Cmjv05i04p0402

Page 1

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.402-411

JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

On Fourier Series Involving I-Function of Two Variables S. S. Srivastava and Anshu Singh Department of Mathematics, Govt., P. G. College, Shahdol, M. P., INDIA. (Received on: August 22, 2014) ABSTRACT The subject of Fourier series for generalized hypergeometric functions occupies a outstanding place in the literature of special functions and boundary value problems. Certain double Fourier series of generalized hypergeometric functions play an vital role in the improvement of the theories of special functions and two-dimensional boundary value problems. Looking vital role of Fourier series in the literature of special functions and boundary value problems, in this paper, we establish some new Fourier series involving I-function of two variables.In section (4.3), we establish some new Fourier series involving Ifunction of two variables, while some particular cases, relevant to the present discussion, are given at the end of the section (4.3). Keywords: Fourier series for generalized hypergeometric functions, special functions, boundary value problems, double Fourier series of generalized hypergeometric functions, Dirichlet's conditions, the Jordan's theorem.

The I-function of one variable introduced by Saxena2, will be represented as follows:

4.1 INTRODUCTION 4.1.1. Definition of I Function:

[(aj, αj)1, n], [(aji, αji)n + 1, pi] ] = (1/2πi) ∫ θ (s) xs ds I pm,, qn : r [x| [(b i i j, βj)1, m], [(bji, βji)m + 1, qi] L

θ (s) =

m Π Γ(b – β s) j j j=1 r Σ i=1

qi Π

n Π Γ(1 – a + α s) j j j=1

Γ(1 – bji + βjis)

j=m+1

(1.3.1)

pi Π

, Γ(aji – αjis)

j=n+1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

403

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

integral is convergent, when (B >0, A ≥ 0), where n

pi

m

qi

Β = Σ αj – Σ αji + Σ βj – Σ βji ,

A=

j=1

j=n+1

pi

qi

j=1

(1.3.2)

j=m+1

Σ αji – Σ βji ,

j=1

j=1

s = (aj – 1 – v) | αj j = 1, 2, …, n; v = 0, 1, 2, …. lie to the left hand side

|arg x| < ½ Bπ, ∀ i ∈ (1, 2, …, r). pi (i = 1, 2, …, r); qi(i = 1, 2, …, r); m, n are integers satisfying 0 ≤ n ≤ pi, 0 ≤ m ≤ qi, (i = 1, 2, …, r); r is finite αj, βj, αji, βji are real and positive and aj, bj, aji, bji are complex numbers such that αj (bh + v) ≠ Bh (aj – 1 – k), for v, k = 0, 1, 2, ….. h = 1, 2, …, m; j = 1, 2, …, r; L is a contour runs from σ – i∞ to σ + i∞ (σ is real), in the complex s-plane such that the poles of

I[x ]=I

0, n

: m1, n1

:m2,

n2

x

[|

pi, qi: r : pi´, qi´: r´ :pi´´, qi´: r´´ y

s = (bj + v) | βj j = 1, 2, …, m; v = 0, 1, 2, …. and right of L. The I–function of two variables introduced by Sharma & Mishra9, will be defined and represented as follows:

[(aj; αj, Aj)1, n], [(aji; αji, Aji) n + 1, pi] [(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´´] ∫ ∫ φ1(ξ, η) θ2(ξ) θ3(η)xξ yη dξ dη, = 1 . 2

(2πω)

where φ1 (ξ, η) =

L1 L2

n Γ ( 1 − aj + αjξ + Ajη) Π j=1 r

pi

,

qi

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

θ2 (ξ) =

j = n+ 1

j=1

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

qi´

Σ [Π i´ = 1

j=m +1 1

,

pi´

Γ (1 − dji´ + δji´ξ) Π Γ (cji´ − γji´ξ) ] j=n +1 1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

(1.3.3)

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

m2 Π

j=1

θ3 (η) =

r ´´

n2

Γ (fj − Fjη) Π Γ (1 − ej + Ejη)

j=1

,

qi´´

pi´´

Σ [Π i´´ = 1

pi´´

Γ (eji´´ − Eji´´η) ]

