Kaniadakis statistics

Template:Short description Template:Multiple issues Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann–Gibbs statistical mechanics, based on a relativistic generalization of the classical Boltzmann–Gibbs–Shannon entropy (commonly referred to as Kaniadakis entropy or κ-entropy). Introduced by the Greek Italian physicist Giorgio Kaniadakis in 2001, κ-statistical mechanics preserve the main features of ordinary statistical mechanics and have attracted the interest of many researchers in recent years. The κ-distribution is currently considered one of the most viable candidates for explaining complex physical, natural or artificial systems involving power-law tailed statistical distributions. Kaniadakis statistics have been adopted successfully in the description of a variety of systems in the fields of cosmology, astrophysics, condensed matter, quantum physics, seismology, genomics, economics, epidemiology, and many others.

Mathematical formalism

The mathematical formalism of κ-statistics is generated by κ-deformed functions, especially the κ-exponential function.

κ-exponential function

Plot of the κ-exponential function ${\exp }_{\kappa }(x)$ for three different κ-values. The solid black curve corresponding to the ordinary exponential function $\exp (x)$ ($\kappa =0$).

The Kaniadakis exponential (or κ-exponential) function is a one-parameter generalization of an exponential function, given by:

${\exp }_{\kappa }(x)=\{\begin{array}{cc}(\sqrt{1+{\kappa }^2{x}^2}+\kappa x{)}^{\frac{1}{\kappa }}& \text{if }0<\kappa <1.\\ \exp (x)& \text{if }\kappa =0,\end{array}$

with ${\exp }_{-\kappa }(x)={\exp }_{\kappa }(x)$.

The κ-exponential for $0<\kappa <1$ can also be written in the form:

${\exp }_{\kappa }(x)=\exp (\frac{1}{\kappa }\text{arcsinh}(\kappa x)).$

The first five terms of the Taylor expansion of

${\exp }_{\kappa }(x)$

are given by:

${\exp }_{\kappa }(x)=1+x+\frac{x^2}{2}+(1-{\kappa }^2)\frac{x^3}{3!}+(1-4{\kappa }^2)\frac{x^4}{4!}+\cdots$

where the first three are the same as a typical exponential function.

Basic properties

The κ-exponential function has the following properties of an exponential function:

${\exp }_{\kappa }(x)\in {ℂ}^{\mathrm{\infty }}(ℝ)$

$\frac{d}{dx}{\exp }_{\kappa }(x)>0$

$\frac{d^2}{d{x}^2}{\exp }_{\kappa }(x)>0$

${\exp }_{\kappa }(-\mathrm{\infty })=0^{+}$

${\exp }_{\kappa }(0)=1$

${\exp }_{\kappa }(+\mathrm{\infty })=+\mathrm{\infty }$

${\exp }_{\kappa }(x){\exp }_{\kappa }(-x)=-1$

For a real number $r$, the κ-exponential has the property:

$[{\exp }_{\kappa }(x){]}^r={\exp }_{\kappa /r}(rx)$.

κ-logarithm function

Plot of the κ-logarithmic function ${\ln }_{\kappa }(x)$ for three different κ-values. The solid black curve corresponding to the ordinary logarithmic function $\ln (x)$ ($\kappa =0$).

The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,

${\ln }_{\kappa }(x)=\{\begin{array}{cc}\frac{x^{\kappa }-x^{-\kappa }}{2\kappa }& \text{if }0<\kappa <1,\\ \ln (x)& \text{if }\kappa =0,\end{array}$

with ${\ln }_{-\kappa }(x)={\ln }_{\kappa }(x)$, which is the inverse function of the κ-exponential:

${\ln }_{\kappa }({\exp }_{\kappa }(x))={\exp }_{\kappa }({\ln }_{\kappa }(x))=x.$

The κ-logarithm for $0<\kappa <1$ can also be written in the form:

${\ln }_{\kappa }(x)=\frac{1}{\kappa }\sinh (\kappa \ln (x))$

The first three terms of the Taylor expansion of ${\ln }_{\kappa }(x)$ are given by:

