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The study describes a general argument analysis technique for holomorphic and meromorphic complex functions in several variables, or simply

The argument principle for one-variable complex functions is one of the fundamental concepts and theories based on the Cauchy integral, which paves the way for establishing numerous results in complex analysis, just mentioning Rouché’s theorem and the open mapping theorem as well as the maximum modulus principle, besides many others [

This note tries to contrive a slightly more general argument analysis technique for holomorphic and meromorphic

The paper is arranged as follows. Section

Notations and terminologies herein are standard, whose exact definitions can be found in most textbooks about complex analysis [

In what follows,

Notations and terminologies about discs and Cauchy contours on

Disc and its boundary on the complex plane

To understand our discussion for

Poly-disc and its poly-circular boundary in the space

To facilitate our statements, we recall additional notations and collect facts about

Let

It is said that

The Cauchy line integral for one-variable complex functions is modified with regard to the

To see the definition and properties about the total and partial derivatives of the

Furthermore, it follows readily that, for a class of

The following lemma plays a key role in establishing partial and local argument relationships for holomorphic

Suppose that

By the Taylor series expansion formula for

Now let

By Lemma

In what follows, for discussion brevity and without loss of generality, only one of the isolated zeros is considered directly and explicitly. In this sense, the subscript

Now we collect some facts about the complex logarithm for one-variable functions and then extend them to

Accordingly, for a

If we treat

In this section, we explain the partial and local argument analysis in

Consider

By applying the Cauchy line integral as modified in (

Furthermore, noting also that, on the same logarithm branch, the following argument increment relation is satisfied:

Finally, we can claim the following partial argument relation:

Based on (

Consider a

Assertion (

Remarks about Theorem

In the above discussion, it is our underlying assumption that the Cauchy contour orientation is specified as clockwise, and the corresponding loci orientations are determined accordingly. Indeed, the orientations of the Cauchy contours and those of the corresponding loci can be self-defined such that each and all the argument increments are nonnegative as appropriately, when isolated zeros are concerned.

If the

For each entry-order of the zero

The argument incremental formula in the second relation of (

Different from the argument principle for one-variable complex functions,

Isolated singularities such as poles of partially meromorphic

Let

Consider a

Under the given assumptions, it remains only to show that, for the pole

To this end, we notice by the definition about

Remarks about Theorems

Since Theorem

In general, isolated zeros and poles of meromorphic

The minus “−” in (

In this section, argument analysis for several

As explained in the remarks about Theorems

In this case, we examine the 2-variable complex function:

The corresponding loci based on the above two poly-discs are illustrated in Figures

Loci for the 2-variable complex function

In this case, we examine the 2-variable complex function:

The corresponding loci with respect to the same poly-discs in Section

Loci for the 2-variable complex function

In this case, we examine the 2-variable complex fraction:

The corresponding loci in terms of the poly-disc

Loci for the 2-variable complex function

Now we examine the 2-variable complex fraction:

Loci for the 2-variable complex function

Carefully examining

Loci for the 2-variable complex function

Finally, by comparing

This pure mathematical note is aiming at creating a more general complex analysis technique for argument-related features in complex functions in several variables. The main contributions, namely, Theorems

Carefully examining the methodological aspects of the suggested technique, one might assert that the technique can be interpreted as decomposing the original

The data for the numerical simulation are available at request.

The authors declare that they have no conflicts of interest.

The study is completed jointly under support of the National Natural Science Foundation of China under Grant no. 61573001 and no. 61703137.

_{1}-gain analysis for positive 2-D systems with state delays in the Roesser model