International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 753916, 21 pages

http://dx.doi.org/10.1155/2012/753916

## On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

^{1}Facultad de Ciencias, Universidad Autónoma de Baja California, Km. 103 Carretera Tijuana-Ensenada, 22860 Ensenada, BC, Mexico^{2}Grupo Alximia SA de CV, Departamento de Investigación, Ryerson 1268, Zona Centro, 22800 Ensenada, BC, Mexico^{3}Departamento de Matemáticas, Universidad de Sonora, 83000 Hermosillo, SON, Mexico^{4}División Multidisciplinaria de la UACJ en Cuauhtémoc, Universidad Autónoma de Ciudad Juárez, 32310 Ciudad Juárez, CHIH, Mexico

Received 26 March 2012; Accepted 16 May 2012

Academic Editor: Mihai Putinar

Copyright © 2012 Alvaro Alvarez-Parrilla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new technique for solving a certain class of systems of autonomous ordinary differential equations over is introduced ( being the real or complex field). The technique is based on two observations: (1), if has the structure of certain normed, associative, commutative, and with a unit, algebras over , then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over , is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

#### 1. Introduction

Throughout this work, will stand for a field, usually the real or the complex field .

Consider the autonomous ordinary differential equation with having certain regularity conditions.

In general the solution is not easy to obtain since this is usually a system of coupled differential equations. There is a vast literature regarding the solution of ordinary differential equations by different means and in particular by techniques utilizing *generalized analytic functions,* see for instance [1–14]. These include applications to the three-dimensional Stokes problem, solutions of planar elliptic vector fields with degeneracies, the Dirichlet problem, multidimensional stationary Schrödinger equation, among others [3, 5, 6, 12, 13]. In particular, the technique that we present is of interest for people working on vector fields with singularities. For instance, in order to gain insight into the behaviour of analytic vector fields, correct visualization of vector fields in the vicinity of their singular set is required, in the case of visualization of two-dimensional complex analytic vector fields with essential singularities the usual methods only provide partial results (see [15–17]), whilst the technique which we promote provides accurate and correct solutions [18, 19]. These questions arise naturally in discrete and continuous dynamical systems (see [20–22]).

As a first step in obtaining a solution to (1.1), we notice that, if can be given the structure of a certain algebra , it is possible to reduce this system to a single autonomous differential equation with being a function -differentiable with respect to the variable in the algebra .

Having done so, we proceed to show that it is possible to solve (1.2) by extending a geometric technique introduced in [18, 19] in the context of complex analytic vector fields related to Newton vector fields which are first studied by [23]. This technique is based upon the construction of two functions, which, are respectively, constant and linear on the trajectories which are the solutions of (1.2).

The paper is organized as follows. In Section 2 the algebras in question are introduced, in particular we introduce the notion of normed, associative, commutative, and with a unit, finite dimensional algebra over , showing in Section 2.1 that these have a *first fundamental representation* into the algebra of matrices over , . Furthermore in Section 2.2 *normal algebras* are defined and their corresponding tensor products are constructed. This is standard material which can be found in [24–26] but is presented here for completeness.

In Section 3 we give the definition of -differentiability, and proceed to show that if the family of matrices is linearly equivalent to a subset of an algebra in , that is, the image of the first fundamental representation of an algebra with respect to the canonical basis of , that is, , then in fact is -differentiable on . The problem of determining if a map is -differentiable for some algebra is treated indirectly in [27], where the conditions that are given ensure the existence of an algebra such that the set of relations are the generalized Cauchy-Riemann equations for , ([27, 28] show that these equations give a criterion for -differentiability). Furthermore [29] considers the case when , where he proves that every analytic map (in the usual sense) which is -differentiable has an expansion in power series (see [29, pp. 646 and 653]).

In Section 4 we show that for normal algebras it is possible to express the function in terms of a single variable in the algebra, and hence there is a function that represents . The analogy being the algebra of complex numbers, where is the variable and the -differentiable functions being the analytic functions. We also show that there is a differentiable operator which has the property that , where is the Lorch derivative of (see [26]), hence providing a framework for the usual calculus of one variable.

