K- bianalytic functions and their related theorems

In this paper, we present the concept of K-l analytic functions on the complex plane and discuss the properties of the K-l analytic functions. Then we give the expression of K-l analytic functions and proved the Cauchy integral formula of K-l analytic functions. Also, the power series expansion theorem and the Fourier series of K-l analytic function were proved in this study.

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Introduction

Analytic functions have been applied in many fields, e.g., theoretical physics, astromechanics, and fluid mechanics. Over the past two hundred years, systematic theories in analytic functions have been established through the efforts of many researchers. In 1983, Professor Jianding Wang first proposed the semi-analytic function theory, which leads to establishment of the conjugate analytic function theory [1] . These two theories have been successfully applied to electric fields, magnetic fields, fluid mechanics, elastic mechanics and other fields. Based on these theories, many researchers continued this study and expanded the problem into a series of new fields, bianalytic functions, complex harmonic functions, polyanalytic functions (k analytic functions), and the corresponding boundary value problems, differential equations, integral equations, etc [2][3][4][5] . Recently, some researchers have proposed the concept of Kanalytic function and studied its analytical properties (see [6][7][8] ). Zhang jianyuan [6] concluded a necessary and sufficient conditions of K-analytic function. In [7] , the author deducts the relation between K-analytic function and K-integral, the relation between K-analytic function and K-harmonic function. Zhang jianyuan [8] used the series theory to propose the power series expansion of the K-analytic function and its zero isolation and uniqueness. In this research, we explained the concept of K-bianalytic analytical function, and we derived the K-bianalytic functions expression. Then, we proved the Cauchy theorem and Cauchy integral formula of K-bianalytic function. In the last part, we discussed the power series expansion and the Fourier series expansion of K-bianalytic functions. 2．Preparatory knowledge Definition 1 [6] (definition of K-Complex number) Let function 𝑓(𝑧) be defined in area D, we say that the form 𝑧(𝑘) = 𝑥 + 𝑖𝑘𝑦(𝑘 ∈ 𝑅, 𝑘 ≠ 0) is k-Complex number of 𝑧 = 𝑥 + 𝑖𝑦.

Definition 2 [6] (definition of K-derivative) Let function 𝑓(𝑧) be defined in a neighborhood of 𝑧 0 , 𝑧(𝑘) = 𝑥 + 𝑖𝑘𝑦(𝑘 ∈ 𝑅, 𝑘 ≠ 0), if the limit lim ∆𝑧→0 ∆𝑓 ∆𝑧(𝑘) = lim ∆𝑧→0 𝑓(𝑧)-𝑓(𝑧 0 ) 𝑧(𝑘)-𝑧 0 (𝑘) is exist, then we call function 𝑓(𝑧) is k-differentiable at 𝑧 0 , and this limit is defined as the k-derivative of function 𝑓(𝑧) at 𝑧 0 , denoted as 𝑓 (𝑘) ′ (𝑧 0 ) or

．

When function 𝑓(𝑧) is K-derivable at every point in area D, 𝑓(𝑧) is called K-derivative in D. and its k-derivable is denoted as 𝑓 (𝑘) ′ (𝑧), (𝑧 ∈ 𝐷)． Definition 3 [6] (definition of K-analytic function) If function 𝑓(𝑧) is k-differentiable in region D, 𝑓(𝑧) is K-analytic in region D. All of functions parsed in k-in area D is denoted as 𝐹(𝐷(𝑘))． Lemma 1 [6] (Necessary and sufficient conditions of K-analytic) the function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) is K-analytic in region D, ⟺ 𝑢, 𝑣 is differentiable in region D, and the condition of C-R-K holds, where condition C-R-K is

Lemma 2 [7] The function 𝑓(𝑧) is parsed in the single connected region D of z plane, and C is any closed curve in D, then we have the identity ∮ 𝑓(𝑧)(𝑧 -𝑎) 𝑛-1 (𝑘)𝑑𝑧(𝑘) 𝐶 = 0.

Lemma 3 [8] (power series expansion of K-analytic) Suppose function 𝑓(𝑧) is K-analytic in region D, and B(𝑘): |(𝑧 -𝑎)(𝑘)| < 𝑅 ⊂ 𝐷, we have

, and the expansion is unique. Corollary Let D be a bounded n+1 connected region surrounded by segmented smooth curve 𝐶 = 𝐶 0 + 𝐶 where 𝑎 is an arbitrary point in complex plane.

K-bianalytic functions and their related theorems

Below, we will use the Cauchy theorem of K-bianalytic function to prove the Cauchy integral formula of K-bianalytic function.

To this end, we first prove a lemma. Thus there is

Then we can know that 𝑓(𝑧)𝜙(𝑧) is K-bianalytic in D. Theorem 3 (Cauchy integral formula of K-bianalytic functions) Let D be a bounded region surrounded by a closed curve or segmented smooth curve 𝐶 = 𝐶 0 + 𝐶 are continuous on

Proof : For every 𝑧 ∈ 𝐷, we make a weekly 𝐶 𝜌 : (𝜁 -𝑧)(𝑘) = 𝜌𝑒 𝑖𝜃 (0 < 𝜃 < 2𝜋), so that its surrounding area

is bianalytic in 𝐷 -𝐷 𝜌 according to hypothesis and lemma 4, and the following can be obtained by theorem 2

Since 𝐶 𝜌 : (𝜁 -𝑧)(𝑘) = 𝜌𝑒 𝑖𝜃 , let 𝜌 → 0, we also conclude that

Substituting ( 2) into (1), we have

By Theorem 3, we can get the power series expansion theorem of K-bianalytic functions.

When 𝑛 ≥ 1, we can see that 𝑓(𝑧)(𝑧 -𝑎) 𝑛-1 (𝑘) is K-analytic in region D，and applying Lemma 2, we have

Add form (5) to form (6), we have

Subtracting ( 5) from ( 6) is obtained. The uniqueness is obvious, theorem is proved.

1. Jianding Wang. Semi-analytic function conjugate analytic function, M. Beijing University of Technology Press, 1988
2. Zhao Zhen, Bianalytic functions, complex harmonic functions and their basic boundary value problem, J. Journal of Beijing Normal University, Natural Sci. 1995; 31(2):175-179.
3. Wang Minghua. Some properties of bianalytic functions and its Hilbert boundary value problem, J. Journal of Beijing Normal University, Natural Sci. 1998; 34(1):13-20.
4. Yang Pi-wen, The k-regular Function and Some Boundary Value Problems, J. Journal of Sichuan Normal University, Natural Sci. 2001; 24(1):5-8.
5. Tang Huo, Li Shuhai. Fourier series of k analytic function, J. pure mathematics and Applied Mathematics. 2009, 25(1):83-86
6. Zhang Jianyuan, K-analytic function and its existence condition J Journal of Yunnan University of Nationalities: Natural Science Edition. 2007; 16(4):298-302.
7. Zhang Jianyuan. K-integral of complex functions J. Journal of Yunnan Normal University: Natural Science Edition. 2009; 29(1):24-28.
8. Zhang Jianyuan. Expansion of K-analytic function in Power series J. Journal of Dali University. 2009; 8(4):14-18.
9. Zhong Yuquan. Complex function theory M. Beijing: Higher Education Press, 2004.