Double integrals and half-range Fourier series of H-function

Yashwant Singh Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 9( Version 1), September 2014, pp.27-32
www.ijera.com 27 | P a g e
On Some Double Integrals of H -Function of Two Variables and
Their Applications
Yashwant Singh, Vineeta Malsariya
Department of Mathematics, Government College, Kaladera, Jaipur, Rajasthan, India
Department of Mathematics, J.J.T.University, Chudela, Jhunjhunu, Rajasthan, India
Abstract
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
Key words: H -function of two variables, Half-range Fourier series, H -function, Multiple half-range Fourier
series.
(2000 Mathematics Subject Classification: 33C99)
I. Introduction
The H -function of two variables will be defined and represented by Singh and Mandia [7] in the following
manner:
           
          1 2 2 3 2 1, 1 1, 2 2 1, 2 1, 3 3 1, 3
1 1 2 2 2 2
1, 1 1, 2 2 1, 2 1, 3 3 1, 3
, : , : , , ; , , ; , , , , ; , ,
, : , ; ,
, ; , , , , ; , , , , ;
,
j j j p j j j n j j n p j j j n j j n p
j j j q j j m j j j m q j j m j j j m q
o n m n m n a A c K c e E R e E
x x
y p q p q p q y b B d d L f F f F S
H x y H H
  
  
 
 
 
     
 
=  
1 2
2 1 2 3
1
, ( ) ( )
4 L L
x y d d           

   (1.1)
Where
 
 
   
1
1 1
1
1
1
1 1
1
,
1
n
j j j
j
p q
j j j j j j
j n j
a A
a A b B
  
  
     

  
   

      

 
(1.2)
 
    
    
2 2
2 2
2 2
1 1
2
1 1
1
1
j
j
n m
K
j j j j
j j
p q
L
j j j j
j n j m
c d
c d
   
 
   
 
   
    

    
 
 
(1.3)
 
    
    
3 3
3 3
3 3
1 1
3
1 1
1
1
j
j
n m
R
j j j j
j j
p q
S
j j j j
j n j m
e E f F
e E f F
 
 
 
 
   
    

    
 
 
(1.4)
Where x and y are not equal to zero (real or complex), and an empty product is interpreted as unity
, , , i i i j p q n m are non-negative integers such that 0 , ( 1,2,3; 2,3) i i j j  n  p o  m  q i  j  . All the
1 1 2 2 ( 1,2,..., ), ( 1,2,..., ), ( 1,2,..., ), ( 1,2,..., ), j j j j a j  p b j  q c j  p d j  q
RESEARCH ARTICLE OPEN ACCESS

3 3 ( 1, 2,..., ), ( 1, 2,..., ) j j e j  p f j  q are complex parameters.
2 2 0( 1,2,..., ), 0( 1,2,..., ) j j   j  p   j  q (not all zero simultaneously), similarly
3 3 0( 1,2,..., ), 0( 1,2,..., ) j j E  j  p F  j  q (not all zero simultaneously). The exponents
3 2 2 3 3 3 ( 1,2,..., ), ( 1,..., ), ( 1,2,..., ), ( 1,..., ) j j j j K j  n L j  m  q R j  n S j  m  q can take on non-negative
values.
The contour 1 L is in  -plane and runs from ito i. The poles of   2 ( 1,2,..., ) j j  d   j  m lie
to the right and the poles of     2 1 1 ( 1,2,..., ), 1 ( 1,2,..., )
K j
j j j j j   c   j  n   a    A j  n
to the left of the contour. For 2 ( 1, 2,..., ) j K j  n not an integer, the poles of gamma functions of the
numerator in (1.3) are converted to the branch points.
The contour 2 L is in  -plane and runs from ito i. The poles of   3 ( 1,2,..., ) j j  f  F j  m
lie to the right and the poles of
    3 1 1 ( 1,2,..., ), 1 ( 1,2,..., )
Rj
j j j j j   e  E j  n   a    A j  n
to the left of the contour.
For 3 ( 1, 2,..., ) j R j  n not an integer, the poles of gamma functions of the numerator in (1.4) are converted to
the branch points.
The functions defined in (1.1) is an analytic function of x and y , if
1 2 1 2
1 1 1 1
0
p p q q
j j j j
j j j j
U    
   
