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Most of the circuits introduced so far have been analyzed in time domain. This means that the input to the circuit, the circuit variables, and the responses have been presented as a function of time. All the input functions such as unit step, ramp, impulse, exponential, sinusoidal, etc. have been introduced as a time-dependent variable, and their effects on circuits have been identified directly as a function of time. This required utilization of differential equations and solutions in time domain. However, high-order circuits result in high-order differential equations, which, considering the initial conditions, sometimes are hard to solve. In addition, for circuits which are exposed to a spectrum of frequencies such as filters, the time domain analysis is a limiting factor.

This paper presents a novel approach, based on the theory of hyperplanes, for mode identification of linear systems. The proposed approach can operate on either a set of ordinary differential equations (converted to diagonal form, if needed) or a set of partial fractions derived from a synthesized transfer function of the system under analysis. For either format, the linear system is structured to have as unknown variable a vector containing the residues. Singular value decomposition is initially used to identify an initial sparsity of the residue vector where the number of nonzero values corresponds to the pre-defined order of the dominant poles (eigenvalues) under search. An algorithm based on geometrical search of hyperplanes is used to optimize the selection of the nonzero residue locations, minimizing the residual of the zero residue hyperplanes. Finally, a recalculation of the residues is carried out by using the obtained optimal sparsity.

In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].

In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.

For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.

Summary The Z-transform of a complex time signal (or the analytic signal of a real signal) is equal to the Z-transform of a prediction error divided by the Z-transform of the prediction error operator. This inverse is decomposed into a sum of partial fractions, which are used to obtain impulse response operators formed by non-causal filters that complex-conjugate symmetric coefficients. The time components are obtained by convolving the filters with the original signal, and the peak frequencies, corresponding to the poles of the prediction error operator, are used for mapping the time components into frequency components. For non-stationary signals, this decomposition is done in sliding time windows, and the signal component values, in the middle of each window, are attributed to the peak value of its frequency response that corresponds to the pole of this partial fraction component. The result is an exact, but non-unique, time–frequency representation of the input signal. A sparse signal decomposition can be obtained by summing along the frequency axis in patches with similar characteristics in the time–frequency domain. The peak amplitude frequency of each new time component is obtained by computing a scalar prediction error operator in sliding time windows, resulting in a sparse time–frequency representation. In both cases, the result is a time–frequency matrix where an estimate of the frequency content of the input signal can be obtained by summation over the time variable. The performance of the new method is demonstrated with excellent results on a synthetic time signal, the LIGO gravitational wave signal and seismic field data.

In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since the smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.

It is proved that the sums $$ \sum_{k=1}^{n} \frac{1}{(z-a_{k})^{2}}\mspace{2mu}, \qquad \operatorname{Im}a_{k} < 0, \quad n \in \mathbb{N}, $$ are dense in all Hardy spaces $$H_{p}$$ , $$1

A dependable approach for inverting two classes of rational Laplace transforms, involving regular polynomials and partial sums with non-integer exponents is developed. Such types of Laplace transforms frequently emerge as the system transfer functions in the analysis of feedback control loops or process dynamics. The proposed method systematically translates the Laplace inversion problem into an integer or fractional order differential equation and yields the analytical inverse Laplace transform function utilizing the Adomian decomposition method. The method is especially useful in dealing with high-order rational transfer functions; where approaches based on the partial fraction expansion and the residue inversion theorem lose their practicality. The ready-to-use inversion formulas are presented and their usability and reliability is demonstrated through a number of case studies.

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $$\wp (z|\tau )$$ as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups $$\Gamma _0(N)$$ and $$\Gamma _1(N)$$ .

Graphical abstract, Highlights • It presents a partial fraction decomposition of non commensurate systems. • Suitable inversion of each fraction is done in two ways: series and integer/fractional decomposition., A study of non-commensurate fractional linear system is done in a parallel way to the commensurate case. A partial fraction decomposition is accomplished using a recursive procedure. Each partial fraction is inverted in two different ways. The decomposition integer/fractional is done also. Some examples are presented.

