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Journal articles on the topic 'Expansion de Taylor'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Kitahara, Kazuaki, and Taka Aki Okuno. "A Note on Two Point Taylor Expansion III." International Journal of Modeling and Optimization 4, no. 4 (August 2014): 287–91. http://dx.doi.org/10.7763/ijmo.2014.v4.387.

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2

Grünwald, Eva, Ydalia Delgado Mercado, and Christof Gattringer. "Taylor and fugacity expansion for the effective ℤ3 spin model of QCD at finite density." International Journal of Modern Physics A 29, no. 32 (December 30, 2014): 1450198. http://dx.doi.org/10.1142/s0217751x1450198x.

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Different series expansions in the chemical potential μ are studied and compared for an effective theory of QCD which has a flux representation where the complex action is overcome. In particular we consider fugacity series, Taylor expansion and a modified Taylor expansion and compare the outcome of these series to the reference results from a Monte Carlo simulation in the flux representation where arbitrary μ is accessible. It is shown that for most parameter values the fugacity expansion gives the best approximation to the data from the flux simulation, followed by our newly proposed modified Taylor expansion. For the conventional Taylor expansion we find that the results coincide with the flux data only for very small μ.

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3

Shimomura, Tetsu, and Yoshihiro Mizuta. "Taylor expansion of Riesz potentials." Hiroshima Mathematical Journal 25, no. 3 (1995): 595–621. http://dx.doi.org/10.32917/hmj/1206127635.

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4

Stanković, B. "Taylor Expansion for Generalized Functions." Journal of Mathematical Analysis and Applications 203, no. 1 (October 1996): 31–37. http://dx.doi.org/10.1006/jmaa.1996.0365.

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5

Chu, M. S., T. H. Jensen, and P. M. Bellan. "General Taylor configuration expansion revisited." Physics of Plasmas 6, no. 5 (May 1999): 1495–99. http://dx.doi.org/10.1063/1.873401.

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6

Kulchitski, O. Yu, and D. F. Kuznetsov. "The unified Taylor-Ito expansion." Journal of Mathematical Sciences 99, no. 2 (April 2000): 1130–40. http://dx.doi.org/10.1007/bf02673635.

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7

Gerritzen, L. "Taylor expansion of noncommutative polynomials." Archiv der Mathematik 71, no. 4 (October 1, 1998): 279–90. http://dx.doi.org/10.1007/s000130050265.

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8

Chouquet, Jules. "Taylor Expansion, Finiteness and Strategies." Electronic Notes in Theoretical Computer Science 347 (November 2019): 65–85. http://dx.doi.org/10.1016/j.entcs.2019.09.005.

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9

Hong. "Approximation of a Warship Passive Sonar Signal Using Taylor Expansion." Journal Of The Acoustical Society Of Korea 33, no. 4 (2014): 232. http://dx.doi.org/10.7776/ask.2014.33.4.232.

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10

Stanković, B. "Asymptotic Taylor expansion for Fourier hyperfunctions." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 453, no. 1960 (May 8, 1997): 913–18. http://dx.doi.org/10.1098/rspa.1997.0050.

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11

Pettersson, Roger. "Stratonovich–Taylor expansion and numerical methods∗." Stochastic Analysis and Applications 10, no. 5 (January 1992): 603–12. http://dx.doi.org/10.1080/07362999208809294.

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12

Yang, Tonghai. "Taylor expansion of an Eisenstein series." Transactions of the American Mathematical Society 355, no. 7 (February 27, 2003): 2663–74. http://dx.doi.org/10.1090/s0002-9947-03-03194-5.

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13

Wang, Hongyue, Yun Zhang, Changyong Feng, and Xin M. Tu. "On the Misuse of Taylor Expansion." Biometrics 71, no. 4 (December 2015): 1195. http://dx.doi.org/10.1111/biom.12425.

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14

Ernzerhof, Matthias. "Taylor-series expansion of density functionals." Physical Review A 50, no. 6 (December 1, 1994): 4593–607. http://dx.doi.org/10.1103/physreva.50.4593.