In Section 5 we start by showing that in this context the differential equation (1.1) takes the form with being a function -differentiable with respect to the variable in the algebra , and being a certain singular set, where the solutions are not defined.

We then proceed to show that the geometric technique, introduced in [18, 19], of finding two functions and which are constant and linear on the trajectories which are solutions of the differential equation, can be extended to the case of (1.3). We end the section and the paper with an example.

#### 2. Algebras

We introduce -algebras (see, e.g., [24]).

*Definition 2.1. *A *-algebra* (or algebra over ) is a finite dimensional -linear space on which is defined a bilinear map that is associative and commutative, and there is a unit element in that satisfies for all .

An element is called *regular* if there exist a unique element in denoted by called *inverse* of such that . An element which is not regular is called *singular*. If and is regular, the quotient will mean .

##### 2.1. Algebras and Their Fundamental Representations

We define the first fundamental representation.

If is an ordered basis of an algebra , the product between the elements of is given by
where for are called the *structure constants* of . The *first fundamental representation* of associated to is the isomorphism defined by
where is the matrix whose entry is for . The commutativity and the associativity of are equivalent to the identities (1), for all and (2), for all , respectively.

Using we assign to the norm induced from the operator norm in (see [24]). In this way each algebra is a normed algebra, that is, there exists a norm satisfying for all and .

*Example 2.2. *Let be the linear space with the product between the elements of the standard basis given in the following equation:
that extends to the product in given by

The product between the elements of define the *structure constants * for . So, with the product given is an algebra .

##### 2.2. Normal Algebras and Their Tensor Products

Let be matrices , each of one the following four types:
where
and . In this case we will say that the matrix given by
is in its *normal form*. We will associate to an algebra of matrices of dimension over which contains and use the following nomenclature: The first block will be called *real simple block*, the second *real Jordan block*, the third *simple complex block,* and the fourth *complex Jordan block*.

For , let be the linear sections defined by substituting the matrix in the block of the matrix and taking the other entries as zero. The real Jordan block may be written in the following way , where is the identity and is a nilpotent matrix of order . The simple complex block may be written in the form , where is a diagonal matrix and is a matrix with . The complex Jordan blocks may be written in the form , where is the identity and is a matrix with and is a nilpotent matrix of order .

We define the matrices whose entries are in the block as follows.(i)If is a real simple block, , in this case . (ii)If is a real Jordan block, and for . (iii)If is a simple complex block, and . (iv)If is a complex Jordan block, , , , and for .

Observe that the product of the matrices and is the zero matrix if . The products of and for are as follows.(i)If is a real simple block, . (ii)If is a real Jordan block, then for , where when . (iii)If is a complex simple block, then (iv)If is a complex Jordan block, then for , where when .

The commutativity of the elements in the set with respect to the matrix product, follows from the well-known result: For a Jordan canonical form the diagonal matrix commutes with the nilpotent matrix .

Moreover, the -linear space spanned by is an -algebra, as is claimed in the following proposition.

Proposition 2.3. *The set is a base for an n-dimensional linear space which is an algebra with respect to the matrix product, and its first fundamental representation with respect to is the identity isomorphism. *

*Proof. *Let be the first fundamental representation of associated to . As the algebra is generated by , (considered only when this exists, i.e., ), and (considered only in the case when is a Jordan complex block), in order to prove that is the identity, we only need to prove that , but for these three cases the equality is trivial. So, is the identity isomorphism.

*Definition 2.4. *Given a matrix in its normal form, one will call the algebra in , as constructed above, an *-normal algebra (containing **)*.

*Definition 2.5. *Two matrix algebras and in are *linearly equivalent* if there exists an invertible matrix such that .

For a proof of the following result, see [30].

Proposition 2.6. *Let and be algebras. There exists a product in satisfying
**
where and denote the products in and , respectively. The product is associative and .*

Therefore, the finite tensor product of algebras is an algebra.

