** Research article **

** Available online www.ijsrr.org **

** ISSN: 2279–0543 **

**International Journal of Scientific Research and Reviews **

**International Journal of Scientific Research and Reviews**

**An Integral Representation of Bicomplex Dirichlet Series **

**Jogendra Kumar **

Dept. of Mathematics, Govt. Degree College, Raza Nagar, Swar(Rampur) –244924, India

**ABSTRACT **

In this paper, we have defined the **Bicomplex Dirichlet Series**

1

n n

n

e )

(

f and investigate its

region of convergence. We have also obtained an integral representation of **Bicomplex Dirichlet Series **

1

n n

n

e )

(

f .

**KEYWORDS:**

Bicomplex numbers, Bicomplex Gamma Function, Bicomplex Riemann Zeta Function,
Complex Dirichlet Series, Euler Product

**2010 AMS SUBJECT CLASSIFICATION:**

11F66, 30G35, 32A30, 32A10, 13A18.
***Corresponding author **

**Dr. Jogendra Kumar **

Department of Mathematics,

Govt. Degree College,

Raza Nagar, Swar, Rampur(UP) – 244924, INDIA.

**1. INTRODUCTION **

The set of Bicomplex Numbers defined as:

} i i = i i , 1 -= i = i and i ≠ i , C ∈ x , x , x , x : x i i + x i + x i + x { =

C_{2} _{1} _{1} _{2} _{2} _{3} _{1} _{2} _{4} _{1} _{2} _{3} _{4} _{0} _{1} _{2} _{1}2 _{2}2 _{1} _{2} _{2} _{1}

Throughout this paper, the sets of complex and real numbers are denoted by

### C

_{1}and

### C

_{0}, respectively. For details of the theory of Bicomplex numbers, we refer to

**1, 2, 3, 4**. We shall use the notations

### C

### (

### i

_{1}

### )

and### C

### (

### i

_{2}

### )

for the following sets:

### C

### (

### i

_{1}

### )

### =

### {

### u

### +

### i

_{1}

### v

### :

### u

### ,

### v

### ∈

### C

_{0}

### }

;### C

### (

### i

_{2}

### )

### =

### {

###

### +

### i

_{2}

###

### :

###

### ,

###

### ∈

### C

_{0}

### }

**1.1 Idempotent Elements: **

**1.1 Idempotent Elements:**

Besides 0 and 1, there are exactly two non – trivial idempotent elements inC , denoted as 2 e and 1 e and 2

defined as 2 i i + 1 =

e1 1 2 and

2 i i -1 =

e2 1 2 .

Note that

### e

_{1}

### +

### e

_{2}

### =

### 1

and### e

_{1}

### e

_{2}

### =

### e

_{2}

### e

_{1}

### =

### 0

### .

**1.2 Cartesian Idempotent Set: **

**1.2 Cartesian Idempotent Set:**

### )}

### i

### (

### C

### ×

### )

### i

### (

### C

### ∈

### )

### ,

### (

### ,

### e

### +

### e

### =

### :

### C

### ∈

### {

### =

### e

### )

### i

### (

### C

### +

### e

### )

### i

### (

### C

### =

### )

### i

### (

### C

### ×

### )

### i

### (

### C

### =

### C

_{2}

_{1}

_{e}

_{1}

_{1}

_{1}

_{1}

_{2}

###

_{2}

###

1###

_{1}2

###

_{2}1

###

2###

_{1}

_{1}

### )}

### i

### (

### C

### ×

### )

### i

### (

### C

### ∈

### )

### ,

### (

### ,

### e

### +

### e

### =

### :

### C

### ∈

### {

### =

### e

### )

### i

### (

### C

### +

### e

### )

### i

### (

### C

### =

### )

### i

### (

### C

### ×

### )

### i

### (

### C

### =

### C

_{2}

_{2}

_{e}

_{2}

_{2}

_{1}

_{2}

_{2}

###

_{2}

###

###

_{1}

_{1}

###

_{2}

_{2}

###

_{1}

###

_{2}

_{2}

_{2}

**1.3 Idempotent Representation Of Bicomplex Numbers:**

**1.3 Idempotent Representation Of Bicomplex Numbers:**

**(I**)

### C

### (

### i

_{1}

### )

### -

_{ idempotent representation of Bicomplex Number Throughout this paper }

### C

### (

### i

_{1}

### )

-idempotentrepresentation of Bicomplex Number is given by

2 2 1 1 2 3 2 1 4 1 1 3 2 1 4 1 2 2 1 1 1 2 1 1 2 2 1 4 1 3 2 2 1 1 e + e = e )] x + x ( i + ) x -x [( + e )] x -x ( i + ) x + x [( = e ) z i + z ( + e ) z i -z ( = z i + z = ) x i + x ( i + ) x i + x ( =