Γ (1 − fji´´ + Fji´´η) Π

j=m +1 2

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

pi

404

qi

poles of Γ(dj−δjξ) (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. Also R = Σ αji + Σ γji´ − Σ βji − Σ δji´ < 0,

qi´´

S = Σ Aji + Σ Eji´´ − Σ Bji − Σ Fji´´ < 0,

j=1

j=1

pi U=

qi

m1

qi´

n1

Σ αji − Σ βji + Σ δj − Σ

j=n+1

j=1

pi

V=− Σ

j =1

qi

m2

j=1

j =1

pi´

δji´ + Σ γj − Σ

j = m1 + 1

q

i´´

Aji − Σ Bji − Σ Fj − Σ

j=n+1

and

j=1

j =1

n2 F

j = m2 + ji´´ 1

γji´ > 0,

(1.3.4)

j = n1 + 1

j =1

pi´´

+ Σ Ej − Σ j =1

Eji´´ > 0,

j = n2 + 1

| arg x | < ½ Uπ, | arg y | < ½ Vπ. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

(1.3.5)

405

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

ŕśą cos (ux) áˆşsin x/2áˆť ŕ°­ I[

3.1.2 REVIEW OF LITRATURE 1

2

Rainville , Mishra , Bajpai , 6 Taxak , Sharma , Carslaw and Jaeger7 and others have evaluated certain number of Fourier series involving generalized hypergeometric functions. 5

4.2 FORMULA USED: From Rainville1: , , 1 − x 1 + x P (x)P (x) dx = 0, if m ≠n, ( ) ( )

= !( ) ( ), if m = n; (4.2.1) where Re(a) > − 1, Re(b) > − 1. The following orthogonality properties given in2:

‍ ׏‏e dx = ‍ ׏‏e

π, π, 0,

cos nx dx =

e sin nx dx =

m = n; m = n = 0; m ≠n;

(4.2.2)

,

; ;

(4.2.4)

provided either both m and n are odd or both m and n are even integers. We also use formulae mentioned below:

, ( ) ŕśą áˆş1 − yáˆť ŕ°Ž áˆş1 + yáˆť P (y) I[ ] dy డ๥

=

[ ] , ; , ; ,

. I , : ; , : ; , : ´´ !

డ๥ ‌‌,‌...: ŕ°Ž , ,‌‌, ŕ°Ž , :‌‌,‌‌. [ |‌...,‌‌: ŕ°Ž, ,‌‌..,

], ఎ , :‌‌,‌‌.

(2.3.4) provided that k > 0, ω2 > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

]dx

, ; , ; ,

= √(Ď€) I , : ; , : ; , : ´´

‌‌,‌...: , ,‌‌.., , :‌‌,‌‌.

[ |‌...,‌‌:

, ,‌‌, , :‌‌,‌‌.

], (2.3.3)

provided that h > 0, ω1 > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

( !)

(sinθ) cosuθ I[

] dθ

, ; , ; ,

= âˆšĎ€ cos ( ) I , : ; , : ; , : ´´

‌‌,‌...:‌‌.., ," , ," :‌‌,‌‌. ‌...,‌‌: ," , ," ,‌‌.:‌‌,‌‌.

[ |

],

(2.3.5) provided that Ď > −1, δ > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

π/2, m = n; π, m = n = 0; (4.2.3) 0, m ≠n; ,

3,4

/ ŕ°Žŕąž

( !)

(sinθ) sinuθ I[

] dθ

, ; , ; ,

= âˆšĎ€ sin ( ) I , : ; , : ; , : ´´

‌‌,‌...:‌‌.., ," , ," :‌‌,‌‌. ‌...,‌‌: ," , ," ,‌‌.:‌‌,‌‌.

[ |

],

(2.3.6) provided that Ď > −1, δ > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

ŕśą y J áˆşyáˆť I[

= [

డఎ๥

] dy

, ; , ŕ°Ž ; ŕ°Ż , ŕ°Ż 2 I , ŕ°­ : ; ŕ°Ž ᇲ , ᇲ : ᇲ ; ᇲᇲ , ᇲᇲ : ´´ ŕą&#x; ŕą&#x; ŕą&#x; ŕą&#x; ŕą&#x; ŕą&#x;

డఎ๥ ‌‌,‌...:‌‌..,‌‌:‌‌,‌‌. |‌...,‌‌: ಚజಕ, ,‌‌.., ಚడಕ, :‌‌,‌‌. ], ŕ°Ž

ŕ°Ž

(2.3.9)

provided that k > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

406

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

sin x e F [ (

I[

) $

: ( ) " ´ :#( ) ] ´F ´[ ! ´ ]

.

