Int. Res. Adv. 2018, 5(2), 46-51
. Article .
INTEGRATED
RESEARCH ADVANCES Certain results on entire functions defined by bicomplex Dirichlet series Jogendra Kumar1* Government Degree College Raza Nagar Swar, Rampur, Uttar Pradesh –244924, INDIA Received on: 10-JUNE-2018, Accepted and Published on:20-SEPT-2018
ABSTRACT
In this work, we have introduced and studied the Bicomplex version of Complex Dirichlet Series
f (s ) a n e n s
. We
n 1
have derived condition for which the sum function of the Bicomplex Dirichlet Series f ()
n en
represents an
n 1
entire function. The Entireness of sum and Hadamard product of two Entire Bicomplex Dirichlet Series are also discussed..
Keywords: Dirichlet Series, Entire Dirichlet Series, Riemann Zeta Function, Hadamard Product
idempotent elements in C2 , denoted as
INTRODUCTION
Throughout this paper, the set of Bicomplex numbers is denoted by
C2 and
the sets of complex and real numbers are
C1
C
denoted by and 0 , respectively. For details of the theory of bicomplex numbers.1-3 The set of Bicomplex Numbers defined as:
C 2 = {x1 + i1 x 2 + i 2 x 3 + i1i 2 x 4 : x1 , x 2 , x 3 , x 4 ∈ C0 , i1
≠
i 2 and i12 = i 22 = -1, i1 i 2 = i 2 i1}
We shall use the notations following sets:
C ( i1 )
and
C ( i2 )
for the
C ( i1 ) {u i1v : u, v C 0 } C ( i 2 ) { i 2 : , C0 } 1.1 IDEMPOTENT ELEMENTS:
Besides 0 and 1, there are exactly two non – trivial
and
e2
and
1 i1i 2 1 i1i 2 and e 2 2 2 e1 e2 1 and e1e2 e2e1 0.
defined as e1 Note that
e1
1.2 CARTESIAN IDEMPOTENT SET:
Cartesian idempotent set X determined by denoted as X=
X 1 e X 2
X1 and X 2
is
and is defined as
{
X 1 × e X 2 = ∈ C 2 : = 1 e 1 + 2 e 2 , ( 1 , 2 )∈ X 1 × X 2
C 2 = C(i1 ) ×e C(i1 ) = C(i1 ) e1 + C(i1 )e 2 = { ∈ C 2 : = 1 e1 + 2 e 2 , (1 , 2 ) ∈ C(i1 ) × C(i1 )} C 2 = C(i 2 ) × e C(i 2 ) = C(i 2 ) e1 + C(i 2 )e 2 = { ∈ C 2 : = 1 e1 + 2 e 2 , (1 , 2 ) ∈ C(i 2 ) × C(i 2 )} 1.3 IDEMPOTENT REPRESENTATION OF BICOMPLEX NUMBERS
Dr. Jogendra Kumar, Assistant Professor, Government Degree College Raza Nagar Swar, Rampur, Uttar Pradesh –244924, INDIA Email:
[email protected] Cite as: Int. Res. Adv., 2018, 5(2), 46-51.
(I) C(i1 ) -idempotent representation of Bicomplex Number is given by
= z 1 + i 2 z 2 = ( z 1 - i 1 z 2 ) e1 + ( z 1 + i 1 z 2 ) e 2 = 1 e1 + 2 e 2
©IS Publications ISSN 2456-334X http://pubs.iscience.in/ira Integrated Research Advances
}
Int. Res. Adv., 2018, 5(2), 46-51
46
Kumar (II) C( i 2 ) -idempotent Number is given by
representation
of
Bicomplex
= ( x 1 + i 2 x 3 ) + i1 (x 2 + i 2 x 4 ) = w 1 + i1 w 2
the Dirichlet series
= ( w 1 - i 2 w 2 )e1 + ( w 1 + i 2 w 2 )e 2 = 1 e1 + 2 e 2
convergent for
Note 1.1: Out of the two idempotent representation, we use
12
z1 z 2 =
x
2 1
2
1 2 2 2 12 = 2
σ 0 Re (s) of conditional
divergence, the region convergence and the line
Re (s) σ 0 ).
is called the abscissa of absolute
The quantity
f (s) a n e n s
and the line
n 1
Re( s) is called the line of absolute convergence.