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15

Hosono, Toshio. "Numerical algorithm for Taylor series expansion." Electronics and Communications in Japan (Part I: Communications) 69, no. 6 (1986): 10–18. http://dx.doi.org/10.1002/ecja.4410690602.

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16

Maybank, S. J. "Optical flow and the Taylor expansion." Pattern Recognition Letters 4, no. 4 (September 1986): 243–46. http://dx.doi.org/10.1016/0167-8655(86)90004-8.

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17

Joyal, Pierre. "Un théorème de préparation pour fonctions à développement Tchébychévien." Ergodic Theory and Dynamical Systems 14, no. 2 (June 1994): 305–29. http://dx.doi.org/10.1017/s0143385700007896.

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AbstractIn this article, we prove a preparation theorem for functions which admit a certain type of expansion called Chebychev expansion. Taylor expansions are particular cases of Chebychev expansions. The result is based on an approach essentially different from those used for the classical preparation theorems. It has applications in bifurcation theory of vector fields.

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18

SHUTTLEWORTH, I. G. "IMPROVEMENTS TO THE RADIUS OF CONVERGENCE OF DIRECT INVERSION TECHNIQUES IN HELIUM ATOM SCATTERING." Surface Review and Letters 15, no. 05 (October 2008): 519–23. http://dx.doi.org/10.1142/s0218625x08011718.

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Recent studies (I. G. Shuttleworth, Surf. Rev. Lett.14(2) (2007) 321) have demonstrated a direct method of diffraction pattern inversion for the helium atom scattering (HAS) experiment. The method requires the expansion of the Patterson function as a multivariate Taylor series to form a set of simultaneous equations. Benchmark tests of the procedure show that incomplete Taylor expansions introduce inconsistency into the set of simultaneous equations, whereas larger Taylor expansions attract significant numerical errors during their solution. The current work replaces the conventional matrix inversion techniques with a multidimensional Newton–Raphson algorithm. Tests have shown that the Newton–Raphson procedure removes the Taylor series and numerical limitations from the inversion technique for any realistic surface.

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19

Liu, Hongbo, Ye Ji, and Aboul Ella Hassanien. "Image Color Transfer Approach by Analogy with Taylor Expansion." International Journal of System Dynamics Applications 2, no. 2 (April 2013): 43–54. http://dx.doi.org/10.4018/ijsda.2013040103.

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The Taylor expansion has shown in many fields to be an extremely powerful tool. In this paper, the authors investigated image features and their relationships by analogy with Taylor expansion. The kind of expansion could be helpful for analyzing image feature and engraftment, such as transferring color between images. By analogy with Taylor expansion, the image color transfer algorithm is designed by the first and second-order information. The luminance histogram represents the first-order information of image, and the co-occurrence matrix represents the second-order information of image. Some results illustrate the proposed algorithm is effective. In this study, each polynomial in the Taylor analogy expansion of images is considered as one of image features which help in re-understanding images and its features. By using the proposed technique, the features of image, such as color, texture, dimension, time series, would be not isolated but mutual relational based on image expansion.

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20

Eke, A., and P. Jackreece. "Taylor series expansion of nonlinear integrodifferential equations." American Journal of Scientific and Industrial Research 2, no. 3 (June 2011): 376–80. http://dx.doi.org/10.5251/ajsir.2011.2.3.376.380.

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21

Arter, Wayne. "Beyond linear fields: the Lie–Taylor expansion." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2197 (January 2017): 20160525. http://dx.doi.org/10.1098/rspa.2016.0525.

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The work extends the linear fields’ solution of compressible nonlinear magnetohydrodynamics (MHD) to the case where the magnetic field depends on superlinear powers of position vector, usually, but not always, expressed in Cartesian components. Implications of the resulting Lie–Taylor series expansion for physical applicability of the Dolzhansky–Kirchhoff (D–K) equations are found to be positive. It is demonstrated how resistivity may be included in the D–K model. Arguments are put forward that the D–K equations may be regarded as illustrating properties of nonlinear MHD in the same sense that the Lorenz equations inform about the onset of convective turbulence. It is suggested that the Lie–Taylor series approach may lead to valuable insights into other fluid models.