${\ln }_{\kappa }(1+x)=x-\frac{x^2}{2}+(1+\frac{{\kappa }^2}{2})\frac{x^3}{3}-\cdots$

following the rule

${\ln }_{\kappa }(1+x)=\sum _{n=1}^{\mathrm{\infty }}{b}_n(\kappa )(-1{)}^{n-1}\frac{x^n}{n}$

with $b_1(\kappa )=1$, and

$b_n(\kappa )(x)=\{\begin{array}{cc}1& \text{if }n=1,\\ \frac{1}{2}(1-\kappa )(1-\frac{\kappa }{2})...(1-\frac{\kappa }{n-1}),+\frac{1}{2}(1+\kappa )(1+\frac{\kappa }{2})...(1+\frac{\kappa }{n-1})& \text{for }n>1,\end{array}$

where $b_n(0)=1$ and $b_n(-\kappa )=b_n(\kappa )$. The two first terms of the Taylor expansion of ${\ln }_{\kappa }(x)$ are the same as an ordinary logarithmic function.

Basic properties

The κ-logarithm function has the following properties of a logarithmic function:

${\ln }_{\kappa }(x)\in {ℂ}^{\mathrm{\infty }}({ℝ}^{+})$

$\frac{d}{dx}{\ln }_{\kappa }(x)>0$

$\frac{d^2}{d{x}^2}{\ln }_{\kappa }(x)<0$

${\ln }_{\kappa }(0^{+})=-\mathrm{\infty }$

${\ln }_{\kappa }(1)=0$

${\ln }_{\kappa }(+\mathrm{\infty })=+\mathrm{\infty }$

${\ln }_{\kappa }(1/x)=-{\ln }_{\kappa }(x)$

For a real number $r$, the κ-logarithm has the property:

${\ln }_{\kappa }(x^r)=r{\ln }_{r\kappa }(x)$

κ-Algebra

κ-sum

For any $x,y\in ℝ$ and $|\kappa |<1$, the Kaniadakis sum (or κ-sum) is defined by the following composition law:

$x\stackrel{\kappa }{\oplus }y=x\sqrt{1+{\kappa }^2{y}^2}+y\sqrt{1+{\kappa }^2{x}^2}$,

that can also be written in form:

$x\stackrel{\kappa }{\oplus }y=\frac{1}{\kappa }\sinh (\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa x)+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa y))$,

where the ordinary sum is a particular case in the classical limit $\kappa \to 0$: $x\stackrel{0}{\oplus }y=x+y$.

The κ-sum, like the ordinary sum, has the following properties:

$\text{1. associativity:}(x\stackrel{\kappa }{\oplus }y)\stackrel{\kappa }{\oplus }z=x\stackrel{\kappa }{\oplus }(y\stackrel{\kappa }{\oplus }z)$

$\text{2. neutral element:}x\stackrel{\kappa }{\oplus }0=0\stackrel{\kappa }{\oplus }x=x$

$\text{3. opposite element:}x\stackrel{\kappa }{\oplus }(-x)=(-x)\stackrel{\kappa }{\oplus }x=0$

$\text{4. commutativity:}x\stackrel{\kappa }{\oplus }y=y\stackrel{\kappa }{\oplus }x$

The κ-difference $\stackrel{\kappa }{\ominus }$ is given by $x\stackrel{\kappa }{\ominus }y=x\stackrel{\kappa }{\oplus }(-y)$.

The fundamental property ${\exp }_{\kappa }(-x){\exp }_{\kappa }(x)=1$ arises as a special case of the more general expression below: ${\exp }_{\kappa }(x){\exp }_{\kappa }(y)=ex{p}_{\kappa }(x\stackrel{\kappa }{\oplus }y)$

Furthermore, the κ-functions and the κ-sum present the following relationships:

${\ln }_{\kappa }(xy)={\ln }_{\kappa }(x)\stackrel{\kappa }{\oplus }{\ln }_{\kappa }(y)$

κ-product

For any $x,y\in ℝ$ and $|\kappa |<1$, the Kaniadakis product (or κ-product) is defined by the following composition law:

$x\stackrel{\kappa }{\otimes }y=\frac{1}{\kappa }\sinh (\frac{1}{\kappa }\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa x)\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa y))$,

where the ordinary product is a particular case in the classical limit $\kappa \to 0$: $x\stackrel{0}{\otimes }y=x\times y=xy$.