) &

],

(2.3.7) provided that h and k are positive integers, P < Q (or P = Q + 1 and |c| < 1), U´ < V´ (or U´ = V´ + 1 and |d| < 1), no one of the β, and δ- ´ are zero negative integer and Re (ω) > 0 and and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

0 !

=

] dθ

, ; , ; , âˆšĎ€ I , : ; , : ; , : ´´

1 ‌‌,‌...: , ,‌..,( , ):‌‌,‌‌. [ |‌...,‌‌: , ,‌‌.,( , ):‌‌,‌‌. ],

(2.3.12) provided that Re(3 − 2u) > 0, n = 0, 1, 2, ‌, h > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

0 !

sin (2n + 1)θ sinθ I[

=

,-/,. ,-/,‌‌:‌‌,‌‌. ¹ ‌...,‌‌:‌‌..,( ,-):‌‌,‌‌.

sin (2n + 1)θ sinθ I[

‌‌,‌...: , ,‌..,( , ):‌‌,‌‌.

, ‌‌,‌...:.

[ |‌...,‌‌: , ,‌‌.,( , ):‌‌,‌‌. ], (2.3.13)

*´ (

[ +|

, ; , ; ,

, ; , ; ,

] dθ

= âˆšĎ€ I , : ; , : ; , : ´´

I , : ; , : ; , : ´´

0 (!/ )

] dx

#% &$& %'´ ( &( =âˆšĎ€e / ∑+ ,* '( ) ! " *!

cos nθ sin θ/2 I[

] dθ

, ; , ; , âˆšĎ€ I , : ; , : ; , : ´´

1 ‌‌,‌...: , ,‌..,( , ):‌‌,‌‌. [ |‌...,‌‌: , ,‌‌.,( , ):‌‌,‌‌. ],

(2.3.12) provided that Re(3 − 2u) > 0, n = 0, 1, 2, ‌, h > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

provided that Re(1 − 2u) > 0, n = 0, 1, 2, ‌, h > 0 and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5). 4.3 FOURIER SERIES 2

( ) 0 ( /) 3 ****(sin ) . (1 − y). I[ ] * * 4 + ∑ ,* cos rx

* √

=

,

P* [

, ; , ; ,

(y)I , : ; 1, 1: ; , : ´´

ଶషౡ ா

ŕ°­

‌‌,‌...:ቀమା୵భ,ୌበ,áˆşŕŹľŕŹžŕ­ľŕ°Ž ି୲,ŕ­Šáˆť,‌‌‌,áˆşŕ­ľŕ°­ ,ŕ­Śáˆť,(ଶାŕ­&#x;ାୠା୵మ ା୲,ŕ­Š):‌‌,‌‌.

ŕŽ—|‌...,‌‌:áˆşŕ­ľŕ°­ŕŹžŕ­°,ŕ­Śáˆť,áˆşŕŹľŕŹžŕ­ľŕ°ŽŕŹžŕ­&#x;,ŕ­Šáˆť,‌‌..,áˆşŕ­ľŕ°­ ି୰,ŕ­Śáˆť,(ଵା୵మ,ŕ­Š):‌‌,‌‌.

],

(4.3.1) provided that h > 0, k > 0, Re(a) > − 1, Re(b) > − 1, U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5). Proof of (4.3.1) To establish (4.3.1), let

๎ ( )ŕ°Žŕąž ( )డ๥

f(x, y) = (sin ) ŕ°­ (1 − y) ŕ°Ž I[

ŕ°Ž

, = ∑+ (y). ,* A ,* cos rx P*

]

(4.3.9) Equation (4.3.9) is valid, since f(x, y) is defined in the region 0 < x < Ď€, − 1 < y < 1. The problems concerning the possibility of expressing a function f(x, y) as double Fourier series expansion are many and cumbersome. However, convergence of

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

407

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

almost all double Fourier series expansions is covered by two-variables analogues of well known Dirichlet's conditions and the Jordan's theorem. In this respect, a brief discussion given by Carslaw and Jaeger7, together with the references indicated in7 provide a good coverage of the subject. Multiplying both sides of (4.3.9) by , (1 − y) (1 + y) P2 (y), integrating with respect to y from − 1 to 1, and using (2.3.4) and (4.2.1), we obtain x , ; , ; , 2. (sin ) . I , : ; , : ; , : ´´ 2

,5 ( ) ‌‌,‌...:6 78 <,:;,‌‌‌,(5797=78 7<,:):‌‌,‌‌. [ ] +|‌...,‌‌:6 78 79,:;,‌‌,( 78 ,:):‌‌,‌‌.