*Definition 2.7. *The *complexification* of an -algebra is the -algebra . One calls an algebra which is the complexification of an -normal algebra a *-normal algebra*.

As usual, if the context is clear, one will drop the from the name and refer to the -normal algebra just as a normal algebra. The following proposition and its corollary follow from a straightforward calculation.

Proposition 2.8. *Let and be and -dimensional matrix algebras in and , respectively, and and be invertible matrices. Then, one has
*

Corollary 2.9. *The tensor product of matrix algebras linearly equivalent to normal algebras is algebras which are linearly equivalent to the tensor product of normal algebras. *

So, by Corollary 2.9 the algebras linearly equivalent to the tensor product of normal algebras are closed under the tensor product.

The following result shows that the first fundamental representation, with respect to an appropriate base, of a tensor product of normal algebras is the inclusion of the algebra in the corresponding matrix space.

Proposition 2.10. *Let and be and -dimensional -algebras, and and be first fundamental representations associated to the basis and , respectively. Then, defined by is the first fundamental representation of associated to the base . *

*Proof. *Let be a base of and let be a base of . We use the notations and for the matrices and , respectively, for , . The set
is a base for (see [30]). In order to find the structure constants of we take the products
from which we obtain a first fundamental representation of , where is the matrix whose entry is given by , where and are the entries and of and for and , respectively.

On the other hand we have that . The tensor product of and is given by
from which we see that in the position the element appears. Thus, we have the equalities of matrices , for . Therefore, .

Corollary 2.11. *If is a matrix algebra in which is a tensor product of -normal algebras, then there exists a base of in which the corresponding first fundamental representation is the identity isomorphism, that is, for all . *

*Proof. *We have that , where is a -normal algebra for every . Obviously, for every we can consider a base for as that given in Proposition 2.3, and taking the corresponding tensor products of these basis we obtain a base for . By Propositions 2.3 and 2.10 we have that the first fundamental representation of associated to is the identity.

#### 3. Differentiability on Algebras

In a paper published in 1893, Sheffers laid a foundation for a theory of analytic functions on algebras, see [27, 29] and references therein.

Differentiability on algebras is a stronger concept than the usual differentiability over . If is a differentiable map in the open set , we denote by its Jacobian matrix at the point , in the standard base of . We also use the notation for the Jacobian matrix at the point of with respect to a base .

The following definition was introduced in [26].

*Definition 3.1. *Let be an algebra and let be a map defined in the open set . One says that is *-differentiable* at if there exists an element , which one calls the *-derivative of ** at *, satisfying
where denotes the product in of with . If is -differentiable at all the points of , one says that is *-differentiable on * and one calls the map assigning to the point the *-derivative of *, or *Lorch derivative of **. *

It follows (see, e.g., [26–29]) that a map is -differentiable at if and only if and is continuous as a function of , where is a base of and is the first fundamental representation of the algebra associated to .

In fact, in this case, is the image of under the map , that is, if is -differentiable, then where and for .

If , , and is defined by then is -differentiable and its differential at is given by , thus, the relation between the Jacobian matrix of and the -differential of is .

The matrix equation (3.2) is equivalent to the equations for all . Using (3.4) and the associativity of the algebra, we can obtain appearing in [29, p. 646].

*Remark 3.2. *It should be noted that in the context of algebras equations (3.5) play the same role as the Cauchy-Riemann equations in the case of one complex variable and thus serve as a criterion for analyticity, see [27, 28, 31–33].

Suppose we consider a basis of , where for , , it can be proved that , where . If we denote by the image of under the first fundamental representation of associated to , we have for , that .

*Remark 3.3. *For the differentiation of algebras the usual properties of differentiation of functions from to remain true. Furthermore, the usual rules of differentiation of functions of one variable are satisfied in the case of algebras, therefore polynomial functions, rational functions, and those expressed by means of convergent power series as the exponential, trigonometric, and other usual functions are differentiable in algebras.

