Nikolai E. Korotkov, Alexander N. Korotkov · 2020
Integrals Related to the Error Function presents a table of integrals related to the error function, including indefinite and improper definite integrals. Most of the formulas in this book have not been presented in other tables of integrals or have been presented only for some special cases of parameters or for integration only along the real axis of the complex plane. Many of the integrals presented here cannot be obtained using a computer (except via an approximate numerical integration). Additionally, for improper integrals, this book emphasizes the necessary and sufficient conditions for the validity of the presented formulas, including trajectory for going to infinity on the complex plane; such conditions are usually not given in computer-assisted analytical integration and often not presented in the previously published tables of integrals. Features The first book in English language to present a comprehensive collection of integrals related to the error function Useful for researchers whose work involves the error function (e.g., via probability integrals in communication theory). Additionally, it can also be used by broader audience.
Cited by 15
Herbert E. Salzer · Journal of the Franklin Institute · 1955
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Meraj Mustafa (585271), Muhammad A. Farooq (585272), Tasawar Hayat (433317) et al. · Figshare · 2015
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Amil Daraz (9667034), Abdul Basit (174463), Guoqiang Zhang (131884) · Figshare · 2023
<div><p>Demand of Power electronics converter is increased due to the rapid increase of green re-sources. The distributed nature of uncontrolled renewable energy resources are connected with the grid via power electronics converter. DC-DC converters are high-frequency current conversion circuits that use transformers, inductors, and capacitors to smooth switching noise into regulated DC voltages. In this study, a comparative performance analysis of Proportional integral Derivative (PID) and fuzzy logic controller (FLC) for DC-DC Boost Converter is presented. PID-based controller for dynamic boosting of DC-DC boost converter was replaced by Fuzzy Inference system (FIS) to study the system behavior. Five different membership functions were performed to observe the response of the system for DC -DC boost converter with FIS controller. The performances of FIS-based controller and PID-based controller for DC-DC boost converter were compared in detail. Also, the proposed techniques for the switching mode of DC-DC converter have been applied. The performance analysis of both PID and fuzzy controller was simulated using MAT LAB. The results of FIS based approach are better than PID based controller in terms of transient responses.</p></div>
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Tzeng, Jengnan, Horng-Shing Lu, Henry, Li, Wen-Hsiung · Figshare · 2011
The solid line refers to the case where the points in each group are chosen randomly, while the dash line refers to the case where the points in each group are chosen from neighbors. In all cases, is fixed to 200.<b>Copyright information:</b>Taken from "Multidimensional scaling for large genomic data sets"http://www.biomedcentral.com/1471-2105/9/179BMC Bioinformatics 2008;9():179-179.Published online 4 Apr 2008PMCID:PMC2375126.
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Andrea N. Ceretani, Natalia N. Salva, Domingo A. Tarzia · Applied Mathematics and Computation · 2018
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Kent W. Staley · PhilSci-Archive (University of Pittsburgh) · 2007
This paper analyzes Deborah Mayo's error-statistical (ES) account of scientific evidence in order to clarify the kinds of "material postulates" it requires and to explain how those assumptions function. A secondary aim is to explain and illustrate the importance of the security of an inference. After finding that, on the most straightforward reading of the ES account, it does not succeed in its stated aims, two remedies are considered: either relativize evidence claims or introduce stronger assumptions. The choice between these approaches turns on the value attached to two aims of inquiry that are in tension: drawing strong, informative conclusions and reasoning securely.
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Ahmed A. Soliman, A.H.A. Ali · International Journal of Computer Mathematics · 2006
The present paper is devoted to the development of a new scheme to solve the one-dimensional time-dependent Burgers' equation locally on sub-domains, using similarity reductions for partial differential equations. Each sub-domain is divided into three grid points. The ordinary differential equation deduced from the similarity reduction can be integrated and is then used to approximate the flux vector in the Burgers' equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two different test cases and are compared with other numerical results.
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Kumari Sreeja, Suresh Nair, V. Madhukar Mallayya et al. · International Journal of Scientific and Innovative Mathematical Research · 2016
In this paper we give a rational approximation of an alternating series using remainder term of the series. For that we shall introduce a correction function to the series. The correction function plays a vital role in series approximation. Using correction function we shall deduce an error function to the series.
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Wenzheng Dong, Lanling Zeng, Shunli Ji et al. · Image and Vision Computing · 2024
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Duncan, Glenn William · Mspace (University of Manitoba) · 1967
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Hugues de Lassus Saint-Geniès, David Defour, Guillaume Revy · HAL (Le Centre pour la Communication Scientifique Directe) · 2015
We present a new table-based algorithm for the evaluation of the sine and cosine functions, which removes two sources of error in the final step of the evaluation. The tabulation of exact values for sines and cosines is realized thanks to a selection of good Pythagorean triples. A method for pre-computing such selections for tables indexed by up to 10 bits in reasonable time and memory consumption is shown, along with a theoretical efficiency comparison between our proposed scheme and the current state of the art of table-based algorithms.