( 2 )

= ∑+ A ,2 ( 2 ) ( 2 ) cos rx . (4.3.10) Multiplying both sides of (4.3.10) by cos(ux), integrating with respect to x from 0 to Ď€, and using (2.3.3) and the orthogonality property of cosine functions; we get A>,<

28 7 a + b + 2v + 1 Γ a + b + v + 1 = Γ a + v + 1 âˆšĎ€

, ; , ; ,

I , : ; 1, 1: ; , : ´´

ாଶషౡ

‌‌,‌...:áˆşŕŹľ/ଶା୵భ,ŕ­Śáˆť,(ଵା୵మି୴,ŕ­Š),‌‌‌,áˆşŕ­ľŕ°­ ,ŕ­Śáˆť,(ଶାŕ­&#x;ାୠା୵మ ା୴,ŕ­Š):‌,‌‌. ] ŕŽ—|‌...,‌‌:áˆşŕ­ľŕ°­ŕŹžŕ­ł,ŕ­Śáˆť,áˆşŕŹľŕŹžŕ­ľŕ°ŽŕŹžŕ­&#x;,ŕ­Šáˆť,‌‌..,áˆşŕ­ľŕ°­ ି୳,ŕ­Śáˆť,(ଵା୵మ,ŕ­Š):‌‌,‌‌.

, (4.3.11) except that A0,v is one-half of the above value. From (4.3.9) and (4.3.11), the formula (4.3.1) is obtained. [

( !)

****(sinθ) I[ ]

=

, ; , ; ,

I √ , : ; , : ; , : ´´

‌‌,‌...:‌‌.,( / ,"):‌‌,‌‌. ] ‌...,‌‌: ," ,‌.:‌‌,‌‌.

[ | +

; ,

∑+ I , ; , √ , : ; , : ; , : ´´

&

&

‌‌,‌...:‌‌,( ,"),( ,"):‌‌,‌‌. ] ," , ," ,‌..:‌‌,‌‌. ‌...,‌‌:

[ |

cos(Ď€r/2) cosrθ, (4.3.2) provided that δ is a positive number and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5). Proof of (4.3.2) To establish (4.3.2), let

( !)

f(θ) = (sinθ) I[ 3?

cosrθ, (4.3.12) Equation (4.3.12) is valid since f(θ) is continuous and of bounded variation in the interval (0, Ď€), when Ď > 0. Multiplying both sides of (4.3.12) by cos (uθ) and integrating with respect to θ from 0 to Ď€, we get =

+

]

∑+ C

( !)

(sinθ) cos (uθ) I[ 3?

cos (uθ) dθ

= cosuθ dθ.

]dθ

+ ∑+ C cos rθ

Now using (2.3.5) and orthogonality property of cosine functions, we have C =

√

, ; , ; ,

cos I , : ; , : ; , : ´´

ಙజํ

ಙడํ

‌‌,‌...:‌‌.., ఎ ,! , ఎ ,! :‌‌,‌‌. ] భజಙ ಙ ‌...,‌‌: ,! , ,! ,‌‌.:‌‌,‌‌.

[ |

ŕ°Ž

(4.3.13)

ŕ°Ž

From (4.3.12) and (4.3.13), the result (4.3.2) is obtained. **** ( !)