1/ 2
1.5 ENTIRENESS OF COMPLEX DIRICHLET SERIES:
C 2 becomes a modified Banach algebra, in the sense that
. 2
is absolutely
Re(s) , and not absolutely convergent for Re(s) (this region comprise the region Re (s) σ 0 of
convergence of the series
x 22 x 32 x 24
f (s) a n e n s n 1
C(i1 ) -idempotent representation. All the results also proved with the help of C(i 2 ) -idempotent representation technique. The norm in C 2 is defined as 2
such that
To every Dirichlet series, there exists a number
THEOREM 1.1:4
… (1.1) For the complex Dirichlet Series
a n e
ns
n 1
1.4 COMPLEX DIRICHLET SERIES:4-6
A
Dirichlet
series
If
is
a
series
of
the
form
f (s) a n e ns
where
{λ n } is a strictly monotonically
n 1
increasing and unbounded sequence of positive real numbers, and s i t is a complex variable.
{λ n } of exponent is to be emphasized, such a series is called a complex Dirichlet series of type λ n .4 When the sequence
A Dirichlet series of the type n is a power series in
given by
f (s) a n e
ns
____ log a n n , Then σ 0 σ lim lim n λn
____
n 1
Corollary 1.1: For a Dirichlet Series f (s)
σ 0 σ lim
A Dirichlet series of type
log n is the Generalized
Riemann Zeta function is given by
f (s) a n n
s
that the Dirichlet series
f (s) a n e
ns
Hence,
σ 0 σ lim
Corollary
1.2:
log a n n
The
f (s) a n e n s
Complex
Dirichlet
Series
represents an Entire function iff
n 1 1
an
n
0 as n .
Proof :
converges for
For entireness of
n 1
σ 0 is called the abscissa of convergence of the series, and the Re(s) σ 0 is called the line of convergence.
f (s) a n e n s n 1
lim
log a n n
lim log a n an
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n n lim 1 n n
Re (s) σ 0 and diverges for Re (s) σ 0 . The number line
n
n n lim
n 1
Abscissae of convergence and absolute convergence: To every Dirichlet series, there exists a number σ 0 such
log a n
Proof:
n 1
a n e n s n 1
e s is
a n (e s ) n
.
1
n
lim 1
n
log a n n
lim a n
1
n
0
0 as n
Int. Res. Adv., 2018, 5(2), 46-51
47
Kumar
f (s) a n e n s represents an Entire function if
Hence
n 1
an
1
n
1
n 2 n (1n ) 2 ( 2 n ) 2 ( 3n ) 2 ( 4 n ) 2
1n 4 n 2 n 3n Also
0 as n .
, 2
2
2
n (1n ) ( 2 n ) ( 3n ) ( 4 n ) 2. BICOMPLEX DIRICHLET SERIES:
In this paper we discuss a Bicomplex Dirichlet Series of type n, which is a Bicomplex Power Series in
f ( ) n e
n 1
where { α n } is a sequence of bicomplex numbers and a bicomplex variable. Note that,
1
ξ is
n 1
2
n e n 1 n e n e1 2 n e n n 1
n 1
e2
2
n iff 1n 4 n 2 n 3n
1 1 lim
As, n
n (e )
n 1
n 1 n
e
n
Hence,
2 2 lim Hence,
2
log 2 n
log 1 n
and
n [cf. Cor. 1.1]
n
1 1 2 2 lim
log n n
THEOREM 2.2:
1
1
2
The Bicomplex Dirichlet series
2
f () f ( ) e1 f ( ) e 2
f ( ) n e
Where,
n
is a Bicomplex Dirichlet
f ( ) 1ne n
1
,
2
f ( 2 ) 2 n e n
2
are Complex Dirichlet Series.
