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2. Integration of Rational Functions

Author

Rathour, Laxmi, primary, Obradovic, Dragan, additional, Khatri, Kejal, additional, Tiwari, Shiv Kant, additional, Mishra, Lakshmi Narayan, additional, and Mishra, Vishnu Narayan, additional

Published
2023
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3. Integration of Rational Functions

Author

Rathour, Laxmi, Obradovic, Dragan, Khatri, Kejal, Tiwari, Shiv Kant, Mishra, Lakshmi Narayan, Mishra, Vishnu Narayan, Rathour, Laxmi, Obradovic, Dragan, Khatri, Kejal, Tiwari, Shiv Kant, Mishra, Lakshmi Narayan, and Mishra, Vishnu Narayan

Abstract

A rational function can always be integrated, that is, the integral of such a function is always an elementary function. The integration procedure is complex and consists of four steps: elimination of the common zero-points of the numerator and denominator, reduction to a true rational function, decomposition into partial fractions and integration of the obtained expressions using direct integration, substitution method or partial integration method. Integrating rational functions is important because integrals of rational functions of trigonometric functions as well as integrals of some irrational functions are reduced to integrals of rational functions by appropriate transformations.

Published
2023

16. Degree of Approximation by the $$ \left( {T.\,E^{\,1} } \right) $$T.E1Means of Conjugate Series of Fourier Series in the Hölder Metric

Author

Khatri, Kejal and Mishra, Vishnu

Abstract

We extend the results of Singh and Mahajan (Int J Math Math Sci 2008:9, 2008) which in turn generalizes the result of Lal and Yadav (Bull Cal Math Soc 93:191–196, 2001). In this paper, we determine the degree of approximation of functions $$ \widetilde{f\,} \in H_{\omega } , $$f~∈Hω,a new Banach space using $$ \left( {T.\,E^{\,1} } \right) $$T.E1summability means of conjugate series of Fourier series. Also, some corollaries have also been deduced from our main theorem and particular cases.

Published
2019
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27. APPROXIMATION OF FUNCTIONS BELONGING TO THE GENERALIZED LIPSCHITZ CLASS BY C¹ ⋅ Np SUMMABILITY METHOD OF CONJUGATE SERIES OF FOURIER SERIES.

Author

Narayan Mishra, Vishnu, Khatri, Kejal, and Narayan Mishra, Lakshmi

Subjects
*FOURIER series, *FOURIER analysis, *HARMONIC functions, *APPROXIMATION theory, *FUNCTIONAL analysis
Abstract

In the present study, a new theorem on the degree of approximation of function f, conjugate to a periodic function f belonging to weighted W(Lr, ξ(t))-class using semimonotonicity on the generating sequence {pn} has been established. [ABSTRACT FROM AUTHOR]

Published
2014

28. Degree of Approximation of Functions ... ∊ Hω Class by the (Np · E¹) Means in the Hölder Metric.

Author

Mishra, Vishnu Narayan and Khatri, Kejal

Subjects
*APPROXIMATION theory, *MATHEMATICAL functions, *FOURIER series, *MATHEMATICS theorems, *FOURIER analysis
Abstract

A new estimate for the degree of approximation of a function ... ∊ Hω class by (Np · E¹) means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem. [ABSTRACT FROM AUTHOR]

Published
2014
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29. HYPERGEOMETRIC REPRESENTATION FOR BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS.

Author

MISHRA, VISHNU NARAYAN, KHAN, HUZOOR H., KHATRI, KEJAL, and MISHRA, LAKSHMI NARAYAN

Subjects
HYPERGEOMETRIC series, REPRESENTATIONS of algebras, OPERATOR theory, SAMPLING errors, INTEGRALS
Abstract

In the present paper, we introduce and study the mixed summationintegral type operators having Baskakov and Beta basis functions in summation and integration, respectively. First, we estimate moments of these operators using hypergeometric series. Next, we obtain an error estimation in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. [ABSTRACT FROM AUTHOR]

Published
2013

30. Product Summability Transform of Conjugate Series of Fourier Series.

Author

Narayan Mishra, Vishnu, Khatri, Kejal, and Narayan Mishra, Lakshmi

Subjects
*SUMMABILITY theory, *FOURIER series, *TOPOLOGICAL degree, *MATHEMATICAL transformations, *GENERALIZATION, *MATHEMATICAL functions, *MATHEMATICAL analysis
Abstract

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging to Lipξ(t)-class by (E, 1)(C, 1) product summability means of conjugate series of Fourier series has been generalized for the weighted W(Lr,ξ(t)), (r ≥ 1),(t > 0)-class, where ξ(t) is nonnegative and increasing function of t, by En¹Cn¹ which is in more general form of Theorem 2 of Nigam and Sharma (2011). [ABSTRACT FROM AUTHOR]

Published
2012
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