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22

Fukushima, Toshio. "Taylor series expansion of prismatic gravitational field." Geophysical Journal International 220, no. 1 (October 7, 2019): 610–60. http://dx.doi.org/10.1093/gji/ggz449.

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SUMMARY The exact analytical formulae to compute the gravitational field of a rectangular prism suffer from round-off errors when the evaluation point is outside the Brillouin sphere of the prism. The error magnitude grows cubically with respect to the distance from the prism. This phenomenon is eminent in not only the gravitational potential but also the gravity vector and the gravity gradient tensor. Unfortunately, the issue of error increase is not settled efficiently by the rewriting of the formulae using the addition theorems of the transcendental functions. Besides, the computational labour of the formulae is not small since at least 18 transcendental functions are employed for the potential computation each time. In order to solve these problems, we developed up to the 16th order 3-D Taylor series expansion of all the gravitational field quantities for a uniform rectangular prism of arbitrary dimensions. For instance, the eighth order truncation guarantees the eight digit accuracy of the potential computation of a nearly cubic prism at the cost of 11 per cent of that of the fast computation of the exact formula when the distance is more than 2.7 times longer than the Brillouin sphere radius. By using a simple algorithm, we present an adaptive procedure combining the truncated series of various orders and the exact formulae in order to compute precisely and quickly the gravitational field of an assembly of prisms everywhere. Exactly the same approach is applicable to the geomagnetic field computation.

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23

Choi, Y. H. "Robust adaptive array using Taylor series expansion." Electronics Letters 47, no. 15 (2011): 840. http://dx.doi.org/10.1049/el.2011.0817.

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24

Feng, Hongyinping, and Shengjia Li. "A tracking differentiator based on Taylor expansion." Applied Mathematics Letters 26, no. 7 (July 2013): 735–40. http://dx.doi.org/10.1016/j.aml.2013.02.003.

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25

Manzonetto, Giulio, and Domenico Ruoppolo. "Relational Graph Models, Taylor Expansion and Extensionality." Electronic Notes in Theoretical Computer Science 308 (October 2014): 245–72. http://dx.doi.org/10.1016/j.entcs.2014.10.014.

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26

De Ridder, Hilde, Hennie De Schepper, and Frank Sommen. "Taylor Series Expansion in Discrete Clifford Analysis." Complex Analysis and Operator Theory 8, no. 2 (April 12, 2013): 485–511. http://dx.doi.org/10.1007/s11785-013-0298-2.

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27

Yin-nian, He. "Taylor expansion method for nonlinear evolution equations." Applied Mathematics and Mechanics 26, no. 4 (April 2005): 522–29. http://dx.doi.org/10.1007/bf02465392.

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28

Alexakis, A. "Rotating Taylor–Green flow." Journal of Fluid Mechanics 769 (March 13, 2015): 46–78. http://dx.doi.org/10.1017/jfm.2015.82.

