γ Γ Ď€ ,

(6)

where Re a < l − m − n. 3. INTEGRAL ∞

ŕ°­

(4) ∞

y x; a, b

∑ !

x Ďƒ e ๎ y x; a, 1 dx

(3)

x Ďƒ e y x; a, 1 I Ρ dx

The following results are required in our present investigation: Bessel polynomial2

∞

Eji´´ > 0,

| arg x | < ½ UĎ€, | arg y | < ½ VĎ€.

2. RESULT REQUIRED

x

j = n2 + 1

j =1

77

Γ Ďƒ Γ Ďƒ Γ Ďƒ

,

(7)

where Re Ďƒ < 0, Re (a − Ďƒ) < 2, 1, 2, 3, ‌. The integral (7) can easily established by expressing the Bessel Polynomial in the integrand as its series representation (5), interchainging the order of integration and summation, evaluating the resulting integral with the help of3, using Gauss's theorem4.

Ξ Ν

= I , : ; ′ , ′ : ′ ; ′′ , ′′ : ´´

, ; , ; ,

Ξ â€Śâ€Ś,‌...:‌‌., Ďƒ ,Îť :‌‌,‌‌. Ρ|‌...,‌‌: Ďƒ ,Îť , Ďƒ ,Îť ,‌‌:‌‌,‌‌. ,

(8)

provided that Re Ďƒ < 0, Re (a − Ďƒ) < 2, 1, 2, 3, ‌. and U 0, 0, |argΞ| " UĎ€, where U and V are given in (2) and (3). Proof: To establish (8), expressing the Ifunction in the integrand as a Mellin-Barnes type integral (1) and interchainging the order of integrations which is justified due to the absolute convergence of the integrals involved in the process, evaluating the innerintegral with the help of (7) and using the definition of I-function, we arrive at (8). 4. SOLUTION OF THE TIME-DOMAIN SYNTHESIS PROBLEM OF SIGNALS The classical time-domain synthesis problem occurring in electrical network theory is stated as follows5. Given an electrical signal described by a real valued conventional function f(t) on 0 < t < ∞, construct an electrical network consisting of finite number of components

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

78

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.4 (1), 75-79 (2013)

R, C and I which are all fixed, linear and positive, such that the output of fN(t), resulting from a delta-function δ(t) approximates f(t) on 0 < t < ∞ in some sense. In order to obtain a solution of this problem, we expand the function f(t) into a convergent series: f(t) = ∑∞ Ďˆ t

(9)

of real-valued function Ďˆ t . Let every partial sum fN(t) = ∑ Ďˆ t , N = 0, 1, 2, ‌ c 1

Γ Γ π ! π

(10)

∞

possess the two properties: (i) fN(t) = 0, for − ∞ < t < 0. (ii) The laplace transform FN(s) of fN(t) is a rational function having a zero at s = ∞ and all its poles in the left-half s-plane, except possibly for a simple pole at the origin. After choosing n in (10) sufficiently large to satisfy whatever approximation criterion is being used, an orthonormal series expansion may be employed. The Bessel polynomial transformation and (5) yields an immediate solution in the following form: / / f(t) = ∑∞ e y t; a, 1 , c t where

ŕ°­

f t t e ఎ๪ y t; a, 1 dt,

(11)

where Re a < 1 − 2n. This case is an example of the use of hypergeometric integral in an orthonormal series expansion. 5. PARTICULAR SOLUTION OF THE PROBLEM The particular solution of the problem in terms of I-function is as follows: f(t) =

Ď€ ∞ Γ Γ ∑ 1 Ď€ ! , ; , ; , t / e / y t; a, 1 I , : ; ′ , ′ : ′ ; ′′ , ′′ : ´´ Ξ â€Śâ€Ś,‌...:‌‌.., Ďƒ ,Îť :‌‌,‌‌. Ρ|‌...,‌‌: Ďƒ ,Îť , Ďƒ ,Îť :‌‌,‌‌. ,

(12)

provided that Re Ďƒ < 0, Re (a − Ďƒ) < 2, 1, 2, 3, ‌. and U 0, 0, |argΞ| " UĎ€, where U and V are given in (1.3.4) and (1.3.5).

Proof: Let us consider f(t) = t Ďƒ / e / I Ρ Ξ Îť

/ / = ∑∞ e y t; a, 1 . c t Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

(13)

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.4 (1), 75-79 (2013)

79

Equation (13) is valid, since f(t) is continous and of bounded variation in the open interval (0, ∞). Multiplying both sides of (13) by t / e / y t; a, 1 and integrating with respect to t from 0 to ∞, we get % t Ďƒ e / y t; a, 1 I Ρ dt Ξ Îť

∞

/ = ∑∞ e y t; a, 1 dt. c % t ∞

Now using (8) and (6), we get c

Γ Ď€ , ; , ; , I , : ; ′ , ′ : ′ ; ′′ , ′′ : ´´ ! Ď€

Ρ|‌...,‌‌: Ďƒ ,Îť , Ďƒ ,Îť ,‌‌:‌‌,‌‌. , Ξ â€Śâ€Ś,‌...:‌‌.., Ďƒ ,Îť :‌‌,‌‌.

(14)

From (13) and (14), the solution (12) follows immediately. REFERENCES 1. Sharma C. K. and Mishra, P. L.: On the I-function of two variables and its certain properties, ACI, 17, 1-4 (1991). 2. Exton, H. On Orthogonal of Bessel Polynomials. Rev. Mat. Univ. Parma (4) 12, 213-215(1986). 3. Erdelyi, A.: Table of integral transform, Vol. I, McGraw-Hill, New York, (1954). 4. Erdelyi, A. et. al. Higher Transcendental

Functions, Vol. 1 McGraw-Hill, New York, (1953). 5. Exton, H. Handbook of Hypergeometric Integrals, Ellis Horwood Ltd., Chichester, (1978). 6. Srivastava, S. S.: Studies on Some Contributions of Special Functions in Various Discipline, Minor Research Project Submitted to MPCOST, Bhopal, (2008).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)

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