(sinθ) I[ =

]

; ,

∑+ I , ; , √ , : ; , : ; , : ´´

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

408

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014) &

&

‌‌,‌...:‌‌.,( ,"),( ,"):‌‌,‌‌. ] ," , ," ,‌‌:‌‌,‌‌. ‌...,‌‌:

[ |

Proof of (4.3.4)

sin(Ď€r/2) sinrθ, (4.3.3) provided that δ is a positive number and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5). Proof of (4.3.3): To prove (4.3.3), let

( !)

f(θ) = (sinθ) I[ = ∑+ C sin rθ,

] (4.3.14)

Multiplying both sides of (4.3.14) by sin (uθ) and integrating with respect to θ from 0 to Ď€, then using (2.3.6) and orthogonality property of sine functions, we obtain , ; , ; ,

C = sin I , : ; , : ; , : ´´ √

ಙజํ ಙడํ ‌‌,‌...:‌‌.., ŕ°Ž ,! , ŕ°Ž ,! :‌‌,‌‌. భజಙ ಙ ,! , ,! ,‌‌.:‌‌,‌‌. ‌...,‌‌: ŕ°Ž ŕ°Ž

[ |

]

(4.3.15)

]

+ = √ ∑+ + ∑ ,*

#% & $& %' ( &( '() ) ! "* ( *!

, ; , ; ,

( ) = ∑+ dx. + A e

Now using (2.3.7) and (4.2.2), we get A = e / �

√

áˆşÎą" áˆť c áˆşÎł' áˆť+ d+

, ; , ; ,

ŕ°Ž ŕ°Ż ŕ°Ż I , ŕ°­ : ; ŕ°Ž ᇲ , ᇲ : ᇲ ; ᇲᇲ , ᇲᇲ : ´´

ྍβ# ྯ r! áˆşÎ´( áˆť+ t!

ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

: ( ) ] F #% &$& %' ( &(

∑+ √ ,* '() ) ! "* ( *!

[

]

" :#( ) ,( ) ] ! ]I[+

, ; , ; ,

(4.3.4) where n's are either even or odd in addition to the conditions of validity followed by (2.3.9).

ŕą&#x;

(4.3.17) From (4.3.16) and (4.3.17), the Fourier exponential series (4.3.4) is obtained. ****

&

],

ŕą&#x;

ಥజఎŕąž๨జఎ๥๪ ಥజఎŕąž๨జఎ๥๪జభ ,-/,. ,-/,‌‌:‌‌,‌‌. ŕ°Ž ŕ°Ž ಥజఎŕąž๨జఎ๥๪¹๣జభ ‌...,‌‌:‌‌..,( ,-):‌‌,‌‌. ŕ°Ž

, ‌‌,‌...:.

[ +|

=

,-/,. ,-/,‌..:‌‌,‌‌. ¹ ‌...,‌‌:‌‌,( ,-):‌‌,‌‌.

, ‌‌,‌...:.

[ +|

( )ఎಓ %ŕąŒ :&( )ŕ°Žŕąž )ŕą‘ :*( )ఎ๥ ]dx $ŕą? ] 'F ([ !ŕą’ ]I[

(sinx)8 F [

&

e ( / ) I , : ; , : ; , : ´´

"F # [

,+,

)ŕą‘ :* ఎ๥ %ŕąŒ :& ŕ°Žŕąž "F # [ $ŕą? ] 'F ([ !ŕą’ ]

( )ఎಓ

I[

The equation (4.3.16) is valid, since f(x) is continuous and of bounded variation in the interval (0, π). Multiplying both sides of (4.3.16) by eimx and integrating with respect to x from 0 to π, we have imx sin x e

From (4.3.14) and (4.3.15), the formula (4.3.3) follows immediately. **** áˆşsinxáˆť

To prove (4.3.4), let f(x) ŕ°Žŕąž ఎ๥ ఎಓ ] = sin x னିଵ ŕ­”F ŕ­•[ ŕŽ‘ŕąŒ:ŕ­Ą(୹୧୏ଡ଼)ŕŽ’ŕą?] ŕ­™F ŕ­š[ ŕŽ“ŕą‘:ŕ­˘(୹୧୏ଡ଼)ŕŽ”ŕą’]I[ா(୹୧୏ଡ଼) ŕŽ— ି୧୬୶ = ∑ஶ A e . (4.3.16) ŕ­Źŕ­€ ିஶ ŕ­Ź

I , : ; , : ; , : ´´

0& 3(

,5 ‌‌.,:‌‌,‌‌. ‌‌,‌...: [ |‌...,‌‌:‌‌..,‌.:‌‌,‌‌. ]

+ +√ ∑+ ∑ ,*

#% & $& %' ( &( '() ) ! "* ( *! &

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409

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

I , : ; , 678/ cos nx : ; , : ´´

, ; , ; ,

where n's are either even or odd in addition to the conditions of validity followed by (2.3.9).

