Integration of Heterogeneous Complex Variable Function and Combination Formula

2020 4th International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2020) ISBN: 978-1-60595-674-9

Integration of Heterogeneous Complex Variable Function and

Combination Formula

Yun CHEN

1

, Yan-hui YANG

2

and Ji-cheng TAO

3,* 1

Department of mathematics, Nanjing University of Finance and Economics, Jiangsu 210023, China

2

Department of Information and Computer Sciences, China Jiliang University, Hangzhou 310018, China

3

Department of mathematics, China Jiliang University, Hangzhou 310018, China *Corresponding author

Keywords: Heterogeneous Cauchy integral theorem, Heterogeneous Cauchy integral formula, Combination formula, Residue theory.

Abstract. In this paper, we discuss the integration of heterogeneous complex variable function

n k k

a z z f

) (

1 ) (

 

in curve Cjof heterogeneous complex plane HK_{by using heterogeneous Cauchy integral theorem,}

Cauchy integral formula and residue theory. And then obtain two identical equation of combinatorial mathematics by using two different calculation methods.

Introduction

In the references [1, 2], many authors introduce some reaction diffusion models of biology, study the diffusion problem of heterogeneous manifolds with homogeneous bottom space. Considering these heterogeneous problems, they always discuss the heterogeneous plexus space in the homogeneous bottom space, which brings a lot of difficulties to the research for a long time. In order to solve the above problems, the references [3,4,5] proposed a heterogeneous complex variable function theory, and gave the definition of heterogeneous complex domain, heterogeneous analytic function and heterogeneous integral. Based on this, the relationship between the heterogeneous Laplace equation and the heterogeneous Cauchy-Riemann equations is established. Accordingly, we can obtain the heterogeneous Cauchy integral theorem and the Cauchy integral formula. It’s so important to calculate the expansion coefficient when facing the Taylor expansion and Laurent series expansion of the heterogeneous analytic function. In this process, the key problem is the calculation of a common integral C _z_an

z ) (

d

, where the C represents the path of integration in the heterogeneous complex plane HK_{: a circle taking as} a _{center and}  _{as radius. Therefore, with the help of the}

heterogeneous Cauchy integral theorem , the Cauchy integral formula and Residue theory in the references [4,5,6,7], we take two different calculation methods to discuss the above integral and thus obtain two identical equation in combinatorial mathematics.

Preliminaries

Some basic definitions and properties of heterogeneous complex number and heterogeneous complex variable function are given below, which can also be seen in the references [3,4, 5] specifically.

Definition 2.1. The heterogeneous complex numbers field Ck consists, by definition Hk={z|z=a+jb}

R m jb a

z  ,  _{, then,} mama jmb_,z1 a1 jb1 _and z2 a2 jb2_,then

) ( 1 2 2

1 2

1 z a a j b b

z     

If we give a product operation by

1 1 1 a jb

z   _and z₂ a₂  jb_{2, then} z₁z₂ a₁a₂kb₁b₂j(a₁b₂ a₂b₁) _,

And the length of za jb be definited by

2 2

kb a

z_k  

, and then inverse element of

jb a

z  _be 2

1

k z

jb a z  

, hence Hk is a field. It is called heterogeneous complex numbers field. Definition 2.2. Let f be function from Hk to Hk , z0be a fix point in Ck, f be called differentiable

function in z0 _{,if the following limit exists:}

，

0 0

0 0

) ( ) ( lim lim

z z

z f z f z w A

z

z 

 

  

   

and then A be called derivative of f(z) in z0 _{by being denoted by}

) (z₀ f

.

Suppose f be a function in set D, f be called a differentiable function in D, if every zD, )

(z

f _{is differentiable.}

Definition 2.3. Suppose a directed curve Cj _{in Hk starting from} az() _{and ending in} bz()_,

of the form

) (t z

z



t



.

) (z

f _{is defined along} Cj_{, we take some points of division between} a _and b_:

b z z z z

a ₀, ₁,..., _n1, _n

, then Cj _{can be divided into} n _{segmental arcs, for} t



zt1,zt



_{, we}

have

t n

t t

n f z

S 







) (

1



, zt ztzt1.

When the number of points increases, the arc segments are gradually fined. If the limit of sum number Sn _{exists and denote} S_{, then} f(z) _{is said to be integrable along} Cj_{, the value is} S_, Cj _is

integral path. That is

. d ) (





j

C f z z

S

Theorem 2.4. Suppose function f(z) _{is analytical in a simply connected region} D _, C_j is a random contour or closed curve in D_{, then}

Theorem 2.5. Suppose the boundary of domain D in Hk be closed curve Cj_, f(z) _{be an analytic}

function in domain D and continuous function on DDCj



, then



_



j

C _z f i z

f 

 

 d

) ( 2

1 ) (

，（zD）.

Theorem2.6. Suppose the boundary of domain D in Hk, Cjbe closed curve in Hk, f(z) be an

analytic function in domain D and continuous function on DDCj



, then there exists each order derivative for f(z)and



_ 



j

C n

n

z f i

n z

f 

 

 ( ) d

) ( 2

! )

( ₁

) (

, （zD）

Two Integral Methods and Combination Formula

In this section, we will use the theory of complex variable function and the theory of heterogeneous complex variable function in section 2 to deal with important integral

 d , j

k n C

k

z n za 



, where Cj denote the circle zk a  centered at the point ax0jy0 and with radius  in the heterogeneous complex plane. And then obtain two identical equation of combinatorial mathematics by using two different calculation methods if n3.