This research aims to explore students’ difficulties in resolving Nonhomogeneous 2nd Order Ordinary Differential Equations with initial value problems. The method that can be used to solve this equation is the undetermined coefficient and the Laplace transformation. This research is used descriptive method. The subjects of this study were 73 students in the second year of the Mathematics Education. Data is collected through tests and interviews. Data were analyzed descriptive qualitative. The results of data analysis show that in undetermined coefficient method, students difficult in determining the particular solution of non-homogeneous second-order ordinary differential equations. This is due to student errors in the first step especially in determining the characteristics equation. Whereas, for the Laplace transformation method, students most difficulties are in the step of solving the subsidiary equation. This is due to the weakness of students in completing arithmetic operations in the form of fractions and partial fractions.

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation., 12 pages. Full account of the author's plenary talk at Combinatory Analysis 2018: in Honor of George Andrews' 80th Birthday, June 24, 2018

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

In this paper, iterative procedure for partial fraction approximation of vector functions in Laplace domain is presented. First, the essentials of the original formulation of vector fitting methodology for strictly proper rational approximations are outlined. Then a modification is presented for more efficient application to vector functions. Proposed modification reduces the problem of fitting all elements of vector function to fitting only some of them and sum of the rest. After the set of common poles is obtained, the residues are calculated for all functions as usual. Next, the algorithm for adaptive rational approximation of vector functions is thoroughly presented. Proposed technique consists of successive determination of partial fraction approximations with increasing orders until the specified convergence criterion is satisfied. Detailed results of the numerical example are provided to assess accuracy and efficiency of the proposed procedure.

Many efforts have been made in order to identify state-space models of the black-box devices from their terminal admittance/impedance frequency responses. However, the band of the available admittance/impedance frequency responses is some-times relatively narrow, from which some out-of-band critical modes of interest can thus not be intuitively observed. In order to overcome this limitation, the vector fitting (VF) algorithm is used in this paper to identify some out-of-band critical modes of an artificially created rational function. On its basis, the sensitivity index which depicts the effect of the partial fraction term on the in-band fitting error is derived in order to explain the VF’s extrapolation behavior, and its effectiveness is verified on the cases where out-of-band modes of different numbers coexist.

There is a growing interest in fractional calculus and Fractional Order (FO) system modeling in many fields of science and engineering. Utilization of FO models in real-world applications requires practical realization of FO elements. This study performs an analog circuit realization of approximate FO derivative models based on Modified Stability Boundary Locus (M-SBL) fitting method. This study demonstrates a low-cost and accurate analog circuit implementation of M-SBL fitting based approximate model of FO derivative elements for industrial electronics. For this purpose, a 4th order approximate derivative transfer function model of the M-SBL method is decomposed into the sum of first order low-pass filters form by using Partial Fraction Expansion (PFE) method, and the analog circuit design of the approximate FO derivative model is performed. Firstly, by using the final value theorem, authors theoretically show that the time response of the sum of first order low-pass filter form can converge to the time response of fractional order derivative operators. Then, the approximation performance of proposed FO derivative circuit design is validated for various input waveforms such as sinusoidal, square and sawtooth waveforms via Multisim simulations. Results indicate an accurate realization of the FO derivative in time response (an RMSE of 0.0241). The derivative circuit realization of the M-SBL fitting model in the form of the sum of first order low pass filters can yield a better time response approximation performance compared to the Continued Fraction Expansion (CFE) based ladder network realization of the approximate derivative circuit. © 2021 Karabuk University

We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .

In this brief, we consider the minimax design of infinite impulse response (IIR) filter. First, we apply the partial fraction method to decompose the transfer function into a sum of low order fractions, as a result the minimax IIR filter design problem can be formulated with the stability condition as constraints. Second, among different possible decompositions of the transfer function, we prove that the second order decomposition can always achieve good approximation to the optimal solution. Based on the second order decomposition formulation, several numerical experiments have been conducted, and better IIR filters can be designed using the proposed method compared with existing methods in the literature.