,-/,. ,-/,‌..:‌‌,‌‌. ¹ ‌...,‌‌:‌‌,( ,-):‌‌,‌‌.

, ‌‌,‌...:.

[ +|

],

(4.3.5) where n's are either even or odd in addition to the conditions of validity followed by (2.3.9).

Proof of (4.3.6) To prove (4.3.6), let Sin x F [ : 6 ; ] F [ " :#6 ;! ]

Proof of (4.3.5)

,( )

I[+

To establish (4.3.5), let sin x F [ : ( ) ] F [ " :#( )! ]I[,( ) ] +

=

9?

+ ∑+ B cos nx.

(4.3.18)

Multiplying both sides of (4.3.18) by eimx and integrating with respect to x from 0 to Ď€, and using (2.3.7) and (4.2.3), we get B = e / √

�

,+,

áˆşÎą" áˆť c áˆşÎł' áˆť+ d+

ྍβ# ྯ r! áˆşÎ´( áˆť+ t!

, ; , ; ,

ŕ°Ž ŕ°Ż ŕ°Ż I , ŕ°­ : ; ŕ°Ž ᇲ , ᇲ : ᇲ ; ᇲᇲ , ᇲᇲ : ´´ ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

ಥజఎŕąž๨జఎ๥๪ ಥజఎŕąž๨జఎ๥๪జభ ,-/,. ,-/,‌‌:‌‌,‌‌. ŕ°Ž ŕ°Ž ಥజఎŕąž๨జఎ๥๪¹๣జభ ‌...,‌‌:‌‌..,( ,-):‌‌,‌‌. ŕ°Ž

]

(4.3.19)

From (4.3.18) and (4.3.19), the Fourier cosine series (4.3.5) is obtained. **** (sinx)୵ିଵ ŕ­”F ŕ­•[

=

ா(୹୧୏ଡ଼) ŕŽ‘ :ŕ­Ą(୹୧୏ଡ଼) ŕŽ“ :ŕ­˘(୹୧୏ଡ଼) ] ŕŽ’ ] ŕ­™ F ŕ­š [ ŕŽ” ]I[ŕŽ—

&

, ; , ; ,

I , : ; , e / sin nx : ; , : ´´

√

�

,+,

áˆşÎą" áˆť c áˆşÎł' áˆť+ d+

ྍβ# ྯ r! áˆşÎ´( áˆť+ t!

, ; , ; ,

ŕ°Ž ŕ°Ż ŕ°Ż I , ŕ°­ : ; ŕ°Ž ᇲ , ᇲ : ᇲ ; ᇲᇲ , ᇲᇲ : ´´ ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

ŕą&#x;

ಥజఎŕąž๨జఎ๥๪ ಥజఎŕąž๨జఎ๥๪జభ ,-/,. ,-/,‌‌:‌‌,‌‌. ŕ°Ž ŕ°Ž ಥజఎŕąž๨జఎ๥๪¹๣జభ ‌...,‌‌:‌‌..,( ,-):‌‌,‌‌. ŕ°Ž

, ‌‌,‌...:.

[ +|

],

(4.3.21)

From (4.3.20) and (4.3.21), the Fourier sine series (4.3.6) is obtained. **** sinθ I[

0 !

]

= √ ∑+ I , : ; , : ; , : ´´

, ; , ; ,

sin(2r+1)θ,

ಥజఎŕąž๨జఎ๥๪ ಥజఎŕąž๨జఎ๥๪జభ ,-/,. ,-/,‌..:‌‌,‌‌. ŕ°Ž ŕ°Ž ಥజఎŕąž๨జఎ๥๪¹๤జభ ‌...,‌‌:‌‌,( ,-):‌‌,‌‌. ŕ°Ž

, ‌‌,‌...:.

[ +|

Multiplying both sides of (4.3.20) by eimx and integrating with respect to x from 0 to π, and using (2.3.7) and (4.2.4), we get / C = e

‌‌,‌...: , ,‌..,( , ):‌‌,‌‌. [ |‌...,‌‌: , ,‌..‌‌,( , ):‌‌,‌‌. ]

#% & $& %' ( &( ∑+ ∑+ ,*

√ '() ) ! "* ( *!

