Elementary Complex Research Articles

Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy--Riemann Equations

Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy--Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential system embodied by the Cauchy--Riemann equations. Here I show in general how this may be accomplished by purely algebraic means. Several examples are given, for functions with increasing levels of complexity. The development of these ideas within the Mathematica software system is also presented. This approach could easily serve as an alternative in the early development of complex variable theory.

Walks confined in a quadrant are not always D-finite

We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z 2 , and always stay in the quadrant x⩾0, y⩾0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,−1),(−1,2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis.

Ultrasonic waves propagation in absorbing thin plates: Application to paper characterization

Guided wave theory is applied to a thin orthotropic and absorbing plate for low frequency propagation of ultrasonic waves. The aim of this paper is to give some physical interpretations of the non-destructive characterization of paper materials, which are cellulosic fibrous networks. It is shown that the propagation problem reduces to two normal modes of propagation in the plane of the plate. Each of them depends on four complex and independent stiffnesses that are combinations of elementary complex stiffnesses of the media. The imaginary part of these stiffnesses corresponds to a possible mechanism of energy dissipation during the wave propagation for this kind of material. The reverse problem, which gives four complex values, is then numerically solved using a small attenuation assumption. The specially designed experimental set-up has led to the first measurements of tracing paper damping factors. The phase velocity measurements of the plate waves agree with the results already found by several paper researchers. As a particular and new result, the shear wave velocities are found to present a quasi-isotropic repartition in the plane of the paper sheet. It was found that the absorbing phenomenon can occur for each propagation mode in such a material. The attenuation values are small, except for one of them that corresponds to a coupling term in the propagation model. The anisotropy of their repartition is also shown in the case of quasi-longitudinal waves.

Family of unified complex Hadamard transforms

Novel discrete orthogonal transforms are introduced in this paper, namely the unified complex Hadamard transforms. These transforms have elements confined to four elementary complex integer numbers which are generated based on the Walsh-Hadamard transform, using a single unifying mathematical formula. The generation of higher dimension transformation matrices is discussed in detail.

Uniqueness of the Newman–Janis Algorithm in Generating the Kerr–Newman Metric

After the original discovery of the Kerr metric, Newman and Janis showed that this solution could be ``derived'' by making an elementary complex transformation to the Schwarzschild solution. The same method was then used to obtain a new stationary axisymmetric solution to Einstein's field equations now known as the Kerr-newman metric, representing a rotating massive charged black hole. However no clear reason has ever been given as to why the Newman-Janis algorithm works, many physicist considering it to be an ad hoc procedure or ``fluke'' and not worthy of further investigation. Contrary to this belief this paper shows why the Newman-Janis algorithm is successful in obtaining the Kerr-Newman metric by removing some of the ambiguities present in the original derivation. Finally we show that the only perfect fluid generated by the Newman-Janis algorithm is the (vacuum) Kerr metric and that the only Petrov typed D solution to the Einstein-Maxwell equations is the Kerr-Newman metric.

Voltammetric metal speciation in colloidal latex dispersions

Two types of polystyrene latices, one with surface carboxylic groups and the other covered with a polymer shell containing carboxylic and sulphonic groups, were analyzed in terms of their metal ion binding properties. In-situ voltammetric experiments were carried out for zinc(II), cadmium(II) and lead(II) to characterize the thermodynamics and kinetics of their association with the surface groups in the latex particles. The kinetics were investigated experimentally through the time-scale dependencies, and set against theoretical predictions [De Jong, H. G. and Van Leeuwen, H. P., J. Electroanal. Chem., 1987, 234, 17; De Jong, H. G. and Van Leeuwen, H. P., J. Electroanal. Chem., 1987, 235, 1] on the basis of rigorous lability criteria and elementary complex formation rate constants. The core-shell latices appear to form relatively weak and labile complexes. In the case of highly charged latices, the metal/carboxylate surface complexes are stronger and lose lability at a high degree of dissociation.

Exponential Decay of Truncated Correlation Functions Via the Generating Function: A Direct Method

We consider statistical mechanics lattice models where the external field dependent partition function can be represented as a standard polymer system. Using this polymer representation and elementary complex analytic arguments, we obtain upper bounds and give a simple proof on the uniform (in n) exponential decay of the n-point truncated correlation function. We illustrate the method by applying it to the high and low temperature Ising model and to contour models.