The κ-product, like the ordinary product, has the following properties:

$\text{1. associativity:}(x\stackrel{\kappa }{\otimes }y)\stackrel{\kappa }{\otimes }z=x\stackrel{\kappa }{\otimes }(y\stackrel{\kappa }{\otimes }z)$

$\text{2. neutral element:}x\stackrel{\kappa }{\otimes }I=I\stackrel{\kappa }{\otimes }x=x\text{for}I={\kappa }^{-1}\sinh \kappa \stackrel{\kappa }{\oplus }x=x$

$\text{3. inverse element:}x\stackrel{\kappa }{\otimes }\bar{x}=\bar{x}\stackrel{\kappa }{\otimes }x=I\text{for}\bar{x}={\kappa }^{-1}\sinh ({\kappa }^2/\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa x))$

$\text{4. commutativity:}x\stackrel{\kappa }{\otimes }y=y\stackrel{\kappa }{\otimes }x$

The κ-division $\stackrel{\kappa }{\oslash }$ is given by $x\stackrel{\kappa }{\oslash }y=x\stackrel{\kappa }{\otimes }\bar{y}$.

The κ-sum $\stackrel{\kappa }{\oplus }$ and the κ-product $\stackrel{\kappa }{\otimes }$ obey the distributive law: $z\stackrel{\kappa }{\otimes }(x\stackrel{\kappa }{\oplus }y)=(z\stackrel{\kappa }{\otimes }x)\stackrel{\kappa }{\oplus }(z\stackrel{\kappa }{\otimes }y)$.

The fundamental property ${\ln }_{\kappa }(1/x)=-{\ln }_{\kappa }(x)$ arises as a special case of the more general expression below:

${\ln }_{\kappa }(xy)={\ln }_{\kappa }(x)\stackrel{\kappa }{\oplus }{\ln }_{\kappa }(y)$

Furthermore, the κ-functions and the κ-product present the following relationships:

${\exp }_{\kappa }(x)\stackrel{\kappa }{\otimes }{\exp }_{\kappa }(y)={\exp }_{\kappa }(x+y)$

${\ln }_{\kappa }(x\stackrel{\kappa }{\otimes }y)={\ln }_{\kappa }(x)+{\ln }_{\kappa }(y)$

κ-Calculus

κ-Differential

The Kaniadakis differential (or κ-differential) of $x$ is defined by:

${\mathrm{d}}_{\kappa }x=\frac{\mathrm{d}x}{\sqrt{1+{\kappa }^2{x}^2}}$.

So, the κ-derivative of a function $f(x)$ is related to the Leibniz derivative through:

$\frac{\mathrm{d}f(x)}{{\mathrm{d}}_{\kappa }x}={\gamma }_{\kappa }(x)\frac{\mathrm{d}f(x)}{\mathrm{d}x}$,

where ${\gamma }_{\kappa }(x)=\sqrt{1+{\kappa }^2{x}^2}$ is the Lorentz factor. The ordinary derivative $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ is a particular case of κ-derivative $\frac{\mathrm{d}f(x)}{{\mathrm{d}}_{\kappa }x}$ in the classical limit $\kappa \to 0$.

κ-Integral

The Kaniadakis integral (or κ-integral) is the inverse operator of the κ-derivative defined through

$\int {\mathrm{d}}_{\kappa }xf(x)=\int \frac{\mathrm{d}x}{\sqrt{1+{\kappa }^2{x}^2}}f(x)$,

which recovers the ordinary integral in the classical limit $\kappa \to 0$.

κ-Trigonometry

κ-Cyclic Trigonometry

[click on the figure] Plot of the κ-sine and κ-cosine functions for $\kappa =0$ (black curve) and $\kappa =0.1$ (blue curve).