**(II**_{) }

### C

### (

### i

_{2}

### )

### -

idempotent representation of Bicomplex Number Throughout this paper### C

### (

### i

_{2}

### )

-idempotent representation of Bicomplex Number is given by2 2 1 1 2 3 2 2 4 1 1 3 2 2 4 1 2 2 2 1 1 2 2 1 2 1 1 4 2 2 1 3 2 1 e + e = e ] ) x + x ( i + ) x -x [( + e )] x -x ( i -) x + x ( [ = e ) w i + w ( + e ) w i -w ( = w i + w = ) x i + x ( i + ) x i + x ( =

**1.4 Singular Elements: **

**1.4 Singular Elements:**

Non zero singular elements exist inC_{2}. In fact, a Bicomplex number =z_{1}+z_{2}i_{2} is singular if and
only if z_{1}2+z2_{2} =0. Set of all singular elements in C2 is denoted as O . 2

**1.5 Norm:**

**1.5 Norm:**

The normin C_{2} is defined as

###

_{2}

###

122
2
1 z
z
**=**
1 2
2 2
1 2
2
_{ }

** = **

###

2###

1/24 2 3 2 2 2

1 x x x

2

C becomes a modified Banach algebra, in the sense that ,C_{2}, we have, in general,
. 2

**1.6 Complex Dirichlet Series **

**1.6 Complex Dirichlet Series**

**5, 6, 7****: **

**:**

In general, a Dirichlet series is a series of the form

- s

∞

1 = n n

n

e a = ) s (

f

### ∑

………**(1.1)**

where {λ_{n}} is a monotonically increasing and unbounded sequence of real numbers, and s=+it is a

complex variable. When the sequence {λ_{n}} of exponent is to be emphasized, such a series is called a

**Complex Dirichlet series of type **λ_{n}.

If λn =n, then f(s) is a power series in s -e =

z . If λn =logn, then

-s

∞

1 = n n

n a = ) s (

f

### ∑

………**(1.2)**

is called an **Ordinary complex Dirichlet series**.

**2. BICOMPLEX DIRICHLET SERIES: **

The **Bicomplex Dirichlet series** is defined as

** **

### ∑

∞1 = n

-n

n

e =

) (

f

** **...** (2.1)**

where {α_{n}} is a sequence of bicomplex numbers, {λ_{n}} is a strictly monotonically increasing and

unbounded sequence of positive real numbers and ξ∈C_{2} is a bicomplex variable. If λ_{n} =n, then

### ∑

∞1 = n

n -n(e ) =

) (

f is a **power series** in

### e

-. If λ_{n}=logn, then

ξ

-∞

1 = n nn

α

= )

ξ

f(

### ∑

...** (2.2) **
is a **Ordinary Bicomplex Dirichlet Series**.

If α_{n} =1in equation **(3.2)** -ξ

∞

1 = n

n = )

ξ

f(

### ∑

represent**Bicomplex Riemann Zeta Function8, 9, 10, 11**in

that consequence we named -ξ

∞

1 = n n

n

α

= )

ξ

f(

### ∑

a**Generalized Bicomplex Riemann Zeta Function12, 13, 14**.

Note that,

2 n

2 1 n

1

ne ( e )e ( e )e

2 n 1

n

n

1

n n

n

e 2

1 n

n 2 1

1 n

n 1

e e

e

e n1 n2

Now we denote the sum function of the series

1

n n

n

e ,

1

n n

1 _{e} n1 _{ and } _{ }

1

n n

2 _{e} n2 _{ by }

)

ξ

f( ,1f(1ξ)and2f(2ξ)respectively.

Thus f(ξ)1f(1ξ)e_{1}2f(2ξ)e_{2}

Then

1

n n

n e )

ξ

f( is a Bicomplex Dirichlet series and

1

n n

1 1

1_{f}_{(} _{)} _{e} n1 _{ , } _{} _{}_{ }

1

n n

2 2

2 _{n}2

e )

(

f are

Complex Dirichlet series. Throughout, we denote the abscissae of convergence of

1

n n

1 1

1 _{n}1

e )

(

f and

1

n n

2 2

2 _{n}2

e )

(

f by σ_{1} and σ_{2}, and the abscissae of absolute convergence by _{1} and _{2},

respectively.

