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The method of differentiating under the integral sign

Gert Almkvist, Doron Zeilberger
1990 Journal of symbolic computation  
Introduction The method of differentiating under the integral sign can be described as follows.  ...  The Method of Differencing under the Integral Sign The same idea can be used to find a linear recurrence relation satisfied by where Nan, := an+l.  ...  The same method of proof yields that under the condition of the lemma (P(x, y)/Q(x, y))" is always holonomic, and, since holonomicity is closed under multiplication (Zeilberger, to appear) also any product  ... 
doi:10.1016/s0747-7171(08)80159-9 fatcat:dm4rwxvn3fao5cw6gkh5tsogb4

Differentiating under integral sign in Castigliano's theorem

Jozef Rédl
2019 Mathematics in Education Research and Applications  
We showed the derivation of modified Castigliano's theorem via Leibnitz rule of differentiating under integration sign.  ...  The Dormand-Prince numerical integration method was used for the numerical integration. Comparing the exact and numerical solution we got the error of numerical integration solution.  ...  The method of differentiating the functions under integral sign was analyzed by [1, 5, 8, 14] .  ... 
doi:10.15414/meraa.2019.05.01.30-37 fatcat:dctemdmh35dbleajp44y54sy4e

Study of Memory Effect in a Fuzzy EOQ Model with No Shortage

Rituparna Pakhira, Department of Applied Mathematics, University of Calcutta, Kolkata, 700009, West Bengal, India, Uttam Ghosh, Susmita Sarkar
2019 International Journal of Intelligent Systems and Applications  
Here, the fractional order model is defuzzyfied using (i) signed distance method,(ii) graded mean integration method.  ...  The feature of the fractional order derivative and fractional order integration is one of the important tools to realize the beauty of the fractional calculus.  ...  The first author would also like to thank the Department of science and Technology, Government of India, New Delhi, for the financial assistance under AORC, Inspire fellowship Scheme towards this research  ... 
doi:10.5815/ijisa.2019.11.06 fatcat:o2ttvxmybfbm5idwdjunvr6g5e

Some divergent trigonometric integrals [article]

Erik Talvila
2001 arXiv   pre-print
The original error was made when a (startlingly) famous mathematician incorrectly differentiated under the integral sign with some convergent integrals that depend on a paramater.  ...  Some divergent trigonometric integrals have appeared in standard tables for many years, listed as converging. We give a simple proof that these integrals diverge and trace their history.  ...  But, differentiating under the integral sign leads to our divergent integrals (1).  ... 
arXiv:math/0101011v1 fatcat:lc4nyejyujabnktu2zxqj62bui

Some Divergent Trigonometric Integrals

Erik Talvila
2001 The American mathematical monthly  
He then proceeds to differentiate under the integral sign with respect to b. Very bad!  ...  It gives the same incorrect values that are in the tables. DIFFERENTIATION UNDER THE INTEGRAL SIGN.  ... 
doi:10.1080/00029890.2001.11919769 fatcat:4qibbtids5c7biyyakuc4nx2qe

Some Divergent Trigonometric Integrals

Erik Talvila
2001 The American mathematical monthly  
He then proceeds to differentiate under the integral sign with respect to b. Very bad!  ...  It gives the same incorrect values that are in the tables. DIFFERENTIATION UNDER THE INTEGRAL SIGN.  ... 
doi:10.2307/2695797 fatcat:dznbmbtnirbg3d7pspqvvxaati

A Modified Differential Coherent Bit Synchronization Algorithm for BeiDou Weak Signals with Large Frequency Deviation

Zhifeng Han, Jianye Liu, Rongbing Li, Qinghua Zeng, Yi Wang
2017 Sensors  
The results show that the proposed algorithm outperforms the traditional method with a frequency deviation of 50 Hz.  ...  Firstly, a differential coherent approach is used to remove the effect of frequency deviation, and the differential delay time is set to be a multiple of bit cycle to remove the influence of NH code.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/s17071568 pmid:28677619 pmcid:PMC5539494 fatcat:sgjtkpsts5gjjgodnhvjtkk25m

Page 3135 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
“In the fifth chapter we discuss the application of the method of sign-definite functions to the problem of stability of solutions of systems of differential-difference equations.  ...  This reflects our belief that the method of sign-definite functions of a finite number of variables is a universal tool for the study of the stability of solutions for systems of ordinary differential  ... 