The Kaniadakis cyclic trigonometry (or κ-cyclic trigonometry) is based on the κ-cyclic sine (or κ-sine) and κ-cyclic cosine (or κ-cosine) functions defined by:

${\sin }_{\kappa }(x)=\frac{{\exp }_{\kappa }(ix)-{\exp }_{\kappa }(-ix)}{2i}$,

${\cos }_{\kappa }(x)=\frac{{\exp }_{\kappa }(ix)+{\exp }_{\kappa }(-ix)}{2}$,

where the κ-generalized Euler formula is

${\exp }_{\kappa }(\pm ix)={\cos }_{\kappa }(x)\pm i{\sin }_{\kappa }(x)$.:

The κ-cyclic trigonometry preserves fundamental expressions of the ordinary cyclic trigonometry, which is a special case in the limit κ → 0, such as:

${\cos }_{\kappa }^2(x)+{\sin }_{\kappa }^2(x)=1$

${\sin }_{\kappa }(x\stackrel{\kappa }{\oplus }y)={\sin }_{\kappa }(x){\cos }_{\kappa }(y)+{\cos }_{\kappa }(x){\sin }_{\kappa }(y)$.

The κ-cyclic tangent and κ-cyclic cotangent functions are given by:

${\tan }_{\kappa }(x)=\frac{{\sin }_{\kappa }(x)}{{\cos }_{\kappa }(x)}$

${\cot }_{\kappa }(x)=\frac{{\cos }_{\kappa }(x)}{{\sin }_{\kappa }(x)}$.

The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit $\kappa \to 0$.

κ-Inverse cyclic function

The Kaniadakis inverse cyclic functions (or κ-inverse cyclic functions) are associated to the κ-logarithm:

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}}_{\kappa }(x)=-i{\ln }_{\kappa }(\sqrt{1-x^2}+ix)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}}_{\kappa }(x)=-i{\ln }_{\kappa }(\sqrt{x^2-1}+x)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}}_{\kappa }(x)=i{\ln }_{\kappa }\left(\sqrt{\frac{1-ix}{1+ix}}\right)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}}_{\kappa }(x)=i{\ln }_{\kappa }\left(\sqrt{\frac{ix+1}{ix-1}}\right)$.

κ-Hyperbolic Trigonometry

The Kaniadakis hyperbolic trigonometry (or κ-hyperbolic trigonometry) is based on the κ-hyperbolic sine and κ-hyperbolic cosine given by:

${\sinh }_{\kappa }(x)=\frac{{\exp }_{\kappa }(x)-{\exp }_{\kappa }(-x)}{2}$,

${\cosh }_{\kappa }(x)=\frac{{\exp }_{\kappa }(x)+{\exp }_{\kappa }(-x)}{2}$,

where the κ-Euler formula is

${\exp }_{\kappa }(\pm x)={\cosh }_{\kappa }(x)\pm {\sinh }_{\kappa }(x)$.

The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:

${\tanh }_{\kappa }(x)=\frac{{\sinh }_{\kappa }(x)}{{\cosh }_{\kappa }(x)}$

${\coth }_{\kappa }(x)=\frac{{\cosh }_{\kappa }(x)}{{\sinh }_{\kappa }(x)}$.

The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit $\kappa \to 0$.

From the κ-Euler formula and the property ${\exp }_{\kappa }(-x){\exp }_{\kappa }(x)=1$ the fundamental expression of κ-hyperbolic trigonometry is given as follows:

${\cosh }_{\kappa }^2(x)-{\sinh }_{\kappa }^2(x)=1$

κ-Inverse hyperbolic function

The Kaniadakis inverse hyperbolic functions (or κ-inverse hyperbolic functions) are associated to the κ-logarithm:

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(x)={\ln }_{\kappa }(\sqrt{1+x^2}+x)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_{\kappa }(x)={\ln }_{\kappa }(\sqrt{x^2-1}+x)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}_{\kappa }(x)={\ln }_{\kappa }\left(\sqrt{\frac{1+x}{1-x}}\right)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}}_{\kappa }(x)={\ln }_{\kappa }\left(\sqrt{\frac{1-x}{1+x}}\right)$,

in which are valid the following relations:

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(x)=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(x){\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_{\kappa }\left(\sqrt{1+x^2}\right)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(x)={\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}_{\kappa }\left(\frac{x}{\sqrt{1+x^2}}\right)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(x)={\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}}_{\kappa }\left(\frac{\sqrt{1+x^2}}{x}\right)$.

