Partial and Local Argument Properties of Holomorphic and Meromorphic Complex Functions in Several Variables
The study describes a general argument analysis technique for holomorphic and meromorphic complex functions in several variables, or simply -variable complex functions with . Argument analytic relationships for -variable complex functions with significance similar to the argument principle for one-variable ones are retrieved partially and locally. More precisely, argument analysis in -variable complex functions is carried out one-by-one in terms of each and all variables, namely, partially, so that argument-principle-like relations are established in poly-disc neighborhoods of the variable domains, namely locally. The technique is applicable graphically with loci plotting, independent of Cauchy integral contour and locus orientations; it is also numerically tractable without loci plotting via argument incremental integration. Numerical examples are included to illustrate the main results.
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 [1, 2]. As an important application of the argument principle, it is frequently employed for locating isolated zeros/poles or singularities of complex functions in terms of distribution and multiplicity; in particular, Nyquist-type stability criteria including classical and generalized ones in various system and control problems are typical results [3–5]. Lately, the argument principle has further been exploited for structural and spectral aspects of linear dynamical systems in terms of controllability/observability and positive realness . When multidimensional [7–12], nonlinear [6, 13], hybrid , and functional models [15, 16] are dealt with, argument features of complex functions in several variables are becoming indispensable and inevitable. Unfortunately, however, the research situation for argument analysis in -variable complex functions with is extremely poor. As a matter of fact, no systematic studies about the related topics can be found in the complex analysis literature, to the best of the authors’ knowledge.
This note tries to contrive a slightly more general argument analysis technique for holomorphic and meromorphic -variable complex functions. Several argument properties related to -variable complex functions that are not known up to now are claimed and proved rigorously, which are in form similar to the argument principle for one-variable complex functions but in some partial and local sense. More specifically, by the suggested technique, argument analysis in -variable complex functions can be completed in a variable-by-variable fashion, namely, partially in variables, so that relations in between zero/pole distribution and argument evaluations are established in various poly-disc neighborhoods, namely, locally in zero/pole distribution. It is worth mentioning that the argument analysis technique is exploitable graphically through loci plotting, independent of Cauchy integral contour and locus orientations, and it is also implementable merely via argument incremental integration, dispensable of loci plotting.
The paper is arranged as follows. Section 2 collects preliminaries, notations, and terminologies about -variable complex functions. Novel argument relationships for holomorphic and meromorphic -variable complex functions are claimed and proved in Section 3. Numerical examples are sketched in Section 4, whereas conclusion is given in Section 5.
2. Preliminaries to -Variable Complex Functions
2.1. Notations and Terminologies
In what follows, and denote the sets of all real and complex numbers, respectively. and , respectively, stand for the -dimensional real and complex Euclidean spaces; or equivalently, and . is the set of all naturals, while the set of strictly positive naturals is denoted by . The boundary of an open and connected set is denoted by . With , we denote and thus means that . As usual, and means the complex conjugate.
Notations and terminologies about discs and Cauchy contours on are explained briefly as in Figure 1, in order to prepare us for generalizing the argument principle for one-variable complex functions to several-variable cases. For describing the disc and the Cauchy contour of Figure 1 with formulas, we writewhere is the radius of the disc and is the disc center. In other words, is open and connected and accordingly is the boundary of , which is a simple and closed contour. will be used as the integration contour in the Cauchy integral for argument analysis.
To understand our discussion for -variable complex functions, we need to introduce the so-called poly-contours defined in as in Figure 2. Similar to those in (1), we writewhere, for , is the radius of the disc and is the center of on the -th complex plane. Simply speaking, the pair of and are defined in the same way as to their counterparts in Figure 1 and (1) but on a specific complex plane accordingly. Also, are all open and connected sets with disc radius sufficiently small. All radii of the discs are not necessarily the same. To simplify our subsequent notations, we also writewhich are called, respectively, the poly-disc defined with , and their corresponding poly-circular boundaries. Clearly, is an open and connected subset of , while is simple and closed in the individual complex plane sense.
2.2. Facts about -Variable Complex Functions
To facilitate our statements, we recall additional notations and collect facts about -variable complex functions from [2, 17, 18]. To those readers who are familiar with complex functions in several variables and their analysis, this part can be skipped.