The following theorem gives conditions that ensure the existence of an algebra in which is -differentiable.

Theorem 3.4. *Let be a map defined in the open set . is -differentiable for an algebra if and only if the set of matrices
**
is a subset of an algebra which is linearly equivalent to an algebra which is a finite tensor product of normal algebras in . Moreover, is a -linear space and has a base such that the image of the first fundamental representation of associated to is . *

*Proof. *Let be a base for as given in Corollary 2.11. Then is a base for , where is a matrix such that . Because for every , we have , where are functions and .

Now consider the base of defined by , where and is the standard base of . Then, we have that . Thus,
in other words , which means that if we define a product between the elements of using the structure constants of the products of the elements of , we have that is an algebra such that its first fundamental representation associated to is that given by for . Therefore is -differentiable.

#### 4. Reduction to a Variable in the Algebra

Sometimes can be expressed as a function of a variable of an algebra whose image under the first fundamental representation is the tensor product of normal algebras (as in the previous sections). In this section we show some necessary conditions for this to be true and also allow for the expression of in terms of .

Proposition 4.1. *With the same hypothesis as Theorem 3.4, there exists a basis of invertible elements of the algebra such that the following diagram commutes
**
where is the matrix associated to the change of basis , where is the canonical basis of . *

*Proof. *By a change of basis one has the following commutative diagram
where is the matrix associated to the change of basis from to , which was used in the previous theorem, and . Now consider a base of regular elements of , where without loss of generality . Then if is the matrix associated to the change of basis from to , let . Note that the domain of is .

Theorem 4.2. *Let be a -differentiable map over an algebra with first fundamental representation , then can be expressed in a variable , for some regular basis . Moreover there is a partial differential operator such that . *

*Proof. *By Proposition 4.1 there is a basis of regular elements of the algebra , with , without loss of generality we may assume that . Introducing the variable in the base as , we then have , with , where is the matrix associated to the change of basis to .

Denote by the *k-conjugate* of related to , which is defined for by
Then we have
and since
then,
Furthermore one can define the differential operators
which satisfy

If is -differentiable, is also -differentiable, hence
and since
then for
so the right hand expression of does not depend on the conjugated variables. In this way we obtain an expression depending only on , and not on for .

Moreover, by (4.7), (4.10), and (4.11) one has that

In the following example we show how we can reduce the variables of a map by substituting for variables in an algebra.

*Example 4.3. *Let . Then
the is in normal form, hence is -differentiable. If
and is the identity, the multiplication in the associated algebra is given by multiplication of matrices , , and representing, respectively, , where is the canonical basis of . It is easy to see that and do not have inverse in the algebra . Consider the basis of regular elements
we see that the matrix associated to the change of basis from to is
so is transformed to ,
Thus
is the variable in the algebra . Now consider the conjugates
Then
so by substituting this in and simplifying we obtain
the function in the variable of the algebra.

#### 5. Solving Systems of Ordinary Differential Equations by Reduction to One Variable on an Algebra

By following [18, 19], we obtain, as direct corollaries of Theorem 3.4, Proposition 4.1, and Theorem 4.2, the following results.

Corollary 5.1. *Let
**
be as in Theorem 4.2. Then there exists which is -differentiable on , where is a singular set, such that
**
with . *

*Proof. *We need to show that there exists such that
Noticing that
and by Remark 3.3, it follows that
for , where
with being the regular elements of .

*Remark 5.2. *Note that can be many things. For instance if with , then . On the other hand, if is the polynomial function with singular, then may be regular for all , (see [26, p. 418]), in which case .

*Remark 5.3. *As a special case one notices that if then will be a complex analytic function on hence consists of isolated points (the isolated singularities of ). This has been studied in [18, 19].

*Remark 5.4. *In case that , then is an open dense set in . This is true since the set is an open dense set in hence for a continuous one has that is an open dense subset of .

*Remark 5.5. *By the previous remark, in case that , then exists for even though it might be a multivalued function defined on each component of .