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The steady state of a forced Taylor–Green flow is investigated in a rotating frame of reference. The investigation involves the results of 184 numerical simulations for different Reynolds numbers $\mathit{Re}_{F}$ and Rossby numbers $\mathit{Ro}_{F}$. The large number of examined runs allows a systematic study that enables the mapping of the different behaviours observed to the parameter space ($\mathit{Re}_{F},\mathit{Ro}_{F}$), and the examination of different limiting procedures for approaching the large $\mathit{Re}_{F}$ small $\mathit{Ro}_{F}$ limit. Four distinctly different states were identified: laminar, intermittent bursts, quasi-two-dimensional condensates and weakly rotating turbulence. These four different states are separated by power-law boundaries $\mathit{Ro}_{F}\propto \mathit{Re}_{F}^{-{\it\gamma}}$ in the small $\mathit{Ro}_{F}$ limit. In this limit, the predictions of asymptotic expansions can be directly compared with the results of the direct numerical simulations. While the first-order expansion is in good agreement with the results of the linear stability theory, it fails to reproduce the dynamical behaviour of the quasi-two-dimensional part of the flow in the nonlinear regime, indicating that higher-order terms in the expansion need to be taken into account. The large number of simulations allows also to investigate the scaling that relates the amplitude of the fluctuations with the energy dissipation rate and the control parameters of the system for the different states of the flow. Different scaling was observed for different states of the flow, that are discussed in detail. The present results clearly demonstrate that the limits of small Rossby and large Reynolds numbers do not commute and it is important to specify the order in which they are taken.

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29

Zhao, Jun Ying, Xing He Ma, Li Jia, and Gen Xian Zhang. "Study on Microcomputer Inverse Time Over-Current Protection Algorithm of Underground Low-Voltage Power-System." Applied Mechanics and Materials 143-144 (December 2011): 8–13. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.8.

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Analysis of general mathematical model and the traditional anti-time algorithm of over-current protection for microcomputer based on the algorithm proposes look-up table and Taylor expansion. To solve the problem of c=0.02 of general anti-time in microcomputer is difficult to implement directly. This paper chooses the Taylor expansion method and selects the general anti-time and thermal over-load (no storage) anti-time for experiment. The experiment proves that the Taylor expansion method can achieve exactly the characteristics of microprocessor-based inverse time.

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30

Xie, Mingliang, Tingting Kong, Jin Li, and Jiang Lin. "The asymptotic stability of the Taylor-series expansion method of moment model for Brownian coagulation." Thermal Science 22, no. 4 (2018): 1651–57. http://dx.doi.org/10.2298/tsci1804651x.

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In the present study, the linear stability of population balance equation due to Brownian motion is analyzed with the Taylor-series expansion method of moment. Under certain conditions, the stability of the Taylor-series expansion method of moment model is reduced to a well-studied problem involving eigenvalues of matrices. Based on the principle of dimensional analysis, the perturbation equation is solved asymptotically. The results show that the Taylor-series expansion method of moment model is asymptotic stable, which implies that the asymptotic solution is uniqueness, and supports the self-preserving size distribution hypothesis theoretically.

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31

Burtnyak, Ivan, and Anna Malytska. "Taylor expansion for derivative securities pricing as a precondition for strategic market decisions." Problems and Perspectives in Management 16, no. 1 (March 13, 2018): 224–31. http://dx.doi.org/10.21511/ppm.16(1).2018.22.

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The strategy of managing the pricing processes, in particular managing the dynamics of the price of the underlying asset and its volatility, the prices of indices, shares, options, the magnitude of financial flows, in the method of calculating the company’s rating based on the available quotations of securities, is developed. The article deals with the study of pricing and calculating the volatility of European options with general local-stochastic volatility, applying Taylor series methods for degenerate diffusion processes. The application of this idea requires new approaches caused by degradation difficulties. Price approximation is obtained by solving the Cauchy problem of partial differential equations diffusion with inertia, and the volatility approximation is completely explicit, that is, it does not require special functions. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and an analytical expression of the fundamental solution is considerably simplified. Applied an approach to the pricing of derivative securities on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). Thus, the approximate value of options can be calculated as effectively as the Black-Scholes pricing of derivative securities. On the basis of the solved problem, it is possible to calculate their turns step-by-step. This enables to predict the dynamics of the pricing of derivatives and to create a strategy of behavior at options according to the passage of the process. For each approximation, price volatility is calculated, which makes it possible to take into account all changes in the market and to calculate possible situations. The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in the financial markets.

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32

Araujo, Edvaldo S., and Reynam C. Pestana. "Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion." GEOPHYSICS 84, no. 4 (July 1, 2019): T299—T311. http://dx.doi.org/10.1190/geo2018-0463.1.

