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

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[5] (multiplicative difference)	Discrete multiplicative integral[6] (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)},\,$	$\left(1+\frac 1x\right)^a\,$	$\Gamma(x)^a\,$
$a^x\,$	$a^x\ln a\,$	$\frac{a^x}{\ln a}\,$	$a\,$	$a^{\frac{x^2}{2}}\,$	$(a-1)a^x\,$	$\frac{a^x}{a-1}\,$	$a\,$	$a^{\frac{x^2-x}{2}}\,$
$\sqrt[x]{a}\,$	$-\frac{\sqrt[x]{a}\ln a}{x^2}\,$	$x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a\,$	$a^{-\frac{1}{x^2}}\,$	$a^{\ln x}\,$	$a^{\frac{1}{1+x}}-a^{\frac{1}{x}}\,$	$? \,$	$a^{-\frac{1}{x+x^2}}\,$	$a^{\psi(x)}\,$
$\log_a x\,$	$\frac{1}{x \ln a}\,$	$\log_a x^x-\frac{x}{\ln a}$	$\exp \left(\frac{1}{x\ln x}\right) \,$	$\frac{(\log_a x)^x}{e^{\operatorname{li}(x)}}\,$	$\log_a\left(\frac 1x -1\right)\,$	$\log_a \Gamma(x)\,$	$\log_x (x+1)\,$	$?\,$
$x^x\,$	$x^x(1+\ln x)\,$	$?\,$	$ex\,$	$e^{-\frac{1}{4}x^2(1-2\ln x)}\,$	$(x+1)^{x+1}-x^x\,$	$? \,$	$\frac{(x+1)^{x+1}}{x^x}\,$	$\operatorname{K}(x)\,$
$\Gamma(x)\,$	$\Gamma(x)\psi(x)\,$	$?\,$	$e^{\psi(x)}\,$	$e^{\psi^{(-2)}(x)}\,$	$(x-1)\Gamma(x)\,$	$(-1)^{x+1}\Gamma(x)(!(-x))\,$	$x\,$	$\frac{\Gamma(x)^{x-1}}{\operatorname{K}(x)}\,$

See also

• Indefinite product
• Multiplicative calculus
• Product integral

References

1. M. Grossman and R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
2. Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. "Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, 2008.
3. Diana Andrada Filip and Cyrille Piatecki. "A non-Newtonian examination of the theory of exogenous economic growth", CNCSIS – UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010.
4. Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, DOI: 10.1007/s10851-011-0275-1, 2011.
5. H. R. Khatami & M. Jahanshahi & N. Aliev (2004). "An analytical method for some nonlinear difference equations by discrete multiplicative differentiation"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 455—462
6. M. Jahanshahi, N. Aliev and H. R. Khatami (2004). "An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 425—435