     (1.5)
1 3 1 3
1 1 1 1
0
p p q q
j j j j
j j j j
V A E B F
   
     (1.6)
The integral in (1.1) converges under the following set of conditions:
1 1 2 2 2 2 1
1 1 1 1 2 1 1 2 1 1
0
n p m q n p q
j j j j j j j j j
j j n j j m j j n j
    L  K  
         
            (1.7)
1 1 2 2 3 3 1
1 1 1 1 2 1 1 2 1 1
0
n p m q n p q
j j j j j j j j j
j j n j j m j j n j
A A F F S E R E B
         
            (1.8)
1 1
| arg | , | arg |
2 2
x   y   (1.9)
The behavior of the H -function of two variables for small values of | z | follows as:
H[x, y] 0(| x | | y | ),max| x |,| y | 0     (1.10)
Where
1 2
min Re j
j m
j
d

  
  
     
  
1 2
min Re j
j m
j
f
F

 
  
     
  
(1.11)
For large value of | z | ,
    ' ' H[x, y] 0 | x | ,| y | ,min | x |,| y | 0     (1.12)
Where
1 2
1
' max Re j
j
j n
j
c
 K
  
  
    
 
,
1 3
1
' max Re j
j
j n
j
e
R
E

 
  
    
 
(1.13)
Provided that U  0 and V  0.

If we take
2 2 2 3 3 3 1( 1,2,..., ), 1( 1,..., ), 1( 1,2,..., ), 1( 1,..., ) j j j j K  j  n L  j  m  q R  j  n S  j  m  q in
(1.1), the H -function of two variables reduces to H -function of two variables due to [4].
Orthogonality of Sine and Cosine functions:
0;
;
0 2
sin sin m n
m n
mx nxdx




  (1.14)
0;
;
0 2
; 0
cos cos m n
m n
m n
mx nxdx





 
  (1.15)
II. Double Integrals
The following double integrals has been evaluated in this paper
1 1
0 0
(sin x) (sin y) sin rxsinty h(x, y)dxdy
 
     
=
2 2 2 sin sin ( , )
2 2
r t
r t    
       
   
   
(2.1)
1 1
0 0
(sin x) (sin y) sin rx costy h(x, y)dxdy
 
     
=
2 2 2 sin cos ( , )
2 2
r t
r t    
       
   
   
(2.2)
1 1
0 0
(sin x) (sin y) cos rxsinty h(x, y)dxdy
 
     
=
2 2 2 cos sin ( , )
2 2
r t
r t    
       
   
   
(2.3)
1 1
0 0
(sin x) (sin y) cos rxcos ty h(x, y)dxdy
 
     
=
2 2 2 cos cos ( , )
2 2
r t
r t    
       
   
   
(2.4)
Where
         
1 2 2 3 2 2  1,  1,   1,  1,   1, 1 2 2 2 3 3 3
1 1 2 2 2 2 2
1, 1 1, 2 2 1, 2 1, 3 3 1, 3
, : , : , , ; , , ; , , , , ; , ,
(sin )
, : , ; ,
(sin ) , ; , , , , ; , , , , ;
( , )
c j j j j j j j j j j j j j p n n p n n p
d
u x
p q p q p q
v y b B d d L f F f F S
h x y H
  
  
 
 
 
  
 
And
         
1 2 2 3 2 2( )  1,  1, 1
1 1 2 2 2 2
1, 1 1, 2 2 1, 2 1, 3 3 1, 3
, 2: , : , (1 ,2 ;1),(1 ,2 ;1), , ; , , ;
2
2, 4: , ; , 1 1
, ; , , ;1 , , ;1 , , , , ; , , , , ;
2 2
( , )
c d j j j j j j p n
o n m n m n c d a A c K u
p q p q p q v r t
b B c d d d L f F f F S
r t
H
   
 
  


 
  
         
   
   

      2 2 1, 2 1, 3 3 1, 3
, j , j , j , j ; j , j , j n p n n p
c  e E R e E
 
 
 
