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Solving Some Integrals with Maple
Chii-Huei Yu
International Journal of Research in Aeronautical and Mechanical Engineering , 2013,
Abstract: This study uses the mathematical software Maple for the auxiliary tool to evaluate two types of integrals. We can obtain the Fourier series expansions of these two types of integrals by using Euler's formula, generalized DeMoivre's formula and integration term by term. In addition, we provide two integrals to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. Therefore, Maple provides insights and guidance regarding problem-solving methods
Determination of the Electromagnetic Field Created by Line Current and Sheet Current Source at the Earth’s Surface  [PDF]
Ghada M. Sami, Maryam I. Al-Nami
International Journal of Geosciences (IJG) , 2014, DOI: 10.4236/ijg.2014.513129
Abstract: The electromagnetic field that generated by line current and sheet current at the surface of the earth can be expressed in analytical form. The line current created at the earth’s surface by an infinitely long line current is given by the inverse Fourier integrals over a horizontal wave number. The sheet current can be obtained by integrating the line current expansions using a Neumann and Struve functions; these functions have known mathematical properties, including the series expansions. The series expansions are exact with neglecting the displacement currents. Assuming a uniform earth and that there is no propagation, the three nonzero field components can be expressed in terms of the Neumann and Struve functions. The integrals of line current expansions are calculated by using the numerical methods. The results represented graphically and illustrated by figures. Results can be used to evaluate numerical solutions of more complicated modeling algorithms.
Two Initial Value Problems Approach for Solving Singular Perturbations Problems  [PDF]
Awoke Andargie, Yanala Narsimha Reddy
American Journal of Computational Mathematics (AJCM) , 2012, DOI: 10.4236/ajcm.2012.23027
Abstract: In this paper, we presented an initial value approach for solving singularly perturbed two point boundary value problems with the boundary layer at one end (left or right). By employing asymptotic power series expansion, the given singularly perturbed two-point boundary value problem is replaced by two first order initial value problems. To demonstrate the applicability of the present method three linear and two nonlinear problems with left end boundary layer are considered. It is observed that the present method approximates the exact solution very well.
Resolution of Grandi’s Paradox as Extended to Complex Valued Functions  [PDF]
Serdar Beji
Advances in Pure Mathematics (APM) , 2020, DOI: 10.4236/apm.2020.108027
Abstract: Grandi’s paradox, which was posed for a real function of the form 1/(1+ x), has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function 1/(1+ ex ) . Grandi-type complex valued functions such as 1/(i + x) are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function \"\" is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.
Alternative Fourier Series Expansions with Accelerated Convergence  [PDF]
Wenlong Li
Applied Mathematics (AM) , 2016, DOI: 10.4236/am.2016.715152
Abstract: The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.
On the Efficacy of Fourier Series Approximations for Pricing European Options  [PDF]
A. S. Hurn, K. A. Lindsay, A. J. McClelland
Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.517267
Abstract: This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.
On the Equiconvergence of the Fourier Series and Integral of Distributions  [PDF]
A. A. Rakhimov
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.311163
Abstract: We prove equiconvergence of the Bochner-Riesz means of the Fourier series and integral of distributions with compact support from the Liouville spaces.
Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12  [PDF]
Bar？？ Kendirli
Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.68133
Abstract: Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of \"\" and \"\" and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of \"\" and \"\" . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of \"\" and \"\".
Fourier-Series Representation of Discontinuous Functions and Its Physical Applications  [PDF]
Sami M. Al-Jaber, Iyad Saadeddin
Applied Mathematics (AM) , 2019, DOI: 10.4236/am.2019.104017
Abstract: In this work, Fourier-series representation of a discontinuous function is used to highlight and clarify the controversial problem of finding the value of the function at a point of discontinuity. Several physical situations are presented to examine the consequences of this kind of representation and its impact on some widely well-known problems whose results are not clearly understood or justified.
Dribbling Basketball Using Fourier Series  [PDF]
Uwaydah Leith
Journal of Applied Mathematics and Physics (JAMP) , 2024, DOI: 10.4236/jamp.2024.1212252
Abstract: Dribbling a basketball is a fundamental skill in the sport, defined by the rhythmic bouncing of the ball with one hand, regardless of whether the player is stationary or in motion. Mastery of dribbling allows an athlete to maintain control of the ball, maneuver around opponents, and create opportunities for passing, shooting, or driving toward the basket. Additionally, dribbling involves various mathematical principles, such as the physics of motion and the statistical analysis of performance data. One significant mathematical tool in this context is Fourier analysis, which effectively decomposes complex signals, such as the dribbling motion of a basketball, into simpler sinusoidal components. This analysis provides insights into the frequency characteristics of the dribble, enhancing the understanding of a player’s skill and consistency.
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