Partial Fraction Expansion Research Articles

Zero dynamics and root locus for a boundary controlled heat equation

We consider single-input single-output systems whose internal dynamics are described by the heat equation on some domain \(\varOmega \subset {\mathbb {R}}^d\) with sufficiently smooth boundary \(\partial \varOmega \). The scalar input is formed by the Neumann boundary values which are forced to be constant in space; the output consists of the integral over the Dirichlet boundary values. We show that the transfer function admits some partial fraction expansion with positive residues. The location of the transmission zeros and invariant zeros is further analyzed. Thereafter we show that the zero dynamics are fully described by a self-adjoint and exponentially stable semigroup. The spectrum of its generator is proven to be the set of invariant zeros. Finally, we show that any positive proportional output feedback results in an exponentially stable system. We further analyze the root loci: as the proportional gain tends to infinity, the eigenvalues of the generator of the closed-loop system converge to the invariant zeros.

Computation of Dominant Poles and Residue Matrices for Multivariable Transfer Functions of Infinite Power System Models

This paper describes the first reliable Newton algorithm for the sequential computation of the set of dominant poles of scalar and multivariable transfer functions of infinite systems. This dominant pole algorithm incorporates a deflation procedure, which is derived from the partial fraction expansion concept of analytical functions of the complex frequency s and prevents the repeated convergence to previously found poles. The pole residues (scalars or matrices), which are needed in this expansion, are accurately computed by a Legendre-Gauss integral solver scheme for both scalar and multivariable systems. This algorithm is effectively applied to the modal model reduction of multivariable transfer functions for two test systems of considerable complexity and containing many distributed parameter transmission lines.

SYNTHESIS OF GENERALIZED CHEBYSHEV LOSSY BANDSTOP FILTERS WITH NON-PARACONJUGATE TRANSMISSION ZEROS

A systematic procedure is presented for synthesis of generalized Chebyshev lossy bandstop filters with non-paraconjugate transmission zeros. From a lossy scattering parameters with the prescribed reflection zeros, the transformation formulas from the scattering matrix to the admittance matrix are obtained by reconstructing the non-paraconjugate transmission zeros as paraconjugate ones. The canonical transversal array is modeled by partial fraction expansion of the normalized admittance functions, resulting in an increased order of the final network provided there are nonparaconjugate transmission zeros. The methods are simpler and more general than the ones in the literature. So it shows great versatility, and can also accommodate lossless network or a transfer function with symmetrical transmission zeros. To illustrate the proposed synthesis procedure, three typical examples have been carried out to validate the synthesis method.

On Solvents of Matrix Polynomials

This paper is concerned with the construction of right and left solvents (also called block roots) of a matrix polynomial from latent roots and vectors. It addresses also the important case of the existence of a complete set of block roots. Solvents do not always exist, so conditions for the existence of such solvents are discussed. The inverse of a matrix polynomial is obtained as a particular case of the block partial fraction expansion of a related rational matrix. It involves the knowledge of a complete set of solvents and the computation of the inverse of a block Vandermonde matrix. Numerical examples are given to illustrate the two results.

A simple proof of a classical identity

A few well-known infinite partial fraction expansions of some trigonometric functions are derived and Euler's elegant identity ∑∞k = 11/k2 = π2/6 is deduced as a corollary.

A block Hankel generalized confluent Vandermonde matrix

Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, q×q say, in which case the i-th column is given by u(zi), where we write u(z)=(1,z,…,zq−1)⊤. If all the zi (i=1,…,q) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all zi are the same, z say, in which case one could speak of a confluent Vandermonde matrix. Non-singularity is obtained if one considers the matrix V(z) whose i-th column (i=1,…,q) is given by the (i−1)-th derivative u(i−1)(z)⊤.We will consider generalizations of the confluent Vandermonde matrix V(z) by considering matrices obtained by using as building blocks the matrices M(z)=u(z)w(z), with u(z) as above and w(z)=(1,z,…,zr−1), together with its derivatives M(k)(z). Specifically, we will look at matrices whose ij-th block is given by M(i+j)(z), where the indices i,j by convention have initial value zero. These in general non-square matrices exhibit a block-Hankel structure. We will answer a number of elementary questions for this matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on z? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix M(z) and the number of derivatives M(k)(z) that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of matrix polynomials is needed.

A New Proof of Stirling's Formula

A new, simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.

An Easy Pure Algebraic Method for Partial Fraction Expansion of Rational Functions With Multiple High-Order Poles

An easy practical algebraic algorithm was proposed for partial expansion of rational functions with multiple high-order poles. The simple recursive implementation of the proposed method involves neither long division nor differentiation and requires only elementary arithmetic operations. It is suitable for computer or hand calculation of partial expansion of both proper and improper functions with multiple high-order poles with desirable accuracy.

Mathematical modeling of Fabry–Perot resonators: II Uniformly converging multimode equivalent-circuit models

Based on complex-variable analysis of a Fabry-Perot resonator as a multimode nonsymmetric two-port waveguide device, two versions of equivalent-circuit configurations are presented: Starting from a renewed study on single-mode two-pole circuits, we develop two respective multimode equivalent circuits of an almost identical configuration: one for the reflection coefficient and the other for the pass-through transmission coefficient. In the mathematics language of complex-variable analysis, the two models successfully "approximate" the two scattering coefficients through two "uniformly converging" partial-fraction series expansions.