The κ-cyclic and κ-hyperbolic trigonometric functions are connected by the following relationships:

${\mathrm{s}\mathrm{i}\mathrm{n}}_{\kappa }(x)=-i{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{c}\mathrm{o}\mathrm{s}}_{\kappa }(x)={\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{t}\mathrm{a}\mathrm{n}}_{\kappa }(x)=-i{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{c}\mathrm{o}\mathrm{t}}_{\kappa }(x)=i{\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}}_{\kappa }(x)=-i{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}}_{\kappa }(x)\ne -i{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}}_{\kappa }(x)=-i{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}}_{\kappa }(ix)$,

${\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}}_{\kappa }(x)=i{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}}_{\kappa }(ix)$.

Kaniadakis entropy

The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:

$S_{\kappa }\left(p\right)=-\sum _i{p}_i{\ln }_{\kappa }\left(p_i\right)=\sum _i{p}_i{\ln }_{\kappa }\left(\frac{1}{p_i}\right)$

where $p=\{p_i=p(x_i);x\in ℝ;i=1,2,...,N;\sum _i{p}_i=1\}$ is a probability distribution function defined for a random variable $X$, and $0\le |\kappa |<1$ is the entropic index.

The Kaniadakis κ-entropy is thermodynamically and Lesche stable and obeys the Shannon-Khinchin axioms of continuity, maximality, generalized additivity and expandability.

Kaniadakis distributions

A Kaniadakis distribution (or κ-distribution) is a probability distribution derived from the maximization of Kaniadakis entropy under appropriate constraints. In this regard, several probability distributions emerge for analyzing a wide variety of phenomenology associated with experimental power-law tailed statistical distributions.

Kaniadakis integral transform

κ-Laplace Transform

The Kaniadakis Laplace transform (or κ-Laplace transform) is a κ-deformed integral transform of the ordinary Laplace transform. The κ-Laplace transform converts a function $f$ of a real variable $t$ to a new function $F_{\kappa }(s)$ in the complex frequency domain, represented by the complex variable $s$. This κ-integral transform is defined as:

$F_{\kappa }(s)={ℒ}_{\kappa }\{f(t)\}(s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}f(t)[{\exp }_{\kappa }(-t){]}^{s}dt$

The inverse κ-Laplace transform is given by:

$f(t)={ℒ}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)=\frac{1}{2\pi i}\underset{c-i\mathrm{\infty }}{\overset{c+i\mathrm{\infty }}{\int }}{F}_{\kappa }(s)\frac{[{\exp }_{\kappa }(t){]}^s}{\sqrt{1+{\kappa }^2{t}^2}}ds$

The ordinary Laplace transform and its inverse transform are recovered as $\kappa \to 0$.

Properties

Let two functions $f(t)={ℒ}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)$ and $g(t)={ℒ}_{\kappa }^{-1}\{G_{\kappa }(s)\}(t)$, and their respective κ-Laplace transforms $F_{\kappa }(s)$ and $G_{\kappa }(s)$, the following table presents the main properties of κ-Laplace transform:

