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It is well known that input impedance of cross couple oscillator circuit has a fourth order denominator polynomial form in frequency domain. Thus, it is important to design pole frequency by symbolic formula so that it is oscillate as a sinusoidal signal. It should be noted that in order for cross couple to oscillate, all pole frequency should be position on imaginary axis only in pole zero diagram. The graph of time domain response is plotted by using inverse Laplace’s transform of the four unknowns of the partial fraction solution which were derived.

In most cases the transfer function of the circuit comes out as the ratio of a polynomial divided by another one. Each polynomial is in the s (frequency) domain. We start the chapter by practicing how to plot the transfer function, making sense of decay rates and phase changes, and introducing the decibel units. In the chapter we identify the meaning of poles and zeroes and their impact on the magnitude and phase of the transfer function. Then the bulk of the chapter deals with simplifying the transfer function in terms of series of simple fractions, each fraction having a constant in the numerator and a polynomial of order in the denominator. The numerator would have the residue, while the denominator would have the pole. Poles could be real or complex; they could also be of first order or higher ones. We outline how to calculate the residues of each simple fraction, plot the magnitude and phase, and rationalize the behavior of each. We also discuss long division and a simple method to estimate pole locations. The chapter has a lot of examples which can be used as backbone for sequel chapters in the book dealing with actual circuits.

A very simple method is given for finding the residues at multiple poles of a rational function in z−1. Compared to the multiple differentiation formula given in most text books, and several other ...

The optimal modified performance of the multi-input multi-output (MIMO) networked control systems (NCSs) with encoding–decoding, channel noise in the forward channel and packet dropouts, quantization in the feedback channel is investigated in this paper. A new and efficient tracking performance index for the NCSs is presented which prevents variations in the tracking error where there is no integrator in the plant. The optimal modified performance is obtained by the method of coprime factorization and partial fraction. The results demonstrate that the optimal modified performance is related to the locations of the non-minimum phase (NMP) zeros, unstable poles of the given plant as well as their directions. In addition, the modified factor, packet dropouts probability, channel noise and encoding–decoding are also closely related to optimal modified performance of the NCSs. Finally, we present some particular examples to illustrate the theoretical results.

We show that the necessary condition for the convergence of the series of simple partial fractions $$ {\displaystyle \sum_{k=1}^{\infty }{\left(z-{z}_k\right)}^{-1}} $$ in $$ Lp\left(\mathrm{\mathbb{R}}\right) $$ , 1 < p < ∞, is the convergence of the series $$ {\displaystyle \sum_{k=1}^{\infty }{\left|{z}_k\right|}^{-1/q}1{\mathrm{n}}^{-1-\varepsilon}\left(\left|{z}_k\right|+1\right)} $$ , e > 0. In the case 1 < p < 2, we obtain a convergence criterion in terms of the imaginary parts of poles under the condition that all the poles z k = x k + iy k belong to the angle |z k | ≤ α|y k | with a fixed α > 0.

Learning to compute integrals via the various techniques of integration (e.g., integration by parts, partial fractions, etc.) is difficult for many students. Here, we look at how students in a college level Calculus II course develop the ability to categorize integrals and the difficulties they encounter using a card sort-resort activity. Analysis of the data required the use of several non-standard techniques which provided interesting insights into the ways students develop categories in mathematics. One finding of note is that students may need a significant amount of time “off topic” to allow sufficient time to fully organize their schema for integration.

By means of the partial fraction decomposition method, we establish some summations of generalized q-harmonic numbers with finite series. We further explore series of generalized q-harmonic number identities as examples and list the relationship with generalized harmonic number identities.

Pade approximants are rational functions, with a denominator which does not vanish at zero, and whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different barycentric forms, and under a partial fraction form, depending on free parameters. According to the choice of these parameters, Pade-type approximants can be obtained under a barycentric or a partial fraction form.

In the problem of approximating real functions by simple partial fractions of order on closed intervals , we obtain a criterion for the best uniform approximation which is similar to Chebyshev's alternance theorem and considerably generalizes previous results: under the same condition on the poles of the fraction of best approximation, we omit the restriction on the order of this fraction. In the case of approximation of odd functions on , we obtain a similar criterion under much weaker restrictions on the position of the poles : the disc is replaced by the domain bounded by a lemniscate contained in this disc. We give some applications of this result. The main theorems are extended to the case of weighted approximation. We give a lower bound for the distance from to the set of poles of all simple partial fractions of order which are normalized with weight on (a weighted analogue of Gorin's problem on the semi-axis).