According to theorem 2.5, theorem2.6, we have,





2 , 1,

d

0, 1.

j

k n C

k

i n z

n z a

 



  _

 



(3.1)

when n1, 2, by using residue theory, the above equations (3.1) holds. we use residue theory to obtain two combination formula by calculate the left side of (3.1) if n3,





= | ,

j k k k K

C z z  a  z H .

(cos sin ), 0 2

k

z  a   j    

,where

2 2 0 0

0 0

= x x ) y y ) , cos =x x ,sin y y ,

  

 

 

   

（ （

0 0

, .

z x jy ax  jy

Firstly, let F(zk) _{denote the function}

n k a

z )

( 1  _.

Let zei, then iz

dz d i z z z

z

 

 

    

 ,

2 sin

, 2 cos

1 1

1 2 2

1 2

1 1

d 2 [( 1) ( 1)]

d

( ) [( 1) ( 1)]

h(z)d

j

n n

k

n n n

C z

k

z

z k z k z

z

z a k z k

z               







Where





1 2 2

1 1 2 1 2 1 ( ) 1 1 1 n n n n n k z z k h z k k z k     _  _          _   _  _   

, we denoteA by

1 1   k k

and B by _{1 1}

1 ) 1 ( 2   

 n n

n

k  .

Then ), ( ) ( ) ( ) ( ) ( ) 1 ( ) ( 2 )

( _{2 1 2}

2 2 2 1 1 2 2 1 z h z h A z ABz A z Bz A z k z A z z h _n n n n n n n n n            _{ }   

(a)When k 1, that is 0 A1, then in the interior of a circle z 1,

n n n A z A z Bz z h ) ( ) ( ) ( 1   

have 2 poles of order one: z1 A _and

;

2 A

z  n n

n A z A z ABz z h ) ( ) ( ) ( 2 2    

have 2 poles of order one:

A

z3  _and z4  A. _{Calculating their residues}

Therefore,









  



 













 

 



 



1 1



2 2 1 1 1 1 2 1 1 1 2 2 2 C C 2 )) ( Re ) ( Re ) ( Re ) ( Re ( 2 d ) ( 1 2 1 2 2 2 1 1 1 2 2 0 1 3 2 1 2 1 2 2 1 1

1 _{1 2 1 2}

                                                          





n n n m n m m n m m n n m n m n m n n z z z z z z z z z n BA k j n m n m n m n m n n BA k j z h s z h s z h s z h s k j z z h   

When 0 k 1, that is A0, Their residues ar siemilar (a).

Consequently, we can gain two combinatorial identical equations from the above. Theorem3.1. If n1, then









2



1

  



1

 

1







2 2

2 1 1

        _{  }                



m n m

n _{n m n m}











  



 













 

 



. 0 2

) 1 2 ( 1

1 2 2

2 2

1 1

2 1 2

4 1 2

2 C

C 2 1

1 2 2

2

2 1

1 4 1

2 0

2 1 4

2   

     

     

   

 

     

 

 

  

 

   



 



n

m n m

m n m

m n n

m

n m n m

n n

n m n

m n m n

m n n

(3.3)

Summary

In order to further study the Taylor expansion and Laurent expansion of the heterogeneous complex variable function, this paper uses the heterogeneous Cauchy integral formula and the heterogeneous integral theorem in the [3,4,5] of the analytical function , and the residue theory[7] of the complex variable function to calculate a important integral formula of the expansion, thus we establish two combinatorial identities. In addition, we have a lot of further discussion, such as how to establish the classification of singularities, similar residue theorem and the application of the heterogeneous complex number and variable function in physics, mechanics and e.t.c.

Acknowledgement

This research was supported by the Research Program for University Students of China Jiliang University (2019X22101).

References

[1] Allen, L., Bolker, B., Lou, Y., et al. Asymptotic profile of the steady states for an SIS Epidemic Patch Model. SIAM Journal on Applied Mathematics, 67(2007), 1283-1309.

[2] Wang, W. and Zhao, X.Q., A nonlocal and time-delayed reaction-diffusion model of dengue transmission. SIAM Journal and Applied Mathematics, 71(2011) 147-169.

[3] Xuejiao Zhao, Yun Chen and Jicheng Tao, Heterogeneous complex number and heterogeneous complex function, Advances in Applied Mathematics, 6(2017) 69-77.

[4] Yun Chen, Xuejiao Zhao and Jicheng Tao, Integration of heterogeneous complex function, Advances in Applied Mathematics, 6(2017) 153-164.

[5] Jicheng Tao, Yun Chen and Xuejiao Zhao，Heterogeneous Complex Function, 2017 International Conference on Applied Mathematics, Modelling and Statistics Application, May 22-27, Beijing, China, Advances in Intelligent Systems Research (AISR), 141(2017) 345-348.

[6] Zhong yuquan, Complex function theory, Higher Education Press, Beijing, 2013.