Throughout, We denote the abscissae of convergence of the
1
n en
1
and
n1
by
σ1
and
2
2 n en n 1
σ 2 and abscissae of their absolute convergence by
2 2 lim
n 1 , 2
( n) k α n e n ξ . n 1
T HEOREM 2.1:
The,
For the Bicomplex dirichlet Series
f ( ) n e n n 1
1 1 2 2 lim
k
log n
n
lim
1n 4 n 2 n 3n
lim
n 1n i1 2 n i 2 3n i1i 2 4 n n 1 n e1 2 n e 2 1
n (1n 4 n ) i1 ( 2 n 3n )
and
log n 1 n
1 1 lim
log n
Proof:
2
and
n 2 log n
Dirichlet series
Where,
log 1 n
Let 1 , 2 and are the associated abscissae of convergence and absolute convergence of the Bicomplex
1 and 2 , respectively.
if ,
have the same
region of convergence.
1 1 lim
n 1
Complex Dirichlet series
( n)k αn e n ξ
derivative defined by
Proof:
n1
n1
1 1
and
and the kth
n 1
n 1
Series
αn enξ
k
n log n 1
n k log n log 1 n n
log 1 n log n k lim lim n n
n (1n 4 n ) i1 ( 2 n 3n )
Integrated Research Advances
Int. Res. Adv., 2018, 5(2), 46-51
48
Kumar
1
log 1 n log 1 n log n k lim lim 0 lim n n n
lim
Similarly, 2
log
log 1 n
2 2 lim
2 2 lim
log n k 2 n
lim
n
log 2 n
αn e
nξ
and the
α
( n)n k
e n ξ obtained after
n 1
k–times term-by-term integration of
αn e
nξ
have the
n 1
1 n
n
n 1
0
n
2 2 lim
e n ξ .
log 1 1 lim 1
lim
lim
n
0
n ,
as
where
1 n
n
m N , such that
2
2
k lim
n k
n
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1n 2
n
2
2
n
2
2n
1 2
n
2
2
2 2 n 1 n 2 2 n and
1
n
1 n
n
log n n
log 1 n log n k lim lim 0 n n
log 1 n
2
1 2 n n 2
n
Conversely 2
n
n n
n m ,
Now
1
n log 1 n
0
as
0 as n
n
α
n
0
n m
n 1
log n
1 n
n
Given
and
Let 1 , 2 and 1 , 2 are the associated abscissae of convergence and absolute convergence of the Bicomplex
( n)n k
1 n
n
1 n
Proof:
Proof:
n 2 log n
2
1
if and only if
n 1 n e1 2 n e 2 .
Let
1 1 lim
n
as
and
same region of convergence.
log 1 n
is
THEOREM 2.4:
Bicomplex Dirichlet series
n
C 2 –space.
convergent in the entire
The Bicomplex Dirichlet series
f n e
said to be an entire Bicomplex Dirichlet Series if it is
T HEOREM 2.3:
lim
2 .
n
DEFINITION 2.1:
The Bicomplex Dirichlet series
n
2 .
Dirichlet series
lim
n
log 2 n
ENTIRE BICOMPLEX DIRICHLET SERIES
Similarly,
Then,
n k
n
1
n
1 n
0
i.e. Given Such that
1
let as
n
0 1 n
2
and
0
n
1 n
0
n
as
and
n
0 m1 , m 2 N 1
n
1 n
n m1
and
2
n
1 n
n m2
Int. Res. Adv., 2018, 5(2), 46-51
49
Kumar Proof:
m max (m1 , m 2 )
Let Then
1
n m , n
1 n n
2
1 n
and
n
1 n
n
i.e. given
0 as n
f n e
The Bicomplex Dirichlet series
n
f n e
n 1
and
2
f
2
2
n e
n 2
2.1:
The
Bicomplex
Dirichlet
1
and
1 n
n n
2.2:
series
f n e n if and only if
1
n
1 n
Dirichlet
series
is an entire Bicomplex Dirichlet series
0
and
n
2.3:
f n e n iff
n
1 n
The
Bicomplex
n
1 n
and n
1 n
n
1 n
n
1 2 (2) n
() ()
1
1 n
(2) 2 n 2 n n
1 n
0 as
1 n
is an entire Bicomplex
Dirichlet Series. THEOREM 2.7:
0. Dirichlet
h () α n n e n ξ n 1
If Corollary
Bicomplex
2
1 n
2
n n n
2 .