**THEOREM 2.1**

### :

A Bicomplex Dirichlet series
_{e} n

1

n n

converges for ξξ_{0} iff

1 n

e

1

n n

1

converges for 1ξ1ξ_{0} and

2 n

e

1

n n

2

converges for 2ξ2ξ_{0}.

**THEOREM 2.2:**

### If

_{)} _{e} n

( f

1

n n

converges for ξξ_{0} then

_{e} n

1

n n

converges in the region

} ) Re( ) Re( and ) Re( ) Re( : C

{ _{2} 1 1_{0} 2 2_{0}

{ C :x x x x andx x x x }

0 4 0 1 4 1 0 4 0 1 4 1

2

or equivalently in the region

}. ) z Re( ) z Re( ) z Im( ) z Im( and ) z Re( ) z Re( : C

{ _{2} _{1} _{1}0 _{2} 0 _{1} _{1}0

2

**COROLLARY 2.1: **

If
_{e} n

1

n n

diverges for ξξ_{0} then

_{e} n

1

n n

diverges in the region

} ) Re( ) Re( and ) Re( ) Re( : C

{ _{2} 1 1_{0} 2 2_{0}

} x x x x and x x x x : C

{ _{2} _{1} _{4} _{1}0 0_{4} _{1} _{4} _{1}0 0_{4}

or equivalently in the region

}. ) z Re( ) z Re( ) z Im( ) z Im( and ) z Re( ) z Re( : C

{ _{2} _{1} 0_{1} _{2} 0 _{1} _{1}0

2

**THEOREM 2.3: **

The Bicomplex Dirichlet series

1

n n

n

e )

(

f converges in the region

}

σ

)

ξ

( Re and

σ

)

ξ

( Re : C

ξ

{

**3. AN INTEGRAL REPRESENTATION OF BICOMPLEX DIRICHLET SERIES: **

**DEFINITION 3.1: **

Let

###

a,b###

be an interval in C_{0}. A curve C in C is a mapping

_{2}ζ:

###

a,b C_{2}. The trace of C is the set

###

###

ζ(t)C_{2}:ta,b

###

.**THEOREM 3.1**

**15**

** :**

Let :XC_{2}be a continuous function, and let

###

be a curve defined by mappingX ] b , a [

:

. If has continuous derivative ':[a,b]C_{2}, then

( (t))d (t) [ (t)] '(t)dt

b

a

**BICOMPLEX INTEGRALS AND THE IDEMPOTENT REPRESENTATION: **

Let X be domain in C and let _{2} f : XC_{2}, f(ζ)1f(1ζ)e_{1}2f(2ζ)e_{2} be a holomorphic function.

Let be a curve ζ(t)z_{1}(t)i_{2}z_{2}(t), atb whose trace is in X, so that ζ(t)1(t)e_{1}2(t)e_{2}, shows

that there are curves _{1} and _{2}, with traces in X and _{1} X respectively, such that _{2}

1:1ζ1ζ(t) atb

_{2}:2ζ2ζ(t) atb

**THEOREM 3.2 **

**1**

**: **

Under the above mentioned notations and hypothesis, integrals of f, f and _{1}f exists 2

on curves , _{1} and _{2} respectively and

f( )d 1f(1ζ)d(1ζ) e_{1} 2f(2ζ)d(2ζ) e_{2}
2

1

.

**DEFINITION 3.2 **

**15**

**: **

Let 1e_{1}2e_{2}C_{2}, p p_{1}e_{1}p_{2}e_{2}, p_{1},p_{2}C_{0}.

We define

_{2}()_{} ep p1dp

Where is a four dimensional curve in C and _{2} _{1}_{1}(p_{1}), _{2} _{2}(p_{2}) are component curves with

traces in A and _{1} A , such that _{2} _{1}e_{1}_{2}e_{2}.

We have obtained the following result regarding the region of convergence of Bicomplex Gamma

function.

**THEOREM 3.3:** Let z_{1}z_{2}i_{2}C_{2}with Re

###

1 0and Re###

2 0 then 2() converges and2 2 1 1

2()( )e ( )e

.