Page 145 of Annals of Mathematics Vol. 3, Issue 3 [page]

1902 Annals of Mathematics  
Z | aie arid y /0 converges for all values of a and represents the function ¢(@) = @. Can the integral be differentiated under the sign of integration ? and a = 41. The integral 42.  ...  Is it allowable here to differentiate under the sign of integration for all values of a’ If not, and if you bave used the method suggested in Problem 13, have you been careful to make your solution rigorous  ... 

Numerical time integration of lumped parameter systems governed by implicit constitutive relations [article]

Saeid Karimi
2017 arXiv   pre-print
Using this new method, the limitations of the trapezoidal time-integration methods when applied to a non-smooth differential-algebraic equation are highlighted.  ...  Under this scheme, implicit-explicit integration of the governing equations is possible.  ...  A complete description of numerical integration under the proposed methods for cases of linear and nonlinear Bingham models is given.  ... 
arXiv:1711.06352v1 fatcat:txnak6cgwzb47h47za7aeqyiky

The Application of Modern Theories of Integration to the Solution of Differential Equations

T. C. Fry
1921 Annals of Mathematics  
Differentiation and integration under the sign of integration.  ...  The Stieltjes integral, 1(t) = fib f(X)dx(X, t) (27) Differentiation under the sign of integration.  ...  (b) It is found that an important class of Stieltjes integrals may be differentiated and integrated under the sign of integration, provided that, when differentiation is performed, left-hand unilateral  ... 
doi:10.2307/1967439 fatcat:dcfot4ymkvhwjhffbsw4syzvaq

Integrated Guidance and Control Method for the Interception of Maneuvering Hypersonic Vehicle Based on High Order Sliding Mode Approach

Kang Chen, Bin Fu, Yuening Ding, Jie Yan
2015 Mathematical Problems in Engineering  
To estimate the high order derivatives of the sliding manifold which is required in the HOSM method, an Arbitrary Order Robust Exact Differentiator is presented.  ...  This paper focuses on the integrated guidance and control (IGC) method applied in the interception of maneuvering near space hypersonic vehicles using the homogeneous high order sliding mode (HOSM) approach  ...  Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.  ... 
doi:10.1155/2015/648231 fatcat:yponx66255fiddk4vp5rvrlame

The XY

G. H. Rayment, Nagazawa Kamenosuke
1913 Mathematical Gazette  
An integral equation is an equation involving signs of integration just as a differential equation involves signs of differentiation. Thus dy y=l+ xydx is the same as d=xy. y=l when x=0.  ...  An integral equation is an equation involving signs of integration just as a differential equation involves signs of differentiation. Thus dy y=l+ xydx is the same as d=xy. y=l when x=0.  ... 
doi:10.2307/3604373 fatcat:diph4fdxnnexhpzs7r3xtxeriy

From Cagniard's method for solving seismic pulse problems to the method of the differential transform

H.C. Murrell, Abraham Ungar
1982 Computers and Mathematics with Applications  
The technique of the differential transform, which evolved from that of Cagniard, involves an integral-free transform and avoids the integral considerations associated with Cagniard's technique.  ...  The integral transform method of Cagniard involves path deformations in the complex plane, residue calculus and, in some cases, branch cuts.  ...  The crucial point, however, is that it might be difficult or impossible to follow the procedure in Diagram 3.1 if complicated functions A(r), B(T) or C(r) are inserted under the integral signs of (4.3)  ... 
doi:10.1016/0898-1221(82)90050-5 fatcat:en7kk6qm4nbwrpgfwqtlptrxsa

Page 112 of Proceedings of the National Academy of Sciences of the United States of America Vol. 18, Issue 1 [page]

1932 Proceedings of the National Academy of Sciences of the United States of America  
ON THE CHANGES OF SIGN OF THE DERIVATIVES OF A FUNCTION DEFINED BY A LAPLACE INTEGRAL By D. V.  ...  Under certain conditions this method enables one to complete the covariant derivative of a tensor J in avery satisfactory manner.  ... 
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