The
1 n
1
Hence Corollary
n
and
( 2) 2 n 2
f n e n is an entire Bicomplex Dirichlet series if and only if
1 n
n n 2 n
Now,
are entire Complex Dirichlet series. Corollary
n m1
n m , n
is an
entire Bicomplex Dirichlet series if and only if both 1
1 n
n
n m2 Let m max ( m1 , m 2 )
T HEOREM 2.5:
1
0 as
0 m1 , m 2 N
Such that
Now
1
1 n
0 as n and n
2
2n 2n 2 2n 1 n
are two
n
2
n
and
n 1
1 n
n
2 n 2 1n 2n
2 n
g ( ) n e n ξ
Entire Bicomplex Dirichlet Series
2
nξ
n 1
1 2 2
2
2
2
f ( ) α n e
series
is an entire Bicomplex Dirichlet series
f () αn e n ξ
and
g ( ) n e n ξ
n 1
entire
be two
n 1
Bicomplex
Dirichlet
series,
then
the
series
α n n e n ξ
0.
is also an entire Bicomplex Dirichlet
n 1
series. T HEOREM 2.6: THEOREM 2.8:
Let
h () α n n e
nξ
If
n 1
of
f () αn e n ξ
and
g() n e n ξ . If f and g are
n1
n1
entire Bicomplex Dirichlet series, then h is also an entire Bicomplex Dirichlet series.
Integrated Research Advances
be the Hadamard product
α n e n ξ
is an Entire Bicomplex Dirichlet Series,
n 1
then kth derivative defined by
( n)k α n e n ξ
is also
n 1
an Entire Bicomplex Dirichlet Series.
Int. Res. Adv., 2018, 5(2), 46-51
50
Kumar Proof:
f ( ) α n e n ξ
is an Entire Bicomplex Dirichlet
Now,
n 1
1 n
αn
( n) k
Series i.e.
1 n
n
1 n
( n) k α n
Now, k
( n) α n n
αn ( n) k
0 as n
1 n
0
as
k (n) n
1 αn n
k (n) n
n
0
as
If the Bicomplex Dirichlet series
α n e n ξ is an entire n 1
Bicomplex Dirichlet series then the Bicomplex Dirichlet series
n 1
αn e n ξ obtained k ( n)
after
k–times
term-by-term
integration of
α n e n ξ
is also an entire Bicomplex
n 1
Dirichlet series. Proof:
Let
f () α n e n ξ
series i.e.
n
n 1
1 n
1 k n ( n)
αn
1 n
k (n) n
k
0
as
n (n ) n 0
αn e nξ k ( n)
as n
is an entire Bicomplex Dirichlet
series. ACKNOWLEDGMENTS
T HEOREM 2.9:
1 n
αn
n 1 1 n
is an entire Bicomplex Dirichlet
0 as n
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I am heartily thankful to Mr. Sukhdev Singh, Assistant Professor-Mathematics, Lovely Professional University, Punjab and Dr. Mamta Nigam, Assistant ProfessorMathematics,University of Delhi for their encouragement and support during the prepration of this paper. REFERENCES AND NOTES 1. M.E. Luna-Elizarrarás, M. Shapiro, D. C. Struppa, A. Vajiac, Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers, Springer International Publishing, (2015) 2. G. B. Price, An int. to multicomplex space and Functions, Marcel Dekker (1991). 3. Rajiv K. Srivastava,Certain Topological aspects of Bicomplex Space, Bull. Pure and Appl. Math., 2(2),(2008), 222–234. 4. G. H. Hardy and M. Riesz, The General Theory of Dirichlet Series, Cambridge Univ. Press (1915). 5. E. C. Titchmarsh, The Theory of functions, Oxford University press (1960). 6. S. Mandelbrojt, Dirichlet series: Principles and Methods, Dordrecht Holland (1969).
.
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