List of derivatives and integrals in alternative calculi: Difference between revisions
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There are many alternatives to the [[Calculus|classical calculus]] of [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]]; for example, each of the infinitely many non-Newtonian calculi.<ref> |
There are many alternatives to the [[Calculus|classical calculus]] of [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]]; for example, each of the infinitely many non-Newtonian calculi.<ref>{{Cite book |last=Grossman |first=Michael |url={{google books|plainurl=yes|id=RLuJmE5y8pYC}} |title=Non-Newtonian calculus |last2=Katz |first2=Robert |date=1972 |publisher=Lee Press |isbn=0-912938-01-3 |location=Pigeon Cove, Mass. |oclc=308822}}</ref> Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.<ref>{{Cite journal |last=Bashirov |first=Agamirza E. |last2=Kurpınar |first2=Emine Mısırlı |last3=Özyapıcı |first3=Ali |date=1 January 2008 |title=Multiplicative calculus and its applications |journal=[[Journal of Mathematical Analysis and Applications]] |language=en |volume=337 |issue=1 |pages=36–48 |doi=10.1016/j.jmaa.2007.03.081 |via=ScienceDirect}}</ref><ref>{{Cite journal |last=Filip |first=Diana Andrada |last2=Piatecki |first2=Cyrille |date=2014 |title=A non-Newtonian examination of the theory of exogenous economic growth |url=https://www.longdom.org/abstract/a-nonnewtonian-examination-of-the-theory-of-exogenous-economic-growth-4496.html |journal=[[Mathematica Eterna]] |volume=4 |issue=2 |pages=101–117}}</ref><ref>{{Cite journal |last=Florack |first=Luc |last2=van Assen |first2=Hans |date=January 2012 |title=Multiplicative Calculus in Biomedical Image Analysis |journal=[[Journal of Mathematical Imaging and Vision]] |language=en |volume=42 |issue=1 |pages=64–75 |doi=10.1007/s10851-011-0275-1 |issn=0924-9907 |via=SpringerLink}}</ref> |
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The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi. |
The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi. |
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! [[Difference operator|Discrete derivative]] (difference)<br><math>\Delta f(x)</math> |
! [[Difference operator|Discrete derivative]] (difference)<br><math>\Delta f(x)</math> |
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! [[Indefinite sum|Discrete integral]] (antidifference)<br><math>\Delta^{-1} f(x)</math><br>(constant term is omitted) |
! [[Indefinite sum|Discrete integral]] (antidifference)<br><math>\Delta^{-1} f(x)</math><br>(constant term is omitted) |
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! [[Discrete multiplicative derivative|Discrete<br> multiplicative derivative]]<ref> |
! [[Discrete multiplicative derivative|Discrete<br> multiplicative derivative]]<ref>{{Cite conference |last=Khatami |first=Hamid Reza |last2=Jahanshahi |first2=M. |last3=Aliev |first3=N. |date=5–10 July 2004 |title=An analytical method for some nonlinear difference equations by discrete multiplicative differentiation |url=http://faculty.uaeu.ac.ae/hakca/papers/khatami.pdf |conference=Dynamical Systems and Applications, Proceedings |location=Antalya, Turkey |pages=455—462 |archiveurl=https://web.archive.org/web/20110706062336/http://faculty.uaeu.ac.ae/hakca/papers/khatami.pdf |archivedate=6 Jul 2011}}</ref><br>(multiplicative difference) |
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! [[Indefinite product|Discrete multiplicative integral]]<ref> |
! [[Indefinite product|Discrete multiplicative integral]]<ref>{{Cite conference |last=Jahanshahi |first=M. |last2=Aliev |first2=N. |last3=Khatami |first3=Hamid Reza |date=5–10 July 2004 |title=An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration |url=http://faculty.uaeu.ac.ae/hakca/papers/jahanshahi.pdf |conference=Dynamical Systems and Applications, Proceedings |location=Antalya, Turkey |pages=425—435 |archiveurl=https://web.archive.org/web/20110706062316/http://faculty.uaeu.ac.ae/hakca/papers/jahanshahi.pdf |archivedate=6 Jul 2011}}</ref> (indefinite product)<br><math>\prod _x f(x)</math><br>(constant factor is omitted) |
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Revision as of 16:35, 14 November 2022
This article needs additional citations for verification. (January 2010) |
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi.[1] Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.[2][3][4]
The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi.
Table
In the following table is the digamma function, is the K-function, is subfactorial, are the generalized to real numbers Bernoulli polynomials.
Function |
Derivative |
Integral (constant term is omitted) |
Multiplicative derivative |
Multiplicative integral (constant factor is omitted) |
Discrete derivative (difference) |
Discrete integral (antidifference) (constant term is omitted) |
Discrete multiplicative derivative[5] (multiplicative difference) |
Discrete multiplicative integral[6] (indefinite product) (constant factor is omitted) |
---|---|---|---|---|---|---|---|---|
See also
References
- ^ Grossman, Michael; Katz, Robert (1972). Non-Newtonian calculus. Pigeon Cove, Mass.: Lee Press. ISBN 0-912938-01-3. OCLC 308822.
- ^ Bashirov, Agamirza E.; Kurpınar, Emine Mısırlı; Özyapıcı, Ali (1 January 2008). "Multiplicative calculus and its applications". Journal of Mathematical Analysis and Applications. 337 (1): 36–48. doi:10.1016/j.jmaa.2007.03.081 – via ScienceDirect.
- ^ Filip, Diana Andrada; Piatecki, Cyrille (2014). "A non-Newtonian examination of the theory of exogenous economic growth". Mathematica Eterna. 4 (2): 101–117.
- ^ Florack, Luc; van Assen, Hans (January 2012). "Multiplicative Calculus in Biomedical Image Analysis". Journal of Mathematical Imaging and Vision. 42 (1): 64–75. doi:10.1007/s10851-011-0275-1. ISSN 0924-9907 – via SpringerLink.
- ^ Khatami, Hamid Reza; Jahanshahi, M.; Aliev, N. (5–10 July 2004). An analytical method for some nonlinear difference equations by discrete multiplicative differentiation (PDF). Dynamical Systems and Applications, Proceedings. Antalya, Turkey. pp. 455–462. Archived from the original (PDF) on 6 Jul 2011.
- ^ Jahanshahi, M.; Aliev, N.; Khatami, Hamid Reza (5–10 July 2004). An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration (PDF). Dynamical Systems and Applications, Proceedings. Antalya, Turkey. pp. 425–435. Archived from the original (PDF) on 6 Jul 2011.