Using partial fraction decomposition, limit methods and algebraic techniques, finding the absolutely minimal realization matrices for first-degree (multi-linear) nD SISO systems is transformed into solving a system of equations consisting of the all principal minors of an unknown matrix. Without using the symbolic approach by Grobner basis, the necessary and sufficient conditions and construction of the absolutely minimal realizations for the degenerate and non-degenerate of transfer functions are derived. Five examples for first-degree 3D, 4D, and 5D SISO systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure, and computer programs written in Mathematica are also given.

This paper presents an efficient approach to enforce passivity of rational-based fitted systems. The origin of non-passivity is thoroughly reviewed and analyzed. The proposed approach identifies and modifies, from the original non-passive rational approximation, only the partial fractions that generate passivity violations. It is shown in this paper that these partial fractions generally correspond to out-of-band poles. As for proper-type rational approximations, a new method is proposed to force the constant matrix to be positive-definite. The proposed approach is verified via three case studies with highly non-passive conditions.

It is known that for the best uniform approximation of real constants c by simple partial fractions ρn of order n on a real segment it is necessary and sufficient to have an alternance of n + 1 points on this segment for the difference ρn − c, n ≥ n0(c). We propose an algorithm for constructing the alternance and simple partial fraction of the best approximation of constants c.

Sums of products of two Gaussian q-binomial coefficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences.

By means of partial fraction decomposition, derivative operator and inverse series relations, some algebraic identities involving symmetric functions are established. As applications, they are specialized to several summation formulae containing harmonic numbers and q-harmonic numbers.

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature.

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPFs) as suggested by Dolzhenko. Many solved and unsolved extremal problems, related to SPFs, are traced back to works of Boole, Macintyre, Fuchs, Marstrand, Gorin, Gonchar, and Dolzhenko. Now many authors systematically develop methods for approximation and interpolation by SPFs and their modifications. Simultaneously, related problems, being of independent interest, arise for SPFs: obtaining inequalities of different metrics, estimation of derivatives, separation of singularities, etc. In introduction to this survey, we systematize some of these problems. In themain part, we formulate principal results and outline methods to prove them whenever possible.

An asymptotics of the error of interpolation of real constants at Chebyshev nodes is obtained. Some well-known estimates of the best approximation by simple partial fractions (logarithmic derivatives of algebraic polynomials) of real constants in the closed interval [−1, 1] and complex constants in the unit disk are refined. As a consequence, new estimates of the best approximation of real polynomials on closed intervals of the real axis and of complex polynomials on arbitrary compact sets are obtained.

We study uniform approximation of real-valued functions f, f(∞) = 0, on ℝ+ and ℝ by real-valued simple partial fractions (the logarithmic derivatives of polynomials). We obtain a criterion for the best approximation on ℝ+ and ℝ in terms of the Chebyshev alternance. This criterion is similar to the known criterion on finite segments. For the problem of approximating odd functions on ℝ we construct an alternance criterion with a weakened condition on the poles of fractions. We present a criterion for the best approximation by simple partial fractions on ℝ+ and ℝ in terms of Kolmogorov. We prove analogs of the de la Vallee-Poussin alternation theorem.

We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R,S) in terms Macdonald polynomials of type A, are q,t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups, important in the theory of plane partitions. As applications of our results we obtain combinatorial formulas for characters of affine Lie algebras, Rogers-Ramanujan identities for such algebras complementing recent results of Griffin et al., and transformation formulas for Kaneko-Macdonald-type hypergeometric series.

In the chapter, The z-transform, the z-transform is presented. The z-transform is developed starting from the DTFT as a generalization of the Fourier analysis. Extending the set of basis signals at other than the unit-circle, results in the capability to analyze a large set of unbounded signals, which is very useful in stability analysis of systems. Examples of deriving the z-transform of useful signals are given. The properties of z-transform, which make the analysis of signals and systems much simpler, are presented. While the inverse z-transform is defined in terms of contour integration, in practice, the much simpler partial fraction and long-division methods are used. A number of examples of finding the inverse z-transform are presented. The transfer function, which is the ratio of the transform of the output and that of input, is presented with examples. The pole-zero characterization of systems is presented and the stability criterion of systems is given.

