A Short Note on Holomorphic Functions

The meaning of the holomorphic function is having two different ends in a domain that are complex and symmetrical in form. Based on these particular facts about the terminology of mathematics, the study will include an overview of the holomorphic functions. In addition, this current study further includes the discussion regarding the composition of the holomorphic function. The properties of the holomorphic function will be included in the discussion part of this particular study. 

Holomorphic Functions: Overview

In the function that contains a complex value, the single complex variable has been referred to as f. Based on this particular concept, in a holomorphic function at a point z0 the derivatives of f in its domain will be defined as the limit. In simple diction, a complex number z has been taken as a limit that is z0. This particular value has been maintained by the same sequence of each complex value of z. As per the existence of the limit, f will be recognised as the complex differentiable at z0. As per the mathematics, it can be the liner and the product rule, chain rule and quotient rule that have been obeyed by the holomorphic functions. 

Composition of Holomorphic Functions

As per the facts described previously in the study, if a function that is complex is represented by f(x + i y) = u(x, y) + i v(x, y), the equation will be considered as a holomorphic. Based on this scenario, the v and u have been considered as first partial derivatives with respect to x and y. In reference to this particular phenomenon, the f is dependent on z̅ from the complex conjugate of z in a functional manner. On the other hand, if continuity has not been given, u and v have been continuous and the converse will not be necessarily true. 

Properties of Holomorphic Functions

It has been considered that the quotient of two holomorphic functions is holomorphic as per the condition where the dominator is at zero. As per the instances, it can be proposed that g and f are considered holomorphic in a domain U. Based on this particular scenario; the properties will be fg, f-g, f+g and f0g. On the other hand, if one identifies C with the real plane R2, then the holomorphic functions will be of two real variables with the first derivation that is continuous in order to solve a set of two partial equations that are differential to each other. As per these facts, each holomorphic function has been able to be separated into its imaginary and real parts such as f(x+iy) = u(x,y)+iv (x,y). 

Use of Holomorphic Functions

In the field of the analysis of complex functions, holomorphic functions are ubiquitous. In reference to the holomorphic functions, the complex does not refer to the meaning that denotes complex. On the other hand, it refers to the analysis of complex numbers. In the terminology of mathematics, the existence of the derivatives that are complex in the neighbourhood has been considered as an extremely strong condition. This particular condition helps in the implication that showcased a holomorphic function as a differentiable function in an infinite manner. Therefore, the central object of the complex analysis has been represented by the holomorphic functions. 

Singularities of Holomorphic Functions 

Based on the mathematical terminology, in reference to the complex analysis, it can be proposed that the holomorphic function has the holomorphic singularity. This removable singularity has been referred to as a point of the function where the function has been considered an undefined one. On the contrary, the function can be redefined as well with a regular resulting function. 

Cauchy-Riemann’s Conditions

A necessary and sufficient condition has been provided by the Cauchy-Riemann’s Conditions. The conditions represented by Cauchy-Riemann has been mentioned below,

•  It can be stated that a function that is complex is represented by f(z)
• This particular function will be considered a holomorphic in the respect of a point that is z0
• In order to use the differentiable properties, the considered function can be represented as f(z)=X(x, y)+iY(x, y)

Conclusion 

In order to sum up all that has been stated so far in the study, it can be stated that the study has discussed the facts regarding the holomorphic functions. Based on this particular terminology of mathematics, the study has included the properties. The compositions of the holomorphic properties have been described in the study as well along with discussing different examples of the holomorphic functions. In addition, the study has included Cauchy-Riemann’s Conditions and the Singularities of Holomorphic Functions in order to showcase in-depth knowledge based on holomorphic functions.