List of derivatives and integrals in alternative calculi

There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi.{{Cite book |last1=Grossman |first1=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}} Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.{{Cite journal |last1=Bashirov |first1=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 |bibcode=2008JMAA..337...36B |doi-access=free }}{{Cite journal |last1=Filip |first1=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}}{{Cite journal |last1=Florack |first1=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 |s2cid=254652400 |issn=0924-9907 |via=SpringerLink|doi-access=free }}

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;

$\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ is the digamma function,

$\operatorname{K}(x)=e^{\zeta^\prime(-1,x)-\zeta^\prime(-1)}=e^{\frac{z-z^2}{2}+\frac z2 \ln (2\pi)-\psi^{(-2)}(z)}$ is the K-function,

$(!x)=\frac{\Gamma(x+1,-1)}{e}$ is subfactorial,

$B_a(x)=-a\zeta(-a+1,x)$ are the generalized to real numbers Bernoulli polynomials.

class="wikitable"
Function $f(x)$ ! Derivative $f'(x)$ ! Integral $\int f(x) dx$ (constant term is omitted) ! Multiplicative derivative $f^*(x)$ ! Multiplicative integral $\int f(x)^{dx}$ (constant factor is omitted) ! Discrete derivative (difference) $\Delta f(x)$ ! Discrete integral (antidifference) $\Delta^{-1} f(x)$ (constant term is omitted) ! Discrete multiplicative derivative{{Cite conference \|last1=Khatami \|first1=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}} (multiplicative difference) ! Discrete multiplicative integral{{Cite conference \|last1=Jahanshahi \|first1=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}} (indefinite product) $\prod _x f(x)$ (constant factor is omitted)
$a$ \| $0$ \| $ax$ \| $1$ \| $a^x$ \| $0$ \| $ax$ \| $1$ \| $a^x$
$x$ \| $1$ \| $\frac{x^2}{2}$ \| $\sqrt[x]{e}$ \| $\frac{x^x}{e^x}$ \| $1$ \| $\frac{x^2}{2}-\frac x2$ \| $1+\frac 1x$ \| $\Gamma(x)$
$ax+b$ \| $a$ \| $\frac{ax^2+2bx}{2}$ \| $\exp\left(\frac{a}{ax+b}\right)$ \| $\frac{(b+a x)^{\frac{b}{a}+x}}{e^x}$ \| $a$ \| $\frac{ax^2+2bx-ax}{2}$ \| $1+\frac{a}{ax+b}$ \| $\frac{a^x\Gamma(\frac{ax+b}{a})}{\Gamma(\frac{a+b}{a})}$
$\frac 1x$ \| $-\frac{1}{x^2}$ \| $\ln \|x\|$ \| $\frac{1}{\sqrt[x]{e}}$ \| $\frac{e^x}{x^x}$ \| $-\frac{1}{x+x^2}$ \| $\psi(x)$ \| $\frac{x}{x+1}$ \| $\frac{1}{\Gamma(x)}$
$x^a$ \| $ax^{a-1}$ \| $\frac{x^{a+1}}{a+1}$ \| $e^{\frac ax}$ \| $e^{-a x} x^{ax}$ \| $(x+1)^a-x^a$ \| $a\notin \mathbb{Z}^-\,;$ $\frac{B_{a+1}(x)}{a+1},$ $a\in\mathbb{Z}^-\,;$ $\frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)},$ \| $\left(1+\frac 1x\right)^a$ \| $\Gamma(x)^a$
$a^x$ \| $a^x\ln a$ \| $\frac{a^x}{\ln a}$ \| $a$ \| $a^{\frac{x^2}{2}}$ \| $(a-1)a^x$ \| $\frac{a^x}{a-1}$ \| $a$ \| $a^{\frac{x^2+x}{2}}$
$\sqrt[x]{a}$ \| $-\frac{\sqrt[x]{a}\ln a}{x^2}$ \| $x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a$ \| $a^{-\frac{1}{x^2}}$ \| $a^{\ln x}$ \| $a^{\frac{1}{1+x}}-a^{\frac{1}{x}}$ \| $?$ \| $a^{-\frac{1}{x+x^2}}$ \| $a^{\psi(x)}$
$\log_a x$ \| $\frac{1}{x \ln a}$ \| $\log_a x^x-\frac{x}{\ln a}$ \| $\exp \left(\frac{1}{x\ln x}\right)$ \| $\frac{(\log_a x)^x}{e^{\operatorname{li}(x)}}$ \| $\log_a\left(\frac 1x+1\right)$ \| $\log_a \Gamma(x)$ \| $\log_x (x+1)$ \| $?$
$x^x$ \| $x^x(1+\ln x)$ \| $?$ \| $ex$ \| $e^{-\frac{1}{4}x^2(1-2\ln x)}$ \| $(x+1)^{x+1}-x^x$ \| $?$ \| $\frac{(x+1)^{x+1}}{x^x}$ \| $\operatorname{K}(x)$
$\Gamma(x)$ \| $\Gamma(x)\psi(x)$ \| $?$ \| $e^{\psi(x)}$ \| $e^{\psi^{(-2)}(x)}$ \| $(x-1)\Gamma(x)$ \| $(-1)^{x+1}\Gamma(x)(!(-x))$ \| $x$ \| $\frac{\Gamma(x)^{x-1}}{\operatorname{K}(x)}$
$\sin(ax)$ \| $a\cos(ax)$ \| $-\dfrac{\cos(ax)}{a}$ \| $e^{a\cot(ax)}$ \| $?$ \| $\sin(a(x+1))-\sin(ax)$ \| $-\dfrac{1}{2}\csc\left(\dfrac{a}{2}\right)\cos\left(\dfrac{a}{2}-ax\right)$ \| $\cos(a)+\sin(a)\cot(ax)$ \| $?$

class="wikitable"