 
And ( ) stands for the pair of parameters ( ), ( ) .
Also
1 1 1 2
Re( ) 2 min 2 min 0 j j
j m j m
j j
b d
 c c
     
   
          
   
,

1 1 1 2
j m j m
j j
b d
 d c
     
   
          
   
and conditions (1.7), (1.8) and (1.9) are also satisfied.
Proof: If we express the H -function of two variables occurring in the integrand of (2.1) as the Mellin-Barnes
type integral (1.1) and interchange the order of integrations (which is permissible due to the absolute
convergence of the integrals involved in the process), we get
L.H.S. of (2.1)
 
1 2
2 1 2 3
2 1 2 1
0 0
1
, ( ) ( )
4
(sin ) sin (sin ) sin
L L
c d
u v
x rxdx y tydy d d
 
 
   
      

     
 
 
 
 
 
 
Now, applying the result ([5],p.70, eq. (3.1.5)) and equation (1.1), the result (2.1) follows at once.
The remaining integrals can be evaluated similarly.
III. Double Half-Range Fourier Series
The following double half-range Fourier series will be proved:
4
1 1
( , ) 2 sin sin ( , ) sin sin
2 2 m n
m n
f x y m n mx ny    

 
 
 
   
    
   
 (3.1)
4
1 0
( , ) 2 sin cos ( , ) sin cos
2 2 m n
m n
f x y m n mx ny    

 
 
 
   
    
   
 (3.2)
4
0 1
( , ) 2 cos sin ( , ) cos sin
2 2 m n
m n
f x y m n mx ny    

 
 
 
   
    
   
 (3.3)
4
1 1
( , ) 2 cos cos ( , ) cos cos
2 2 m n
m n
f x y m n mx ny    

 
 
 
   
    
   
 (3.4)
Where
1 1 f (x, y) (sin x) (sin y) h(x, y)      and provided that
1 1 1 2
j m j m
j j
b d
 c c
     
   
          
   
,
1 1 1 2
j m j m
j j
b d
 d c
     
   
          
   
and
conditions (1.7), (1.8) and (1.9) are also satisfied.
Proof: To prove (3.1), let
1 1
,
1 1
( , ) (sin ) (sin ) ( , ) sin sin m n
m n
f x y x y h x y A mx ny  
 
 
 
  (3.5)
Which is valid since f (x, y) is continuous and of bounded variation in the open interval (0, ) .
Multiplying both sides of (3.5) by sin rxsinty and integrating from 0 to  with respect to both x and y , it
is seen that
1 1
0 0
(sin x) (sin y) h(x, y)sin rxsin tydxdy
 
      
,
1 1 0 0
sin sin sin sin m n
m n
A mx ny rx tydxdy
   
 
   (3.6)
Now using (2.1), and orthogonal property of sine functions, it follows that
4
, 2 sin sin ( , )
2 2 r t
r t
A r t    
      
    
   
(3.7)
Substituting the value of m,n A from (3.7) to (3.5), the result (3.1) follows at once.
To establish (3.2), put

,
1 0
( , ) sin cos m n
m n
f x y B mx ny
 
 
  (3.8)
Multiplying both sides of (3.8) by sin rxcosty and integrating from 0 to  with respect to both x and y and
using (2.2) and orthogonal properties of sine and cosine functions, we find that
4
, 2 sin cos ( , )
2 2 r t
r t
B r t    
      
    
   
(3.9)
Except that r ,0 B is one-half of the above value.
From (3.8) and (3.9), the series (3.2) follows easily. The result (3.3) can be established in a manner similar to
above.
To establish (3.4), let
,
1 0
( , ) cos cos m n
m n
f x y D mx ny
 
 
  (3.10)
Multiplying both sides of (3.8) by cos rxcosty and integrating from 0 to  with respect to both x and y and
using (2.2) and orthogonal properties of cosine functions, we obtain
4
, 2 cos cos ( , )
2 2 r t
r t
D r t    
      
    
   
(3.11)
Except that 0, ,0 ,t r D D are one-half and 0,0 D is quarter of the above values.
The Fourier series (3.4) now follows from (3.10) and (3.11).
IV. Multiple Integrals
The following multiple integral analogous to (2.1) can be derived easily on following the procedure as
given $2 and taking the help of ([5],p.70,eq.(3.1.5):
1 1 1
1 1 1 1 1
0 0
... (sin ) (sin ) n sin ...sin ( ,..., ) ...
n n n n n x x r x r x h x x dx dx
 