Mathematical modeling of Fabry–Perot resonators: I Complex-variable analysis by uniformly convergent partial-fraction expansion

In the first part of a two-part study on the equivalent-circuit representation of any given Fabry-Perot resonator (FPR) that supports, by nature, infinitely many resonance modes, the complex-variable pole-zero structure of its scattering coefficients is extensively analyzed in general terms through partial-fraction expansion based on a corollary to Mittag-Leffler's theorem for meromorphic functions. By finding the right offset constant in the expansion from the theory, we present two sets of uniformly converging series of partial fractions for the two scattering coefficients. We compare quality of convergence between the two series sets and find that a set obtained by the fraction-reciprocated reflection coefficient for the FPR is relatively better than the other one, which is fortunate for the subsequent work in the second part.

Efficient Recursive Methods for Partial Fraction Expansion of General Rational Functions

Partial fraction expansion (pfe) is a classic technique used in many fields of pure or applied mathematics. The paper focuses on the pfe of general rational functions in both factorized and expanded form. Novel, simple, and recursive formulas for the computation of residues and residual polynomial coefficients are derived. The proposed pfe methods require only simple pure-algebraic operations in the whole computation process. They do not involve derivatives when tackling proper functions and require no polynomial division when dealing with improper functions. The methods are efficient and very easy to apply for both computer and manual calculation. Various numerical experiments confirm that the proposed methods can achieve quite desirable accuracy even for pfe of rational functions with multiple high-order poles or some tricky ill-conditioned poles.

Hearing the Shape of a Triangle

In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general inverse spectral problem we will give a new proof of this fact. The central point of the argument is to show that area, perimeter and the sum of the reciprocals of the angles determine a triangle uniquely. This is proved using convexity arguments and the partial fraction expansion of $\sin^{-2}x$.

Lumped-parameter model of foundations based on complex Chebyshev polynomial fraction

This paper uses the complex Chebyshev polynomials to develop the lumped-parameter (L-P) model of foundations. The method is an important extension to the approach adopting polynomials to model the dynamic properties of the foundation. Using the complex Chebyshev polynomials can reduce the unexpected wiggling in the foundation modeling, which inevitably occurs if using the simple polynomials of high degree. In the present analysis, the normalized flexibility function of foundation is expressed in terms of complex Chebyshev polynomial fraction. Through the partial-fraction expansion the complex Chebyshev polynomial fraction is decomposed into two sets of basic discrete-element models. The parameters in the models are obtained through the least-square curve-fitting to the available analytical or on-site measurement results. The accuracy and validity of the L-P model is validated through the applications to the surface circular foundations, embedded square foundations and pile group foundations, respectively. It is shown that in general the present model has better accuracy and needs fewer parameters than the existing L-P models.

Four derivations of an interesting bilateral series generalizing the series for zeta of 2

We present four derivations of the closed form of the partial fractions expansion This interesting series is a generalization of the series made famous by Euler.

Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

ON HOFFMAN'S RELATION FOR MULTIPLE ZETA-STAR VALUES

The multiple zeta values are natural generalization of Riemann zeta values, which are real numbers first considered by Euler. It is known that there are many linear relations among these values with rational coefficients. Hoffman [Multiple harmonic series, Pacific J. Math.152 (1992) 275–290] gives a class of relations among multiple zeta values, which is called Hoffman's relation. In this paper, we show the corresponding relation for multiple zeta-star values by using partial-fractions expansion as Hoffman did.

Fair and Square Computation of Inverse $ {\cal Z}$-Transforms of Rational Functions

All methods presented in textbooks for computing inverse Z-transforms of rational functions have some limitation: 1) the direct division method does not, in general, provide enough information to derive an analytical expression for the time-domain sequence x(k) whose Z-transform is X(z) ; 2) computation using the inversion integral method becomes labored when X(z)zk-1 has poles at the origin of the complex plane; 3) the partial-fraction expansion method, in spite of being acknowledged as the simplest and easiest one to compute the inverse Z-transform and being widely used in textbooks, lacks a standard procedure like its inverse Laplace transform counterpart. This paper addresses all the difficulties of the existing methods for computing inverse Z -transforms of rational functions, presents an easy and straightforward way to overcome the limitation of the inversion integral method when X(z)zk-1 has poles at the origin, and derives five expressions for the pairs of time-domain sequences and corresponding Z-transforms that are actually needed in the computation of inverse Z -transform using partial-fraction expansion.

Add-on type loop shaping controller design for high precision track following in hard disc drives (HDDs)

In this study, the authors present a systematic design method of the add-on type loop shaping scheme for high precision track following with plant modelling perturbations in HDDs. The proposed controller minimises the standard deviation of the position error signal (PES) of certain closed-loop transfer function while satisfying the robust stability condition in HDDs. Compared with the conventional H ∞ control, the proposed method yields estimated plant states, which can be used for estimator-based controller design. Also, the proposed control method can be implemented in a parallel manner by performing a partial-fraction expansion for easy implementation. The effectiveness and practicality of our work is demonstrated through various experimental results.

A New Differential Operator Method to Study the Mechanical Vibration

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator D in the numerator of the fraction has no effect on input functions (i.e., the derivative operation is removed) because we take the fraction as a whole part in the partial fraction expansion. The method in various variants is widely implemented in related fields in mechanics and engineering. We also point out that the same mistakes in the differential operator method are found in the related references [1-4].

Special values of the Hurwitz zeta function via generalized Cauchy variables

As a continuation of the work of Bourgade, Fujita, and Yor, we show how to recover the extension of the Euler formulae concerning some special values of the Hurwitz zeta function from the product of two, and then N, independent generalized Cauchy variables. Meanwhile, we consider the ratio of two independent generalized Cauchy variables and give another proof of the partial fraction expansion of the cotangent function.