*Remark 5.6. *Further characterization and properties of will be studied elsewhere. In what follows we assume that .

Let , then consider the projection onto

Corollary 5.7. *Let be -differentiable in an algebra with first fundamental representation . Then the solutions to the differential equation,
**
correspond to the level curves of the function
**
moreover the real-valued function,
**
is a linear function of along the solutions of (5.8). *

*Proof. *By Theorem 3.4, Proposition 4.1, and Theorem 4.2, the solutions of (5.8) are in correspondence with the solutions of
On the other hand, Corollary 5.1 shows that
the result follows immediately by applying and to (5.12).

Corollary 5.8. *In particular, to visualize the trajectory that passes through the point at time , one needs only to plot the level curve . Moreover one can find explicitly the point , for , as the intersection of the level curve and the hypersurface . *

*Example 5.9. *Consider the function
where
Let
and notice that
with
Hence belongs to the normal algebra , so the previous results are available.

Thus
with
is just a (complex) analytic function, since
so by letting , we have

By Corollary 5.1 one has
so that
Note that in one has hence the level curves of are in correspondence with the level curves of . So
is a *constant of motion * associated to and the one associated to is
So by Corollary 5.8 the trajectories associated to
are the level curves of . These are presented in Figure 1.

In order to parametrize the solution we need to calculate which turns out to be According to Corollary 5.8 we proceeded to calculate the intersection the level curves and for . The results are shown in Table 1 and as black dots in Figure 1.

#### Acknowledgments

The first author wishes to thank Jesús Muciño-Raymundo for introducing him to the study of complex vector fields and for many interesting, insightful, and productive discussions. This work partially funded by UABC Grant no. 0196, and by CONACYT Grant CB-2010/150532.