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Zaccari, Daniel Gustavo, Alturria Lanzardo, Carmina José, Pérez, Jorge et al. · Conicet · 2019
In this paper, we present two new approximants for the Error Function. The starting point for obtaining them, is to use two alternative integral representations involving improper integrals. Both integrands include the function. Therefore, by replacing by its truncated Taylor's expansion, we obtain a rational approximant for which converges, for each x, to that function. Since these Taylor polynomials have simple roots, the improper integrals can be evaluated with the residues technique of integration in the complex plane, by using an appropriate contour of integration. By just using the roots of the polynomials, we get two new analytical expressions for the Error Function in terms of elementary functions. We show the behaviour of their corresponding errors by giving practical bounds for the absolute and relative errors, respectively.
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Horst Alzer, Man Kam Kwong · Proceedings of the American Mathematical Society · 2014
We prove the following subadditive property of the error function: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="erf left-parenthesis x right-parenthesis equals StartFraction 2 Over StartRoot pi EndRoot EndFraction integral Subscript 0 Superscript x Baseline e Superscript minus t squared Baseline d t left-parenthesis x element-of bold upper R right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mi> π </mml:mi> </mml:msqrt> </mml:mfrac> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mi>x</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mspace width="1em"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mbox {erf}\,(x)=\frac {2}{\sqrt {\pi }}\int _0^x e^{-t^2}dt \quad {(x\in \mathbf {R})}.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be real numbers. The inequality <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="erf left-parenthesis left-parenthesis x plus y right-parenthesis Superscript a Baseline right-parenthesis Superscript b Baseline greater-than erf left-parenthesis x Superscript a Baseline right-parenthesis Superscript b Baseline plus erf left-parenthesis y Superscript a Baseline right-parenthesis Superscript b"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width="thinmathspace"/> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>></mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>b</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mbox {erf}\,\bigl ((x+y)^a\bigr )^b> \mbox {erf}\,(x^a)^b + \mbox {erf}\,(y^a)^b</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> holds for all positive real numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a b less-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">ab\leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>
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Rinku Verma, Pranjali Kekre, Keerti Acharya · jnanabha · 2023
This paper presents a new cryptographic scheme for encryption and decryption by introducing the Laplace transform of error function. Here we have used the concept of congruence relation and modular arithmetic to nd symmetric key, cipher text and Decryption process. The implementation has been done using Matlab.
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Samuel A. Alterovitz, Blaine Johs · Thin Solid Films · 1998
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David S. Watkins · SIAM Journal on Numerical Analysis · 1977
Let u be a smooth function defined on a rectangular grid. One can construct a blended interpolant $Pu$ which interpolates u and its normal derivatives up to a given order on the grid lines. This paper gives bounds for $u - Pu$ with respect to the norms on various Sobolev spaces.
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", Nafya Hameed Mohammed, Nak Eun Cho et al. · Studia Universitatis Babes-Bolyai Matematica · 2022
"The error function takes place in a wide range in the elds of mathe- matics, mathematical physics and natural sciences. The aim of the current paper is to investigate certain properties such as univalence and close-to-convexity of normalized imaginary error function, which its region is symmetric with respect to the real axis. Some other outcomes are also obtained."
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Linsen Huang, Rui Zeng · IEEE Sensors Journal · 2024
The accurate sound field reconstruction, especially in medium- and low-frequency ranges, remains challenging due to limitations in traditional methods relying on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{{1}}$ </tex-math></inline-formula> norm for sparsity promotion. These approaches are vulnerable to noise interference and may underestimate high-amplitude components, affecting reconstruction stability. In this work, we propose a novel sound field reconstruction method utilizing the error function (ERF) as a surrogate for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{{0}}$ </tex-math></inline-formula> norm to improve both the speed and robustness of reconstruction. Comparative analysis with traditional methods, including the Tikhonov regularization method, monotonic two-step iterative shrinkage/thresholding (MTwIST) method, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{{1}}$ </tex-math></inline-formula>-CVX, demonstrates the superior performance of the ERF method in terms of sound pressure accuracy and reconstruction reliability. Monte Carlo simulations, receiver operating characteristic (ROC) analysis, and experimental results further validate that the ERF method achieves improved localization and sound field reconstruction in the low-frequency domain, with better flexibility concerning measurement distance and signal-to-noise ratio (SNR).
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Lawrence Stark · 1968
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