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We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

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33

Srivastava, H. M., and Sébastien Gaboury. "New Expansion Formulas for a Family of theλ-Generalized Hurwitz-Lerch Zeta Functions." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/131067.

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We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.

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34

Dabbaghchian, Setareh, and Reza Saffari. "Dark Energy Constraints on Red-Shift-Based Gravity." ISRN Astronomy and Astrophysics 2013 (February 19, 2013): 1–8. http://dx.doi.org/10.1155/2013/107325.

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We have studied cosmological dynamics in gravity theory via cosmographic parameters. We have changed variables of field equations from time to red-shift and solved the achieved differential equation analytically for . Then we have used Taylor expansion to find general form of function around the present day value of scalar curvature. By introducing we would simplify our calculations; if we consider as a given function we would restrict our answers of . In this paper we offer a linear form of which leads us to a specific function, where is a constant which depends on the present day value of deceleration parameter. As an example, using Taylor expansion coefficients, we have compared our analytically calculated function with reconstructed function for Dark Energy models. To reconstruct action for Dark Energy models, we have used corresponding of each Dark Energy model for calculating Taylor expansion coefficients. As our function is linear, the Taylor expansion coefficients would be a function of present day value of deceleration parameter.

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35

Gülsu, Mustafa, Yalçın Öztürk, and Ayşe Anapali. "A Collocation Method for Solving Fractional Riccati Differential Equation." Advances in Applied Mathematics and Mechanics 5, no. 06 (December 2013): 872–84. http://dx.doi.org/10.4208/aamm.12-m12118.

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AbstractIn this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation is derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.

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36

Yang, Nian, Nan Chen, and Xiangwei Wan. "A new delta expansion for multivariate diffusions via the Itô-Taylor expansion." Journal of Econometrics 209, no. 2 (April 2019): 256–88. http://dx.doi.org/10.1016/j.jeconom.2019.01.003.

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37

Ren, Jin, Jingxing Chen, and Liang Feng. "A Novel Positioning Algorithm Based on Self-adaptive Algorithm of RBF Network." Open Electrical & Electronic Engineering Journal 10, no. 1 (November 21, 2016): 141–48. http://dx.doi.org/10.2174/1874129001610010141.

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Much attention has been paid to Taylor series expansion (TSE) method these years, which has been extensively used for solving nonlinear equations for its good robustness and accuracy of positioning. A Taylor-series expansion location algorithm based on the RBF neural network (RBF-TSE) is proposed before to the performance of TSE highly depends on the initial estimation. In order to have more accurate and lower cost,a new Taylor-series expansion location algorithm based on Self-adaptive RBF neural network (SA-RBF-TSE) is proposed to estimate the initial value. The proposed algorithm is analysed and simulated with several other algorithms in this paper.

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38

SIYANKO, S. "Essentially exact asymptotic solutions for Asian derivatives." European Journal of Applied Mathematics 23, no. 3 (January 10, 2012): 395–415. http://dx.doi.org/10.1017/s0956792511000441.

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In this paper, we will show how to obtain asymptotic solutions for the problem of pricing Asian options. Under the assumption that the underlying follows geometric Brownian motion, we will derive Taylor expansion series for the fixed and floating strike Asian options. While there will be no analytical formulae for calculating expansion coefficients, we will provide relatively simple algorithms for calculating them. The methodology is particularly effective for the case of continuously sampled fixed-strike Asian calls where it takes only seconds to obtain constants for the Taylor expansion series that can converge beyond 10 significant digits. It is needless to say that we need to calculate Taylor expansion constants only once and the option price would be an analytical expression constructed from a cumulative normal distribution function, an exponential function and finite sums.

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39

Ju, Nengjiu. "Pricing Asian and basket options via Taylor expansion." Journal of Computational Finance 5, no. 3 (2002): 79–103. http://dx.doi.org/10.21314/jcf.2002.088.