We prove that for a continuous real-valued function f on the segment [−1, 1] a real-valued simple partial fraction R n with n distinct poles outside the unit disk is a fraction of degree at most n of best approximation and is unique if and only if for the difference f − R n on [−1, 1] there exists a Chebyshev alternance of n + 1 points. The result is applied to the problem on approximation of real constants.

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.

We consider the problem of interpolation and best uniform approximation of constants c ≠ 0 by simple partial fractions ρ n of order n on an interval [a, b]. (All functions and numbers considered are real.) For the case in which n > 4|c|(b − a), we prove that the interpolation problem is uniquely solvable, obtain upper and lower bounds, sharp in order n, for the interpolation error on the set of all interpolation points, and show that the poles of the interpolating fraction lie outside the disk with diameter [a, b]. We obtain an analog of Chebyshev’s classical theorem on the minimum deviation of a monic polynomial of degree n from a constant. Namely, we show that, for n > 4|c|(b − a), the best approximation fraction ρ* n for the constant c on [a, b] is unique and can be characterized by the Chebyshev alternance of n+1 points for the difference ρ* n − c. For theminimum deviations, we obtain an estimate sharp in order n.

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.

In this paper, by the methods of partial fraction decomposition and generating function, we give an algorithm for computing mixed sums of products of l Bernoulli polynomials and k-l Euler polynomials, which are of the formT"n","k^@l(y;l,k-l):=@?j"1+...+j"k=nj"1,...,j"k>=0@?i=1k@l"i^j^"^i(nj"1,...,j"k)@?p=1lB"j"""p(x"p)@?q=l+1kE"j"""q(x"q), where @l=(@l"1,...,@l"k), and @l"1,...,@l"k are nonzero rational numbers. Moreover, some special sums are presented as examples.

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.

We use a trigonometric identity and integral calculus to evaluate several infinite series; in particular, we deduce the corresponding partial fraction decomposition of cotx, cscx, cothx and csch x.

From the theory of modular forms, there are exactly $[(k-2)/6]$ linear relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\ (2\le i \le [k/4])$. We present explicit formulas among these modular forms based on the partial fraction decomposition, and use them to determining a basis of the space of modular forms of weight $k$ on ${\rm SL}_2({\mathbb Z})$., 8 pages

The paper presents a new approach to time-domain modeling of ultra-wideband (UWB) channels with multiple convex obstacles. Rational approximation exploiting the vector fitting algorithm (VF) is used for deriving the closed-form impulse response of a multiple diffraction ray creeping on a cascade of convex obstacles. The VF algorithm is performed with respect to new generalized variables proportional to frequency but also taking into account geometrical parameters of the obstacles. The limits of the approximation domain for the VF algorithm reflect the range of UWB channel parameters that can be met in practical UWB channel scenarios. The results of VF approximation have the form of a finite series of partial fractions in the frequency domain, which then is easily transformed to the time domain. Obtained in this way, impulse response is a sum of exponential functions. As a consequence, in simulations of electromagnetic (EM) wave propagation, we can implement very fast and effective convolution algorithms with any input signal or perform simulations implementing SPICE-like programs.

Let D be the quaternion division algebra over a real closed field F. Then every non-constant polynomial in a skew-polynomial ring D [ t ; σ , δ ] decomposes into a product of linear factors, and thus has a zero in D. This improves [8, Theorem 2] .

The hook length formula for $d$-complete posets states that the $P$-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using $q$-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the $P$-partition generating function for each case as a $q$-integral and then we evaluate the $q$-integrals. Several $q$-integrals are evaluated using partial fraction expansion identities and others are verified by computer., 41 pages, 28 figures

The multipole representation provides an analytical way of Doppler broadening cross sections for neutron interactions and relies on a partial fraction decomposition that represents R-matrix resonance parameters by poles and residues. The windowed multipole method selectively broadens impactful poles and approximates the rest to increase the efficiency of the process. However, this process requires knowledge of the resonance parameters and certain limitations occur when additional channels are opened in the defined resolved resonance region such that only 2/3 of the ENDF/B-VII.1 nuclides can be processed. Some important nuclides needed in reactor applications such as Oxygen-16 and Hydrogen-1 are directly represented in point-wise form in the evaluation nuclear data library hindering the practicality of this approach. This paper presents a new fitting method for nuclear point-wise cross sections data that yields a pole and residue form analogous to the multipole representation. The Vector Fitting technique which originates from the field of signal processing was applied to the fitting of point-wise cross sections. Poles and residues from Vector Fitting were generated for Oxygen-16, Hydrogen-1, Boron-10 and Boron-11 and processed in the windowed multipole format. These new libraries were tested by direct comparison of the microscopic cross sections before and after Doppler Broadening, and by integral comparisons using a typical PWR pin cell from the BEAVRS benchmark. Results indicate that the new libraries are equivalent to the point-wise representation with the added benefit of allowing on-the-fly Doppler broadening without the need for temperature interpolation. Runtimes are approximately ∼28% slower using the multipole representation on this example problem compared to a single temperature ACE file. Keywords: Monte Carlo; Multipole; Doppler broadening; Vector Fitting; OpenMC., U.S. Department of Energy (Grant DE-AC05-00OR22725)