Let denote a -variable scalar complex function that is a holomorphic mapping from to in the sense of , with being a short symbol for the variables . We also write . The definition for holomorphic -variable complex functions can be found in [18, p. 7, Definition 1.2.1]. In rough words, a -variable complex function is said to be partially holomorphic, if each and all the resulting one-variable complex functions are holomorphic after one variable is viewed as variable while all other variables are fixed as constants. Exact definitions are given in [18, p. 13, Definition 1.2.21]. By Remark 1.2.28 of [18, p. 17], every partial holomorphic function is also holomorphic and vice versa. The concept of partially holomorphic -variable complex functions is the starting point for our partial and local argument analysis.
It is said that is an isolated zero of the -variable complex function if , and does not vanish identically in any sufficiently small neighborhood around ; or equivalently, at , while in any sufficiently small neighborhood around .
The Cauchy line integral for one-variable complex functions is modified with regard to the -variable complex function along the contour for a fixed as follows:To rigorously understand (4) and the ensuing discussions, let us introduce what we call the -index order with and write that Then, we are ready to define the -variable power multiplication and the partial derivative with the -index order , respectively, by
To see the definition and properties about the total and partial derivatives of the -variable complex function , let us writeClearly, are the real and imaginary portions of the complex variable , respectively. Then, we define the total derivative bywhere In particular, for every , it is already known by the Cauchy-Riemann equations that . Hence, it is straightforward to see that, for , it holds that
Furthermore, it follows readily that, for a class of -variable complex functions , it always holds thatfor each .
2.3. Taylor Series of Holomorphic -Variable Complex Functions
Lemma 1. Suppose that has an isolated zero at . Then, there exist a neighborhood around , an identically nonvanishing holomorphic function defined on , and a -index order such that , for all .
Proof of Lemma 1. By the Taylor series expansion formula for -variable complex functions, say Corollary 1.5.9 of [18, p. 36], the holomorphic assumption about ensures thatholds true for any and any sufficiently small neighborhood around . Here, means that the -index power superscript changes from to , , , for all . The above Taylor series expansion is convergent compactly and holomorphic on .
Now let , , with (or equivalently, ); the -index order is used. Therefore, we can writewhere is obviously defined and thus . By the definition, is holomorphic and does not vanish identically on . To complete the proof, it remains to show that . On the one hand, if , then , for all and . This says in particular that , which is contradictory to the assumption of . On the other hand, without loss of generality, if and for each , then and thus at , , over all ; this is contradictory to the identically nonvanishing assumption of in . Cases in terms of with can be proved similarly.
Remark 2. By Lemma 1, if the disc radii are sufficiently small and all distinctive isolated roots of are located in different poly-discs, say , always has a Taylor expansion over , where and is holomorphic and identically nonvanishing over . Here, denotes the poly-disc for the -th distinctive and isolated zero of . When all isolated zeros of are examined with respect to the poly-discs sequence , a -variable Taylor series sequence is eventually created for representing on the whole .
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 will be dropped.
2.4. Complex Logarithm for -Variable Complex Functions
Now we collect some facts about the complex logarithm for one-variable functions and then extend them to -variable complex functions. To a complex scalar with and , it is natural to set . However, by this definition, is unique only up to an integer multiple of . In view of this, we define the complex logarithm of aswhere . For each , a branch or a sheet of the complex logarithm of is meant; in particular, the one with corresponds to the principal branch.
Accordingly, for a -variable complex function that does not vanish identically on , the complex function logarithm is defined by In other words, the complex logarithm can also be viewed as a complex phaser whose argument is for some integer .
If we treat as a one-variable complex function in the sense that acts as variable and all other variables are fixed, Theorem 6.2 of [2, p. 100] tells us that is holomorphic in the -variable in each specific logarithm branch. It follows by Remark 1.2.28 of [18, p.17] that is partially holomorphic on and thusis well-defined for each and all .
3. Argument Analysis for -Variable Complex Functions
In this section, we explain the partial and local argument analysis in -variable complex functions.
Consider . Therefore, if we reexpress it as in some poly-disc as in Lemma 1 and Remark 2, together with (11), we can writefor all . Clearly, (17) holds in the -variable sense for a specific while all the non- variables are fixed at some (namely, but ), over which is partially holomorphic and identically nonvanishing as appropriately.