     
= 1 ... 1
1 2 sin ...sin ( ,..., )
2 2
n n n n
n
r r
r r    
        
   
   
(4.1)
Where
1 1 1 2
i
j m j m
j j
b d
 c c
     
   
          
   
; i 1,2,...,n and conditions (1.7), (1.8) and (1.9) are
also hold, and
         
         
2 1
1
1 2 2 3 2 1, 1, 1, 1, 1, 1 2 2 2 3 3 3
1 1 2 2 2 2 2
1, 1 1, 2 2 1, 2 1, 3 3 1, 3
(sin )
, : , : , , ; , , ; , , , , ; , ,
1 , : , ; , (sin ) , ; , , , , ; , , , , ;
( ,..., )
c
j j j p j j j n j j n p j j j n j j n p
cn
n j j j q j j m j j j m q j j m j j j m q
uv x
n p q p q p q x b B d d L f F f F S
h x x H
  
  
 
 



 
 

And
         
1 2 2 3 2 2( 1 ... ) 1,  1,  1, 1 2
1 1 2 2 2 2
1, 1 1, 2 2 1, 2 1, 3 3 1, 3
1,
1
, 2: , : , (1 ,2 ;1) , , ; , , ; ,
2
2, 4: , ; , 1
, ; , , ;1 , , , , ; , , , , ;
2
( ,..., )
c cn j j n j j j j j j p n
j j
j j j q j j j m j j j m q j j m j j j m q
n
n
o n m n m n c a A c K c
u
p q p q p q v r
b B c d d L f F f F S
r r
H
  

  

 
 
 
     
 
 

      2 1, 2 1, 3 3 1, 3
j , j , j , j ; j , j , j n p n n p
 e E R e E
 
 
 
 
 
For j 1,2,...,n.
Multiple integrals analogous to (2.2) to (2.4) can also be written easily.
V. Multiple Half-Range Fourier Series
The following multiple half-range Fourier series analogous to (2.1) can be derived on following lines as
given in $3 [5], using the integral (4.1) and the multiple orthogonal property of sine functions:

1
1
2 ... 1
1
1 1
( ,..., ) 2 ... sin ...sin
2 2
n
n
n n
n
m m
m m
f x x      
  
 
   
    
   
 
  1 1 1 ,..., sin( )...sin( ) n n n  m m m x m x
Where
1 1 1 2
i
j m j m
j j
b d
 c c
     
   
          
   
; i 1,2,...,n and conditions (1.7), (1.8) and (1.9) are
also satisfied , and 1 1 1
1 1 1 ( ,..., ) (sin ) ...(sin ) n ( ,..., )
n n n f x x x x h x x      Similarly the multiple half-range
Fourier series analogous to (2.2), (2.3) and (2.4) can also be solved.
References
[1] Bajpai, S.D., Double and Multiple half-range Fourier series of Meijer’s G -function, Acta
Mathematica Vietnamica 16 (1991), 27-37.
[2] Buschman, R.G. and Srivastava , H.M. ,The H  function associated with a certain class of Feynman
integrals, J.Phys.A:Math. Gen. 23, (1990),4707-4710.
[3] Fox,C., A formal solution of certain dual integral equations, Trans. Amer. Math. Soc. 119, (1965),
389- 395.
[4] Inayat-Hussain, A.A., New properties of hypergeometric series derivable from Feynman integrals :II A
generalization of the H -function, J. Phys. A. Math. Gen. 20 (1987).
[5] Mathai, A.M. and Saxena, R.K.; Generalized Hypergeometric Functions With Applications in Statistics
and Physical Sciences, Springer-Verlag, Berlin, 1973.
[6] Rathi, A.K., A new generalization of generalized hypergeometric functions. Le Mathematiche Fasc.
II52: (1997),297-310.
[7] Singh,Y. and Mandia, H., A study of H -function of two variables, International Journal of
Innovative research in science, engineering and technology,Vol.2,(9),(2013),4914-4921.

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