(4.3.20)

ŕą&#x;

, ‌‌,‌...:.

[ +|

= ∑@ A C sin nx.

]

],

(4.3.6)

(4.3.7)

provided that h is a positive number, 0 ≤ θ ≤ Ď€ and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

410

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

provided that h is a positive number, 0 ≤ θ ≤ Ď€ and U > 0, > 0, |argΞ| < UĎ€, where U and V are given in (1.3.4) and (1.3.5).

Proof of (4.3.7) To prove (4.3.7), let

0 !

f(θ) = sin θ I[ = ∑+ sin(2r + 1)θ C ,

] (4.3.22)

R(1 − 2u) > 0, 0 ≤ θ ≤ Ď€. The equation (4.3.22) is valid, since f(θ) is continuous and of bounded variation in the interval (0, Ď€) when R(1 − 2u) ≼ 0. Multiplying both sides of (4.3.22) by sin(2v + 1)θ and integrating with respect to θ from 0 to Ď€, we have 0 !

:

; sin θ sin(2v + 1)θ I[

] dθ

<

=∑+ C ; sin(2v + 1)θ sin(2r + 1)θ dθ. Now using (2.3.12) and orthogonality property of the sine functions, we get C2 =

, ; , ; ,

I √ , : ; , : ; , : ´´ ŕ°­

‌‌,‌...: -,/ ,‌‌,(-,/):‌‌,‌‌.

ఎ [ |‌...,‌‌: - .,/ ,‌..,( - .,/):‌‌,‌‌. ],

(4.3.23)

From (4.3.22) and (4.3.23), the formula (4.3.7) is obtained. **** sin θ/2 I[ =

0 (!/ )

]

, ; , ; ,

I √ , : ; , : ; , : ´´

‌‌,‌...: , ,‌..:‌‌,‌‌.

[ |‌...,‌‌:‌.., , :‌‌,‌‌.

]

+ √ ∑+ I , : ; , : ; , : ´´

, ; , ; ,

‌‌,‌‌: , ,‌.., , :‌‌,‌‌. [ |‌‌,‌‌: , ,‌‌, , :‌‌,‌‌. ]

cos rθ, (4.3.8)

Proof of (4.3.8): To prove (4.3.8), let us consider 0 (!/ )

f(θ) = sin θ/2 I[ 3?

]

+ ∑+ C cos rθ,

= R(2u) > 0, 0 ≤ θ ≤ Ď€.

(4.3.24)

Multiplying both sides of (4.3.24) by cos vθ and integrating with respect to θ from 0 to Ď€ with the help of (2.3.13) and orthogonality property of the cosine functions, we have C2 =

, ; , ; ,

I √ , : ; , : ; , : ´´

‌‌,‌...: , ,‌‌,( , ):‌‌,‌‌.

[ |‌...,‌‌: 2, ,‌..,( 2, ):‌‌,‌‌. ],

(4.3.25)

From (4.3.24) and (4.3.25), the formula (4.3.8) follows. REFERENCES: 1. Rainville, E. D.: Special Functions, Macmillan, New York, (1960). 2. Mishra Sadhna: Integrals involving Exponential function, Generalized Hypergeometric series and Fox's HFunction and Fourier series for products of Generalized Hypergeometric functions, J. Indian Acad. Math. Vol. 12, No. 1 (1990). 3. Bajpai, S. D.: Fourier series of generalized hypergeometric functions, Proc. Camb. Phil. Soc. 65 (1969), 703707.

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

411

S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 402-411 (2014)

4. Bajpai, S. D.: Double-Fourier CosineSeries For Fox's H-function, Note di Mathematics Vol. XII-n, 143-147 (1993). 5. Taxak, R. L., Fourier series for Fox's Hfunction, Def. Sci. J., Vol. 21, p. 43-48 January (1971). 6. Sharma, C. K.: On Fourier Series For

Generalized Fox's H-functions and Their Applications, Proceeding of National Science Academy, Vol.5, p. 501-507, (1971). 7. Carslaw, H. S. & Jaeger, J. C.: Conduction of Heat in Solids, (2nd Edn.) Clarendon Press, Oxford, (1986).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

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