**PROOF:**

By **Def. 3.2**and

**Th. 3.2**

) (

2

= _{}

dp p e p 1

e p p_{1} 1e_{1} e p2 p_{2} 1e_{2}

###

dp_{1}e

_{1}dp

_{2}e

_{2}

###

11 _{}

_{}

_{}

= _{2} _{2}

0

1 2 p 1

1 0

1 1 p

e dp p

e e dp p

e 2 2

1 1

= (1)e_{1}(2)e_{2}

Now, from the theory of the Gamma function of a complex variable, it is well known that the series

) s (

converges in the half**-**plane Re

###

s 0.Therefore, (1) and (2) converge, respectively, for Re

###

1 0 and Re###

2 0.Hence, _{2}()(1)e_{1}(2)e_{2} converges on {C_{2}:Re

###

1 0 and Re###

0}.Now let, 1 e_{1}2e_{2}z_{1}i_{2}z_{2} and z_{1}x_{1}i_{1}x_{2}, z_{2} x_{3}i_{1}x_{4}

) x x ( i x x z i

z_{1} _{1} _{2} _{1} _{4} _{1} _{2} _{3}

1

and 2z_{1}i_{1}z_{2} x_{1}x_{4}i_{1}(x_{2}x_{3})

4 1 1

x x )

Re( and Re(2)x_{1}x_{4}

Since Re

###

1 0 and Re###

0x_{1}x_{4}0 and x_{1}x_{4}0

x_{1}x_{4}and x_{1}x_{4}

x_{1} x_{4}

Re(z_{1}) Im(z_{2})

Hence, {C2:Re

###

1 0 and Re###

0}

_{ }

_{ }

} z Im z Re : C

{ _{2} _{1} _{2}

.

**THEOREM 3.4 **

**15**

**: **

Let 1e_{1}2e

_{2}z

_{1}z

_{2}i

_{2}C

_{2}with Re

###

z_{1} Im

###

z_{2}. Then

2 2 1 1 2

e ) (

1 e ) (

1 ) ( 1

.

Let μ_{n} logλ_{n} and C_{2}, p p1e1p2e2, p1,p2C0.

Where

###

is a four dimensional curve in C2 and _{1}

_{1}(p

_{1}),

_{2}

_{2}(p

_{2}) are component curves with

traces in A1 and A2, such that 1e12e2.

**THEOREM 3.5: **

Under the above mentioned notations and hypothesis,
_{e}n

n

###

###

_{} p e dp

) (

1 p

n 1

2

provided that

### Re

###

### z

_{1}

###

### Im

###

### z

_{2}and the series on the left is convergent.

**PROOF: **

Let z_{1}i

_{2}z

_{2}C

_{2}such that Re

###

z_{1} Im

###

z_{2}. Then, by

**Th. 3.4**,

2 2 1 1 2 e ) ( 1 e ) ( 1 ) ( 1

...** (3.1) **

Further due to idempotent techniques,

2 1 2 1 1 1 1 e p e p

p 1 2

and _{ }_{n}enp _{}

##

_{ }1

_{n}en1p

##

e_{1}

_{}

_{ }2

_{n}en2p

##

e_{2}

Now, p 1

###

_{n}e p

###

p_{1}1

##

1_{n}e p

##

e_{1}p

_{2}1

##

2_{n}e n2p

##

e_{2}2

1 n 1

n

###

###

_{}p1

_{n}enp dp

=

##

p_{1}1

##

1_{n}e p

##

e_{1}p

_{2}1

##

2_{n}e n2p

##

e_{2}

##

###

dp_{1}e

_{1}dp

_{2}e

_{2}

###

2 1 n 1

###

###

###

###

20 2 p n 2 1 2 1 0 1 p n 1 1

1 e dp e p e dp e

p n 2

2 1 n 1

...** (3.2)**

Now by **(3.1) **and **(3.2)**

###

###

dp e p ) ( 1 p n 1 2 n###

###

###

###

2 0 2 p n 2 1 2 1 0 1 p n 1 1 1 2 2 1 1 e dp e p e dp e p e ) ( 1 e ) ( 1 2 n 2 1 n 1###

###

###

###

2 20 p n 2 1 2 2 1 1 0 p n 1 1 1

1

### p

### e

### d

### p

### e

### )

### (

### 1

### e

### p

### d

### e

### p

### )

### (

### 1

2_{n}

_{2}

1 n 1

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

_{}

_{}

_{}

_{}

2
2
0
p
1
2
n
2
2
1
1
0
p
1
1
n
1
1 _{(} _{)} p e dp e

1 e p d e p ) (

1 2 _{n} _{2}

1 n 1 2 2 n n 2 2 1 1 n n 1

1 ( ) e

)
(
1
e
)
(
)
(
1
2
1
2
n
n
2
1
n
n
1
e
e 2
1
_{}
_{}
n

n _{}_{}_{} _{}
n
n

μn

ne ,

###

nlogn###

**ACKNOWLEDGMENTS **

I am heartily thankful to Mr. Sukhdev Singh, Lovely Prof. Univ., Punjab and all staff of Govt.

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