In this chapter, the problems of the seventh chapter are fully solved, in detail, step-by-step, and with different methods. The subjects include definite integrals, indefinite integrals, substitution rule for integrals, integration techniques, integration by parts, integrals involving trigonometric functions, trigonometric substitutions, integration using partial fractions, integrals involving roots, integrals involving quadratics, applications of integrals, average value, area between curves, and volume of solid of revolution.

Summary We present a new method for obtaining the time-frequency decomposition of a non-stationary signal. The input signal is modeled as a dynamic short-time autoregressive process. From the analytic signal or complex trace, we compute the AR coefficients or prediction error operator (PEO), in sliding time windows. The inverse of the Z-transform of the PEOs can be represented by a sum of partial fractions, each one related to a single pole. Each pole may be used to deflate the PEO, allowing us to rewrite the AR representation of the signal as a sum of signal components. Also, the position of each pole provides the dominant frequency, which is useful to distribute the signal component in the time-frequency domain. The signal components are obtained by convolving the input signal with the reduced PEOs, scaled by the partial fractions coefficients. The new time-frequency signal decomposition method is demonstrated on synthetic data.

In this chapter, the basic and advanced problems of definite and indefinite integrals are presented. The subjects include definite integrals, indefinite integrals, substitution rule for integrals, integration techniques, integration by parts, integrals involving trigonometric functions, trigonometric substitutions, integration using partial fractions, integrals involving roots, integrals involving quadratics, applications of integrals, average value, area between curves, and volume of solid of revolution. To help students study the chapter in the most efficient way, the problems are categorized based on their difficulty levels (easy, normal, and hard) and calculation amounts (small, normal, and large). Moreover, the problems are ordered from the easiest problem with the smallest computations to the most difficult problems with the largest calculations.

As a natural continuation of the study of polynomials, in this chapter we introduce and discuss rational and algebraic expressions in a wide variety of settings. One of the main objectives of this chapter is to present the partial fraction decomposition in complete details; this is accompanied by a few Olympiad level problems. Asymptotes, briefly alluded to in treating hyperbolas in Section 8.4, are fully and rigorously developed here. Another main objective of this chapter is to extend the AM–GM inequality (Sections 5.4, 7.5) to the multivariate harmonic–geometric–arithmetic–quadratic mean inequalities. The AM–GM inequality along with its extensions is a cornerstone of analysis. It has a beautiful geometry which was known to the ancient Greeks, and it appears in a myriad problems such as multivariate extremal problems, factorization problems, etc. Among the literally hundreds of mathematical contest problems involving these means, we chose a representative sample to demonstrate the principal methods. The lesser known permutation (arrangement) inequality is also introduced here pointing out that it implies all the other classical inequalities such as the AM–GM, Cauchy–Schwarz (Sections 5.3, 6.7), and Chebyshev (Section 6.7) inequalities. Finally, we give a detailed (and somewhat more advanced) account on the greatest integer function along with some of Ramanujan’s formulas, and the Hermite identity.

The Z-transform of the product of two functions with a rational transfer function, also known as the Hadamard product of the rational transfer functions, involves a complicated contour integral. It is especially complicated for functions with poles of multiplicity greater than 1. This paper provides a simple and novel approach to this problem in three simple algebraic steps. The first step involves a Partial Fraction Expansion of the rational transfer functions to split up a complicated fraction into simple partial fractions. The problem gets reduced to the sum of the Hadamard product of the partial fractions of the two functions. The principles of generating functions and combinatorics are used to find a generic formula for the Hadamard product of two partial fractions with poles of multiplicity greater than or equal to 1. Applying this formula and adding the result of the product gives us the solution. Based on the results of the generic formula, some properties of the Hadamard product are derived.

We first establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors by constructing a homogeneous kernel function involving hyperbolic functions. Furthermore, we introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then we present several special and interesting Hilbert-type inequalities, in which the constant factors are represented by Bernoulli numbers and by some higher derivatives of trigonometric functions.

Inspired by the Mordell–Tornheim double zeta function, in this paper, we introduce the Eisenstein series of Mordell–Tornheim type. Based on the theory of Cohen series and partial fraction decomposition, we explicitly compute these series. We end by discussing several identities about products of Eisenstein series and their Fourier expansions.