Reaction Dynamics and Chemical Pattern Formation in Capillary Tubes Resulting from the Competition Between Two Elementary Complex Formation Reactions

AbstractA system of competing elementary reactions is investigated experimentally usingthe reaction of xylenol orange with Cr3+ in aqueous solution. The two reagents areinitially separated in a long, thin capillary tube and meet in the center, forming a reactionfront (s). The geometry of the reactor and the initial separation of the reagents makes the system effectively one-dimensional. Aqueous Cr3+ solution has a very rich chemistryand provides two different chemical Cr3+ reactants which compete to react with xylenolorange. Rich spatio-temporal patterns are observed experimentally and are explained by a reaction-diffusion model. Results from exact enumeration simulations predict that when the concentrations of the competing species are very different and the microscopic rate constants of the competing species are such that the majority species reaction rate is much faster than the reaction rate of the minority species, the reaction front splits into two distinct regions. The spatio-temporal patterns generated by theory and experiment agree quantitatively. Also in agreement with the theory are the experimental early time and asymptotic time global rate behaviors, which exhibit multiple crossovers.

A Non-Constant Continuous Function on the Plane Whose Integral on Every Line Is Zero

for every (doubly infinite) line l, where s denotes length measure, then f must be identically zero. This has been known for many years (see [5] for references), but only comparatively recently was it shown that this result fails in the absence of the global integrability condition. In fact, Zalcman [S; pp. 243-244], using a theorem of Arakelian [1; p. 1189] concerning tangential holomorphic approximation on unbounded sets, constructed a non-constant entire function t satisfying (1) on every line 1. The purpose of this note is to give a construction, similar to Zalcman's, but using only elementary complex analysis. As a bonus all the derivatives of the function we produce also have vanishing integrals on all lines. The main idea is to construct a non-constant entire function g whose derivatives satisfy

Numerical Methods for Simultaneous Diagonalization

A Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations is presented. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the off-diagonal entries to zero. Its asymptotic convergence rate is shown to be quadratic and numerically stable. It preserves the special structure of real matrices, quaternion matrices, and real symmetric matrices.

Seizures Induced by Music

Musicogenic epilepsy is a rare disorder. Much remains to be learned about the electroclinical features. This report describes a patient who has been followed at our institution for 17 years, and was investigated with long-term telemetered simultaneous video-EEG recordings. She began to have seizures at the age of 10 years. She experienced complex partial seizures, often preceded by elementary auditory hallucination and complex auditory illusion. The seizures occurred in relation to singing, listening to music or thinking about music. She also had occasional generalized tonic clonic seizures during sleep. There was no significant antecedent history. The family history was negative for epilepsy. The physical examination was unremarkable. CT and MRI scans of the brain were normal. During long-term simultaneous video-EEG recordings, clinical and electrographic seizure activities were recorded in association with singing and listening to music. Mathematical calculation, copying or viewing geometric patterns and playing the game of chess failed to evoke seizures.

Approximate inversion of high‐frequency electromagnetic soundings using complex image theory

Electromagnetic (EM) soundings in an intermediate range between quasi‐static and radar frequencies (e.g., 30 kHz – 300 MHz) were studied with an approximate inversion method for shallow‐layered earth models using complex image theory. The half‐space image theory formulas for calculating forward soundings for any dipole source are easily extended to accommodate a multilayered earth, and only involve elementary complex functions. Approximate inversion using the modified image formulas is compared with more exact numerical integration inversion for horizontal‐layered models to yield the conductivity, thickness, and dielectric permittivity of each layer. A nonlinear least‐squares algorithm was used to obtain parameter and linear parameter error estimates. The results, using known exact forward models and one field data case, indicate that approximate image inversion is more than an order‐of‐magnitude faster than numerical integration inversion. However, approximate inversion could not always resolve deeper layer parameters as well as with exact inversion. Accuracy of both methods, however, depends on the frequency range used. Nevertheless, approximate inversion can be a useful quick reconnaissance tool, and could easily be written for on‐site personal computers.

The Cauchy Integral Formula

Our purpose in this note is to illustrate the Cauchy integral formula and the concept of winding number. The software we use is fiz), ver. 4.0, Martin Lapidus, Lascaux Graphics, Bronx, NY, 1988. Many mathematicians and most students are at a loss to picture the graph of even the most elementary complex function. This tutorial and graphics program enhances geometric intuition about functions of a complex variable by graphing lines, circles, or other figures in a z plane and simultaneously graphing their images in a w =/(z) plane. Other domain options include hyperbolic lines on the Poincare plane and circles on the Riemann sphere. You can construct and save a page with two or more windows, sized, scaled, and positioned as you wish, and with domain options, functions, and operations specified. Once you have con? structed a page, you can request that the operations be carried out at the press of a draw key. The Cauchy integral formula states that

A Visit to the Newtonian N-body Problem via Elementary Complex Variables

Click to increase image sizeClick to decrease image size This article is part of the following collections: Chauvenet Prize: 1980s and 1990s Additional informationNotes on contributorsDonald G. SaariDon Saari received his undergraduate degree from Michigan Technological University and his Ph.D. in mathematics from Purdue University. His Ph.D. dissertation, written under the guidance of Harry Pollard, concerned the qualitative behavior of dynamics in the general Newtonian N-body problem. After a postdoctoral position in the Yale Astronomy Department, Saari moved to Northwestern University where he currently is Professor of Mathematics. His main research interests revolve around applications of dynamical systems to celestial mechanics and physics and to issues coming from the social sciences.