#### References

- L. Bers,
*Theory of Pseudo-Analytic Functions*, Institute for Mathematics and Mechanics, New York University, New York, NY, USA, 1953. - L. Bers, “An outline of the theory of pseudoanalytic functions,”
*Bulletin of the American Mathematical Society*, vol. 62, pp. 291–331, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Düz and K. Koca, “A Dirichlet problem for generalized analytic functions,”
*Selçuk Journal of Applied Mathematics*, vol. 7, no. 1, pp. 9–15, 2006. View at Google Scholar · View at Zentralblatt MATH - S. B. Klimentov, “BMO classes of generalized analytic functions,”
*Vladikavkazskiĭ Matematicheskiĭ Zhurnal*, vol. 8, no. 1, pp. 27–39, 2006. View at Google Scholar - V. Kravchenko, “On the relationship between
*p*-analytic functions and Schrödinger equation,”*Zeitschrift für Analysis und ihre Anwendungen*, vol. 24, no. 3, pp. 487–496, 2005. View at Publisher · View at Google Scholar - A. Meziani, “Elliptic planar vector fields with degeneracies,”
*Transactions of the American Mathematical Society*, vol. 357, no. 10, pp. 4225–4248, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Le Thu Hoai and W. Tutschke, “Associated spaces defined by ordinary differential equations,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 25, no. 3, pp. 385–392, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Tutschke, “Interactions between partial differential equations and generalized analytic functions,”
*Cubo*, vol. 6, no. 1, pp. 281–292, 2004. View at Google Scholar · View at Zentralblatt MATH - W. Tutschke, “Generalized analytic functions in higher dimensions,”
*Georgian Mathematical Journal*, vol. 14, no. 3, pp. 581–595, 2007. View at Google Scholar · View at Zentralblatt MATH - I. N. Vekua,
*Generalized analytic functions*, Pergamon Press, Addison-Wesley, London, UK, 1962. - Z.-l. Xu, “Mixed elastico-plasticity problems with partially unknown boundaries,”
*Acta Mathematicae Applicatae Sinica (English Series)*, vol. 23, no. 4, pp. 629–636, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Zabarankin and P. Krokhmal, “Generalized analytic functions in 3D Stokes flows,”
*The Quarterly Journal of Mechanics and Applied Mathematics*, vol. 60, no. 2, pp. 99–123, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Zabarankin, “The framework of
*k*-harmonically analytic functions for three-dimensional Stokes flow problems. I,”*SIAM Journal on Applied Mathematics*, vol. 69, no. 3, pp. 845–880, 2008. View at Publisher · View at Google Scholar - B. Ziemian, “Generalized analytic functions with applications to singular ordinary and partial differential equations,”
*Dissertationes Mathematicae (Rozprawy Matematyczne)*, vol. 354, 100 pages, 1996. View at Google Scholar · View at Zentralblatt MATH - K. Hockett and S. Ramamurti, “Dynamics near the essential singularity of a class of entire vector fields,”
*Transactions of the American Mathematical Society*, vol. 345, no. 2, pp. 693–703, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Newton and T. Lofaro, “On using flows to visualize functions of a complex variable,”
*Mathematics Magazine*, vol. 69, no. 1, pp. 28–34, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Garijo, A. Gasull, and X. Jarque, “Local and global phase portrait of equation $\dot{z}=\text{}f(z)$,”
*Discrete and Continuous Dynamical Systems A*, vol. 17, no. 2, pp. 309–329, 2007. View at Google Scholar · View at Zentralblatt MATH - A. Alvarez-Parrilla, A. Gómez-Arciga, and A. Riesgo-Tirado, “Newton vector fields on the plane and on the torus,”
*Complex Variables and Elliptic Equations*, vol. 54, no. 5, pp. 440–461, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Alvarez-Parrilla, J. Mucino-Raymundo, S. Solorza-Calderon, and C. Yee-Romero,
*Complex Analytic Vector Fields: Geometry, Dynamics and Visualization*, 2009. - H. E. Benzinger, “Julia sets and differential equations,”
*Proceedings of the American Mathematical Society*, vol. 117, no. 4, pp. 939–946, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. von Haeseler and H. Kriete, “The relaxed Newton's method for rational functions,”
*Random and Computational Dynamics*, vol. 3, no. 1-2, pp. 71–92, 1995. View at Google Scholar · View at Zentralblatt MATH - F. von Haeseler and H. O. Peitgen, “Newton's method and complex dynamical systems,”
*Acta Applicandae Mathematicae*, vol. 13, no. 1-2, pp. 3–58, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Smale, “A convergent process of price adjustment and global Newton methods,”
*Journal of Mathematical Economics*, vol. 3, no. 2, pp. 107–120, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Kreyszig,
*Introductory Functional Analysis with Applications*, John Wiley & Sons, 1978. - W. C. Brown,
*Matrices Over Commutative Rings*, Marcel Dekker, New York, NY, USA, 1993. - E. R. Lorch, “The theory of analytic functions in normed Abelian vector rings,”
*Transactions of the American Mathematical Society*, vol. 54, pp. 414–425, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. A. Ward, “From generalized Cauchy-Riemann equations to linear algebras,”
*Proceedings of the American Mathematical Society*, vol. 4, no. 3, pp. 456–461, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. D. Wagner, “The generalized Laplace equations in a function theory for commutative algebras,”
*Duke Mathematical Journal*, vol. 15, pp. 455–461, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. W. Ketchum, “Analytic functions of hypercomplex variables,”
*Transactions of the American Mathematical Society*, vol. 30, no. 4, pp. 641–667, 1928. View at Publisher · View at Google Scholar - R. S. Pierce,
*Associative Algebras*, Springer, Berlin, Germany, 1982. - L. V. Ahlfors,
*Complex Analysis, An Introduction to the Theory of Analytic Functions of One Complex Variable*, McGraw-Hill, New York, NY, USA, 3rd edition, 1979. View at Zentralblatt MATH - R. E. Greene and S. G. Krantz,
*Function Theory of One Complex Variable*, vol. 40 of*Graduate Studies in Mathematics*, American Mathematical Society, Providence, RI, USA, 2nd edition, 2002. - S. G. Krantz,
*Function Theory of Several Complex Variables*, 1992.