Function

f(x)

! Derivative
$f'(x)$

! Integral
$\int f(x) dx$
(constant term is omitted)

! Multiplicative derivative
$f^*(x)$

! Multiplicative integral
$\int f(x)^{dx}$
(constant factor is omitted)

! Discrete derivative (difference)
$\Delta f(x)$

! Discrete integral (antidifference)
$\Delta^{-1} f(x)$
(constant term is omitted)

! Discrete
multiplicative derivative{{Cite conference |last1=Khatami |first1=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}}
(multiplicative difference)

! Discrete multiplicative integral{{Cite conference |last1=Jahanshahi |first1=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}} (indefinite product)
$\prod _x f(x)$
(constant factor is omitted)

a

| $0$

| $ax$

| $1$

| $a^x$

| $0$

| $ax$

| $1$

| $a^x$

x

| $1$

| $\frac{x^2}{2}$

| $\sqrt[x]{e}$

| $\frac{x^x}{e^x}$

| $1$

| $\frac{x^2}{2}-\frac x2$

| $1+\frac 1x$

| $\Gamma(x)$

ax+b

| $a$

| $\frac{ax^2+2bx}{2}$

| $\exp\left(\frac{a}{ax+b}\right)$

| $\frac{(b+a x)^{\frac{b}{a}+x}}{e^x}$

| $a$

| $\frac{ax^2+2bx-ax}{2}$

| $1+\frac{a}{ax+b}$

| $\frac{a^x\Gamma(\frac{ax+b}{a})}{\Gamma(\frac{a+b}{a})}$

\frac 1x

| $-\frac{1}{x^2}$

| $\ln |x|$

| $\frac{1}{\sqrt[x]{e}}$

| $\frac{e^x}{x^x}$

| $-\frac{1}{x+x^2}$

| $\psi(x)$

| $\frac{x}{x+1}$

| $\frac{1}{\Gamma(x)}$

x^a

| $ax^{a-1}$

| $\frac{x^{a+1}}{a+1}$

| $e^{\frac ax}$

| $e^{-a x} x^{ax}$

| $(x+1)^a-x^a$

| $a\notin \mathbb{Z}^-\,;$ $\frac{B_{a+1}(x)}{a+1},$
$a\in\mathbb{Z}^-\,;$ $\frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)},$

| $\left(1+\frac 1x\right)^a$

| $\Gamma(x)^a$

a^x

| $a^x\ln a$

| $\frac{a^x}{\ln a}$

| $a$

| $a^{\frac{x^2}{2}}$

| $(a-1)a^x$

| $\frac{a^x}{a-1}$

| $a$

| $a^{\frac{x^2+x}{2}}$

\sqrt[x]{a}

| $-\frac{\sqrt[x]{a}\ln a}{x^2}$

| $x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a$

| $a^{-\frac{1}{x^2}}$

| $a^{\ln x}$

| $a^{\frac{1}{1+x}}-a^{\frac{1}{x}}$

| $?$

| $a^{-\frac{1}{x+x^2}}$

| $a^{\psi(x)}$

\log_a x

| $\frac{1}{x \ln a}$

| $\log_a x^x-\frac{x}{\ln a}$

| $\exp \left(\frac{1}{x\ln x}\right)$

| $\frac{(\log_a x)^x}{e^{\operatorname{li}(x)}}$

| $\log_a\left(\frac 1x+1\right)$

| $\log_a \Gamma(x)$

| $\log_x (x+1)$

| $?$

x^x

| $x^x(1+\ln x)$

| $?$

| $ex$

| $e^{-\frac{1}{4}x^2(1-2\ln x)}$

| $(x+1)^{x+1}-x^x$

| $?$

| $\frac{(x+1)^{x+1}}{x^x}$

| $\operatorname{K}(x)$

\Gamma(x)

| $\Gamma(x)\psi(x)$

| $?$

| $e^{\psi(x)}$

| $e^{\psi^{(-2)}(x)}$

| $(x-1)\Gamma(x)$

| $(-1)^{x+1}\Gamma(x)(!(-x))$

| $x$

| $\frac{\Gamma(x)^{x-1}}{\operatorname{K}(x)}$

\sin(ax)

| $a\cos(ax)$

| $-\dfrac{\cos(ax)}{a}$

| $e^{a\cot(ax)}$

| $?$

| $\sin(a(x+1))-\sin(ax)$

| $-\dfrac{1}{2}\csc\left(\dfrac{a}{2}\right)\cos\left(\dfrac{a}{2}-ax\right)$

| $\cos(a)+\sin(a)\cot(ax)$

| $?$

References

External links

[https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]

{{DEFAULTSORT:Derivatives And Integrals Of Elementary Functions In Alternative Calculi}}

Category:Non-Newtonian calculus

Category:Mathematics-related lists

Category:Mathematical tables

List of derivatives and integrals in alternative calculi

Table

See also

References

External links