This paper concerned with realization of fractional order PID (FPID) by electronic circuit utilizing the syntheses of a fractance circuit that can be connected on operational amplifier either in the input or the output feedback of operational amplifier (OPAMP) circuit. Where the fractional order derivative (FD) and fractional order integral (FI) have been approximated to integer order rational transfer function depending on Carlson method. The continuous fractional expansion (CFE) and partial fraction expansion (PE) have been utilized on the (FI) and FD approximated transfer function to synthesis the fractance circuit. The analogue electronic circuit has been simulated using Circuit Wizard to realize the FD, FI and FPID circuits. The time domain response has been tested with sine wave input. The Bode plot for both real FD and FI with that the approximated one has been drawn.

This chapter is devoted entirely to the theory and application of infinite products, and as a consolation prize, we also talk about partial fractions.

We next turn to functions of a complex variable, that are perhaps best known for solving integrals by contour deformation in the complex plane.

In this paper, we develop a novel technique for computing the complex-conjugate zeros of rational transfer functions in partial fraction form, by searching the complex Laplace s-plane for the local minima of the determinant of a block matrix derived from the state-space equations. A higher resolution scan, in the vicinity of each local minima, may be used to increase precision of the zeros. This method can be applied to antennas to extract the zeros from the antenna response. One numerical example and one practical coax patch antenna example have been used to illustrate the efficiency and effectiveness of the proposed method.

Using reduction to polynomial interpolation, we study the multiple interpolation problem by simple partial fractions. Algebraic conditions are obtained for the solvability and the unique solvability of the problem. We introduce the notion of generalized multiple interpolation by simple partial fractions of order ≤ n. The incomplete interpolation problems (i.e., the interpolation problems with the total multiplicity of nodes strictly less than n) are considered; the unimprovable value of the total multiplicity of nodes is found for which the incomplete problem is surely solvable. We obtain an order n differential equation whose solution set coincides with the set of all simple partial fractions of order ≤ n.

We obtain estimates for L p -norms of simple partial fractions in terms of their L r -norms on bounded and unbounded segments of the real axis for various p > 1 and r > 1 (S. M. Nikolskii type inequalities). We adduce examples and remarks concerning sharpness of the inequalities and area of their application.

The paper aims to investigate the convergence of the q-Bernstein polynomials B n , q ( f ; x ) attached to rational functions in the case q > 1 . The problem reduces to that for the partial fractions ( x − α ) − j , j ∈ N . The already available results deal with cases, where either the pole α is simple or α ≠ q − m , m ∈ N 0 . Consequently, the present work is focused on the polynomials B n , q ( f ; x ) for the functions of the form f ( x ) = ( x − q − m ) − j with j ⩾ 2 . For such functions, it is proved that the interval of convergence of { B n , q ( f ; x ) } depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously.

This paper uses partial fraction decompositions to give a direct computation of the logarithmic derivative of the norm in Milnor K-theory for a finite separable extension. This result is useful for computations involving the relative Brauer group in finite characteristic and Witt kernels for function fields in characteristic two. Kato's result that the norm is compatible with the trace under logarithmic differentiation also follows from these tools. When F(x) is rational over F in finite characteristic l, the unramified part of is computed to be .

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.

The coupled nonlinear Schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics, like propagation of pulses. Efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system. In this paper, we introduce a new version of the Cox and Matthews third order exponential time differencing Runge-Kutta (ETD3RK) scheme based on the (1,2)-Pade approximation to the exponential function. In addition, we present its local extrapolation form to improve temporal accuracy to the fourth order. The developed scheme and its extrapolation are seen to be strongly stable, which have ability to damp spurious oscillations caused by high frequency components in the solution. A computationally efficient algorithm of the new scheme, based on a partial fraction splitting technique is presented. In order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear Schrodinger equations and performed several numerical experiments on them. The numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long-range solitary solutions given by coupled nonlinear Schrodinger equations and conserved the conserved quantities mass and energy exactly, to at least five decimal places.