By applying the Cauchy line integral as modified in (4) to relation (17) along the contour in the -variable sense, while all the other variables are viewed as constants in with and as appropriately, we havefor all . In the above, we used the fact that by the Cauchy residue formula. Similarly, it follows that is well defined and equal to zero, since is partially holomorphic in and nonvanishing identically on so that the argument difference of must be zero when runs along the specific contour . This is because the origin does not lay in the interior of the set encircled by the locus , no matter how the other variables are taken in their corresponding discs, except for the disc centers.
Furthermore, noting also that, on the same logarithm branch, the following argument increment relation is satisfied:where and stand for the two sides of a point when runs clockwisely along , while all the other variables are fixed as appropriately.
Finally, we can claim the following partial argument relation:where . In (20), the notation () means that the variable is fixed at a constant , or equivalently . In what follows, denotes the argument incremental of when the integral variable runs clockwisely along and the other variables are fixed as appropriately.
Based on (20), we claim the following results.
Theorem 3. Consider a -variable complex function . For an isolated zero of , one can always fix an open and connected poly-disc , which is a sufficiently small neighborhood around , such that (A1) has no zeros other than in ; (A2) vanishes nowhere on the poly-circular boundary , that is, for any . Then, the -index order about the zero satisfies
Proof of Theorem 3. Assertion (21) follows by rearranging relation (20) into a vector according to . Assumptions (A1) and (A2) guarantee that the Cauchy integrals are well defined in the sense that all the complex logarithms pointwisely throughout the contour are validated and the argument incremental can be calculated.
Remarks about Theorem 3 are summarized as follows:(i)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.(ii)If the -index order possesses one or more 0-entries, this actually says that the -variable complex function has no isolated zeros in the concerned poly-disc . In view of this, it is reasonable to say that may or may not have isolated zeros in , depending on how the poly-disc is defined; or alternatively, the -index order is -related. In this sense, the results of Theorem 3 can only be interpreted locally.(iii)For each entry-order of the zero with respect to a specific variable, the other variables need to be treated as constants. In principle, these fixed ones can be arbitrarily taken as long as they are in the corresponding discs, excluding the disc centers as appropriately. In this sense, the argument equation (21) consists of single-variable relations; or the results of Theorem 3 need to be interpreted partially.(iv)The argument incremental formula in the second relation of (21) can be also obtained simply via numerical integrations, without loci plotting. In this sense, Theorem 3 can be employed graphically with loci plotting or purely numerically without loci plotting. This is also the case for Theorem 4, though we will not mention this point again.(v)Different from the argument principle for one-variable complex functions, loci (or argument conditions) are involved in examining existence and multiplicity of isolated zeros for -variable complex functions. This is also the case for Theorem 4 below.
Isolated singularities such as poles of partially meromorphic -variable complex functions can be dealt with simiarly. This is talked about briefly in the following.
Let mean that the -variable scalar complex function is partially meromorphic (for want of better words, detailed definition can be found in ). This means that, about , when each of the variables is viewed as variable, and all the other variables are fixed as constants, each and all the resulting one-variable functions are meromorphic in the one-variable sense. It is said that a singularity is a pole of if is an isolated zero of . Then, we can prove the following results.
Theorem 4. Consider a -variable complex function . For a pole of , there is always an open and connected poly-disc , which is a sufficiently small neighborhood around , such that (A1) has no pole other than in ; (A2) vanishes nowhere on the poly-circular boundary , that is, for any . Then, the -index order of the pole satisfiesHere, the orientations of the Cauchy contours and the corresponding loci are specified in such a way that the -index order in (21) possesses nonnegative entries.
Proof of Theorem 4. Under the given assumptions, it remains only to show that, for the pole , we can always conclude that andfor some identically nonvanishing holomorphic -variable complex function in any small neighborhood around , and with being the order of the pole in .
To this end, we notice by the definition about that must be an isolated zero of . Hence, there exists a poly-disc , which is a neighborhood of , such that is holomorphic and by Lemma 1, holds over for some -index order and that is nonvanishing identically on . Based on these facts, we observe that for each . The last equation follows readily after letting and , which yields to us the desired relation (23).