A Visit to the Newtonian N-Body Problem via Elementary Complex Variables

The study of how N celestial bodies move under gravitational forces is an old one. If one is willing to acknowledge the work of the ancient astrologers and shepherds ― two groups that carefully plotted the positions of the stars and planets ― then this subject area traces its origins to the earliest reaches of mankind. Indeed, had the title not been already preempted, one might suggest that the study of the N-body problem is «the world's oldest profession». If it isn't the oldest, then, most surely, it is «the second oldest»

Location of essential singularities of a class of analytic functions

A simple method, based on the numerical evaluation of appropriate complex integrals on a closed contour by using appropriate numerical integration rules, is proposed for the location of essential (isolated and non-isolated) singularities of a class of analytic functions in the complex plane. This class of functions consists of those functions where a term in the corresponding Laurent series (in the sum of the negative powers) is missing near the essential singular point and, therefore, it includes odd and even functions among a variety of such functions. The whole approach makes use of the Cauchy residue theorem in elementary complex analysis and requires the values of the analytic function only on a closed contour surrounding the singular point. Numerical results for some elementary transcendental functions are also presented. The present results generalize previous relevant results for zeros and poles of analytic functions and can further be generalized themselves.

The Broken Spiral Theorem

One of the classical problems (Abel's Test) set for students in elementary complex analysis is a generalization of the Alternating Series Theorem. It concerns the series ?00=Oa,,z , where {a,,) is a sequence of positive numbers decreasing to 0 and Izl = 1. The proof that the series converges except possibly at z = 1 is usually done by summation by parts, but this proof does not probe the essence of the result. For many years, I have been using a geometric proof of this theorem, which does not appear in the literature. I offer it here. I exclude the case z = -1, which is the Alternating Series Theorem. I denote the sum of the first n terms of the series by sn, and join consecutive values by a line segment, which will have length a,,. Consecutive segments meet at an angle of T larg(z) j. Let each of these angles be bisected, the bisectors at the ends of the n th segment meeting at a point called c,, so that the angle csn, 1c,,sn, = argz. In each case, the equal sides sn-Icn and cnsn are of length r, = tan, where t = 2 cscjI arg zj. Let Bn be the closed disc with center Cn and radius r,,.

On the zeros of Bessel polynomials

A closed expression for the reciprocal power sums of the difference of zeros of Bessel polynomials is derived using elementary complex analysis. These are sums of the form ∑ k=1 k≠j n (x j−x k) −m; m=1,2..., where x j are the zeros of a Bessel polynomial. Recurrence formulae for sums ∑ i − 1 n x j − m and ∑ i − 1 n x j m are also established.

A Class of Soluble Models for Bound-Free Transitions

We investigate light-induced bound-free transitions in the presence of radiative recombination in models of continuum resonances for which the only singularities of the coupling function are a finite number of simple poles in the complex plane. Using elementary complex analysis, finding the time-dependent solution to the ensuing set of integro-differential equations is reduced to a study of sets of linear algebraic equations and roots of a polynomial. As an example we treat bound-free Rabi flopping for a Lorentzian resonance. We suggest that the extremely restrictive conditions for observing coherent transients, a narrow resonance and a large dipole moment matrix element, may be met in two-electron excitation of an atom.

Elementary complex number analysis of lung models.

Formulae from the complex number method of network analysis are used by respiratory physiologists with increasing frequency. As yet, however, no elementary derivation of these has been found in physiological journals, the reader usually being referred to electronic texts: the derivations in these texts are not directed to respiratory problems and are generally more involved than required for respiratory purposes. The present presentation of the complex number analysis method is elementary, the necessary grounding in the field of complex numbers being given in an Appendix. The analysis is directed first to a one-compartment lung, then to a simple two-compartment lung (effectively the same method applying also to simple multiple-compartment lungs), and finally an indication is given of the analysis of more complex models that include air-compression effects. By gaining an understanding of the underlying principles the respiratory physiologist will improve the extent and depth of application of this powerful method of analysis. In particular, he will be in a position to analyse the body plethysmographic measurement of the non-uniform lung.