By means of partial fraction decomposition, we give simple proofs of Mortenson’s identities first. Then, inspired by them, we derive their q-generalizations and explore further identities of similar type.

We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our $(\infty,2)$- and $(\infty,4)$-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors., We changed the title and the structure of the paper. Several misprints were corrected

In this paper a new approach is suggested for implementing fractional order differentiator in the desired frequency band based on frequency capacitance scaling. The process involves obtaining the rational approximate model of the fractional order differentiator using Matsuda method of approximation and then decomposing it by partial fraction expansion to obtain the circuit parameters (resistance and capacitance). If the frequency band of interest has now to be changed, only the capacitances of the resulting circuit are scaled proportionately. For the choice of the method of approximation and the approach for synthesis, emphasis has been given to accuracy of the model obtained and positive values of resistances and capacitances. The simulations have been performed using OrCAD Capture CIS simulator.

We present a novel technique for computing the zeros of rational transfer functions in partial fraction form, by finding the local minima of the magnitude of the determinant of a block matrix comprised of state-space sub-matrices and the Laplace variable s. In this paper, the technique is developed for systems with real poles and residues, and successfully applied to a 10th order numerical example. In a separate paper, we further develop the technique to systems with complex-conjugate pairs of poles and residues.

In this article, we give a description, via the Cauchy–Fantappie transformation of analytic functionals, of the mutual dualities between the dual Frechet–Schwartz (FS)-space A −∞(Ω) of holomorphic functions in a bounded lineally convex domain Ω of ℂ n (n ≥ 2) with polynomial growth near the boundary ∂Ω, and the (FS)-space of holomorphic functions in the interior of the conjugate set that are in . Then, we prove the existence of countable weakly sufficient sets in A −∞(Ω) and sufficient sets in . Finally, we show a possibility (respectively, the failure) of representating functions from (respectively, A −∞(Ω)) in the form of series of partial fractions.

SUMMARY A conventional block cyclic reduction algorithm operates by halving the size of the linear system at each reduction step, that is, the algorithm is a radix-2 method. An algorithm analogous to the block cyclic reduction known as the radix-q partial solution variant of the cyclic reduction (PSCR) method allows the use of higher radix numbers and is thus more suitable for parallel architectures as it requires fever reduction steps. This paper presents an alternative and more intuitive way of deriving a radix-4 block cyclic reduction method for systems with a coefficient matrix of the form tridiag{ − I,D, − I}. This is performed by modifying an existing radix-2 block cyclic reduction method. The resulting algorithm is then parallelized by using the partial fraction technique. The parallel variant is demonstrated to be less computationally expensive when compared to the radix-2 block cyclic reduction method in the sense that the total number of emerging subproblems is reduced. The method is also shown to be numerically stable and equivalent to the radix-4 PSCR method. The numerical results archived correspond to the theoretical expectations. Copyright © 2013 John Wiley & Sons, Ltd.

For arbitrary natural n ≥ 2 we construct an example of a real continuous function, for which there exists more than one simple partial fraction of order ≤ n of the best uniform approximation on a segment of the real axis. We prove that even the Chebyshev alternance consisting of n+1 points does not guarantee the uniqueness of the best approximation fraction. The obtained results are generalizations of known non-uniqueness examples constructed for n = 2, 3 in the case of simple partial fractions of an arbitrary order n.

In this paper, we present some new results on a frequency weighted balanced truncation technique based on well-known partial-fraction-expansion idea. The reduced order models which are guaranteed to be stable in case of double-sided weighting are obtained by direct truncation. Two sets of simple, elegant and easily computable a priori error bounds are also derived. Relationships between the proposed method and the previous methods based on partial-fraction idea are also derived. The technique is illustrated using a numerical example of a practical application and then compared with other well-known techniques, to show the effectiveness of the method.

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.

Under the assumption that a certain algebraic identity holds for all $ n\in \mathbb{N} $ (it is verified for n ≤ 5), we prove that a real-valued simple partial fraction R n with n simple poles lying outside the unit disk is a simple partial fraction of degree at most n of the best uniform approximation of a continuous real-valued functions f on [−1, 1] provided that for the difference f − R n there is a Chebyshev alternance of n + 1 points on [−1, 1]. The result is applied to the problem of approximation of real constants. Bibliography: 8 titles.

The problem of the best uniform approximation of a real constant c by real-valued simple partial fractions Rn on a closed interval of the real axis is considered. For sufficiently small (in absolute value) c, |c| ≤ cn, it is proved that Rn is a fraction of best approximation if, for the difference Rn − c, there exists a Chebyshev alternance of n + 1 points on a closed interval. A criterion for best approximation in terms of alternance is stated.

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.