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40

Yun, Zhiwei, and Wei Zhang. "Shtukas and the Taylor expansion of L-functions." Annals of Mathematics 186, no. 3 (November 1, 2017): 767–911. http://dx.doi.org/10.4007/annals.2017.186.3.2.

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41

Chen, Chien Sheng, Yung Chuan Lin, Wen Hsiung Liu, and He Nian Shou. "Improved Taylor-Series Expansion for MS Location Estimation." Applied Mechanics and Materials 52-54 (March 2011): 1777–82. http://dx.doi.org/10.4028/www.scientific.net/amm.52-54.1777.

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The objective of wireless location is to determine the mobile station (MS) location in a wireless cellular communications system. Due to the measurement with large errors, location schemes give poorer performance in non-line-of-sight (NLOS) environments. To determine MS location, Taylor-series algorithm (TSA) is commonly used scheme. TSA can give a least-square (LS) solution to a set of simultaneous linearized equations. The disadvantage of TSA is the need for an initial position guess to start the solution process. The iteration process may not converge due to a poor initial estimate of the MS location. To improve the location accuracy with less complexity, the initial MS location selection criterion is proposed. Numerical results demonstrate that the proposed criterion always provides more accurate positioning.

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42

Richter, Hendrik, and Günter Stein. "On Taylor series expansion for chaotic nonlinear systems." Chaos, Solitons & Fractals 13, no. 9 (July 2002): 1783–89. http://dx.doi.org/10.1016/s0960-0779(01)00191-6.

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43

Aoyama, T., T. Matsuo, and Y. Shibusa. "Improved Taylor Expansion Method in the Ising Model." Progress of Theoretical Physics 115, no. 3 (March 1, 2006): 473–86. http://dx.doi.org/10.1143/ptp.115.473.

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44

Ciesielski, M., J. Guillot, D. Gomez-Prado, and E. Boutillon. "High-Level Dataflow Transformations Using Taylor Expansion Diagrams." IEEE Design & Test of Computers 26, no. 4 (July 2009): 46–57. http://dx.doi.org/10.1109/mdt.2009.82.

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45

Zhao, Shanshan, and Ross L. Prentice. "Rejoinder to “On the Misuse of Taylor Expansion”." Biometrics 71, no. 4 (December 2015): 1195. http://dx.doi.org/10.1111/biom.12426.

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46

Liu, Zhen, Bin Wu, and Tian-Chun Ye. "Improved Turbo Decoding With Multivariable Taylor Series Expansion." IEEE Communications Letters 22, no. 1 (January 2018): 37–40. http://dx.doi.org/10.1109/lcomm.2017.2705643.

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47

Wang, Jin, Gwanggil Jeon, and Jechang Jeong. "Deinterlacing Using Taylor Series Expansion and Polynomial Regression." IEEE Transactions on Circuits and Systems for Video Technology 23, no. 5 (May 2013): 912–17. http://dx.doi.org/10.1109/tcsvt.2013.2240914.

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48

Yang, You Liang, Jing Zhang, Fan Wei Meng, and Cui Hong Ma. "RFID Localization Algorithm Based on Taylor Series Expansion." Advanced Materials Research 756-759 (September 2013): 3501–5. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.3501.

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RFID is widely used in indoor positioning, as one of the typical system LANDMARK introduced reference label, used the nearest neighbor data correlation algorithm. This article uses the Taylor series expansion method, by the LANDMARK law get positioning results for further processing, resulting in more accurate positioning results.

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49

Kanwal, R. P., and K. C. Liu. "A Taylor expansion approach for solving integral equations." International Journal of Mathematical Education in Science and Technology 20, no. 3 (May 1989): 411–14. http://dx.doi.org/10.1080/0020739890200310.

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50

Friedrich, Helmut. "The Taylor Expansion at Past Time-like Infinity." Communications in Mathematical Physics 324, no. 1 (October 4, 2013): 263–300. http://dx.doi.org/10.1007/s00220-013-1803-1.

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