Remarks about Theorems 3 and 4 are as follows:(i)Since Theorem 4 is also claimed with poly-discs and the corresponding poly-circular boundaries, it may not be true if the poly-circular boundaries are replaced with the so-called toy contours defined on each different complex plane . This means in rough words that Theorem 4 only applies partially and locally, as seen by Theorem 3.(ii)In general, isolated zeros and poles of meromorphic -variable complex functions cannot be dealt with in the same poly-disc rigorously. This is because the proof arguments for Theorems 3 and 4 are constructed by means of Lemma 1, which is essentially true merely in some local sense. Even if a concerned zero and a pole are located in each other’s small neighborhoods, their Taylor series expansions may not be unified through analytic continuation [2, p.53], noting once again that there exist some ’frozen’ variables.(iii)The minus “−” in (22) is not essential and has no absolute meaning. It is added in (22) simply for the sake of expressing the -index order in accordance with in (21).
4. Numerical Illustrations
In this section, argument analysis for several -variable complex functions is illustrated numerically. The coefficients and in the examples are fixed at and such that and the polynomial has a pair of conjugate zeros at . The discs and have the radii pair and will be specified in the following; accordingly, the Cauchy contours and are meant as in Figure 2. The loci with are plotted by black curves, while the loci with and are in blue and red curves, respectively.
As explained in the remarks about Theorems 3 and 4, zeros and poles of the examples, if any, can be examined directly in a numerical fashion without loci plotting. However, we do implement the results graphically with loci plotting so that the reader can have better understanding.
4.1. 2-Variable Complex Function with Isolated Zeros
In this case, we examine the 2-variable complex function:Therefore, is holomorphic in the whole space . Clearly, has an isolated zero with if the two discs are given byOr has the isolated zero with if the two discs are
The corresponding loci based on the above two poly-discs are illustrated in Figures 3(a) and 3(b), respectively. Based on Theorem 3, the loci encirclements around the origin in Figure 3 and the numerical argument integration exactly coincide with the isolated zero existence and the -index order evaluations determined in the above.
4.2. 2-Variable Complex Function without Isolated Zeros
In this case, we examine the 2-variable complex function: Clearly, is holomorphic in the whole space , and it has no isolated zero, no matter how the disc is defined.
The corresponding loci with respect to the same poly-discs in Section 4.1 are illustrated in Figures 4(a) and 4(b), respectively. Based on Theorem 3, since neither of the two dash-dotted loci in Figures 4(a) and 4(b) encircles around the origin , there is no isolated subzero in the corresponding complex plane; or has no isolated zeros in the specified poly-disc. Our numerical argument integration also reflects this point.
4.3. 2-Variable Complex Function with Poles
In this case, we examine the 2-variable complex fraction: It follows readily that is partially meromorphic in the whole space . Clearly, has no isolated zero, while it has a pole with .
The corresponding loci in terms of the poly-disc with and being as follows are illustrated by Figure 5:Based on Theorem 4, the loci encirclements around the origin in Figure 5 and the numerical argument integration exactly coincide with the pole existence and the -index order determined according to the definition.
4.4. 2-Variable Complex Function with Cancellation of Pole/Nonisolated Zero
Now we examine the 2-variable complex fraction:which is partially meromorphic in the whole space . Clearly, has no isolated zero, while it seems to have a pole with by the formula. However, the loci of Figure 6 show that this is not true.
Carefully examining , we can see that, for , . This in turn implies thatholds true no matter how the disc is defined. In other words, when is investigated on , it reduces to the following 2-variable complex fraction partially and locally:Clearly, has a pole with . This has been illustrated by Figure 6 graphically. If we plot the loci of in Figure 7, the same conclusion follows.
Finally, by comparing and , it is said that pole/zero cancellation occurred in the poly-disc limit sense. However, such cancellation is not removable, since it is due to the existence of nonisolated zero .
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 3 and 4, are several argument relationships expressed in the partial and local sense for holomorphic and meromorphic complex functions in several variables. Major implementation issues are also talked about in the relevant remarks.
Carefully examining the methodological aspects of the suggested technique, one might assert that the technique can be interpreted as decomposing the original -variable complex function to a group of one-variable complex functions, to which the one-variable complex analytical concepts and tools are applied. This study reveals for the first time to what extent the one-variable complex analysis theory can be extended to -variable complex functions, whenever argument-related features are concerned. It is also worth doing to exploit the results for addressing control problems, where multidomain complex/frequency features are encountered, though we did not probe into any details about these points in this paper.
The data for the numerical simulation are available at request.
Conflicts of Interest
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.
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