Properties of the κ-Laplace transform

Property	$f(t)$	$F_{\kappa }(s)$
Linearity	$af(t)+bg(t)$	$a{F}_{\kappa }(s)+b{G}_{\kappa }(s)$
Time scaling	$f(at)$	$\frac{1}{a}{F}_{\kappa /a}(\frac{s}{a})$
Frequency shifting	$f(t)[{\exp }_{\kappa }(-t){]}^a$	$F_{\kappa }(s-a)$
Derivative	$\frac{df(t)}{dt}$	$s{ℒ}_{\kappa }\left\{\frac{f(t)}{\sqrt{1+{\kappa }^2{t}^2}}\right\}(s)-f(0)$
Derivative	$\frac{d}{dt}\sqrt{1+{\kappa }^2{t}^2}f(t)$	$s{F}_{\kappa }(s)-f(0)$
Time-domain integration	$\frac{1}{\sqrt{1+{\kappa }^2{t}^2}}\underset{0}{\overset{t}{\int }}f(w)dw$	$\frac{1}{s}{F}_{\kappa }(s)$
	$f(t)[\ln ({\exp }_{\kappa }(t)){]}^n$	$(-1{)}^n\frac{d^n{F}_{\kappa }(s)}{d{s}^n}$
	$f(t)[\ln ({\exp }_{\kappa }(t)){]}^{-n}$	$\underset{s}{\overset{+\mathrm{\infty }}{\int }}d{w}_n\underset{w_n}{\overset{+\mathrm{\infty }}{\int }}d{w}_{n-1}...\underset{w_3}{\overset{+\mathrm{\infty }}{\int }}d{w}_2\underset{w_2}{\overset{+\mathrm{\infty }}{\int }}d{w}_1{F}_{\kappa }(w_1)$
Dirac delta-function	$\delta (t-\tau )$	$[{\exp }_{\kappa }(-\tau ){]}^s$
Heaviside unit function	$u(t-\tau )$	$\frac{s\sqrt{1+{\kappa }^2{\tau }^2}+{\kappa }^2\tau }{s^2-{\kappa }^2}[{\exp }_{\kappa }(-\tau ){]}^s$
Power function	$t^{\nu -1}$	$\frac{s^2}{s^2-{\kappa }^2{\nu }^2}\frac{{\mathrm{\Gamma }}_{\frac{\kappa }{s}}(\nu +1)}{\nu s^{\nu }}=\frac{s}{s+|\kappa |\nu }\frac{\mathrm{\Gamma }(\nu )}{|2\kappa {|}^{\nu }}\frac{\mathrm{\Gamma }(\frac{s}{|2\kappa |}-\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{s}{|2\kappa |}+\frac{\nu }{2})}$
Power function	$t^{2m-1},m\in Z^{+}$	$\frac{(2m-1)!}{\prod _{j=1}^m[s^2-(2j{)}^2{\kappa }^2]}$
Power function	$t^{2m},m\in Z^{+}$	$\frac{(2m)!s}{\prod _{j=1}^{m+1}[s^2-(2j-1{)}^2{\kappa }^2]}$

The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit $\kappa \to 0$.

κ-Fourier Transform

The Kaniadakis Fourier transform (or κ-Fourier transform) is a κ-deformed integral transform of the ordinary Fourier transform, which is consistent with the κ-algebra and the κ-calculus. The κ-Fourier transform is defined as:

${ℱ}_{\kappa }[f(x)](\omega )=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\int }{\mathrm{\backslash limits}}^{+\mathrm{\infty }}f(x){\exp }_{\kappa }(-x{\otimes }_{\kappa }\omega {)}^i{d}_{\kappa }x$

which can be rewritten as

${ℱ}_{\kappa }[f(x)](\omega )=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\int }{\mathrm{\backslash limits}}^{+\mathrm{\infty }}f(x)\frac{\exp (-i{x}_{\{\kappa \}}{\omega }_{\{\kappa \}})}{\sqrt{1+{\kappa }^2{x}^2}}dx$

where $x_{\{\kappa \}}=\frac{1}{\kappa }\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa x)$ and ${\omega }_{\{\kappa \}}=\frac{1}{\kappa }\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa \omega )$. The κ-Fourier transform imposes an asymptotically log-periodic behavior by deforming the parameters $x$ and $\omega$ in addition to a damping factor, namely $\sqrt{1+{\kappa }^2{x}^2}$.

Real (top panel) and imaginary (bottom panel) part of the kernel $h_{\kappa }(x,\omega )$ for typical $\kappa$-values and $\omega =1$.

The kernel of the κ-Fourier transform is given by:

$h_{\kappa }(x,\omega )=\frac{\exp (-i{x}_{\{\kappa \}}{\omega }_{\{\kappa \}})}{\sqrt{1+{\kappa }^2{x}^2}}$

The inverse κ-Fourier transform is defined as:

${ℱ}_{\kappa }[\hat{f}(\omega )](x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\int }{\mathrm{\backslash limits}}^{+\mathrm{\infty }}\hat{f}(\omega ){\exp }_{\kappa }(\omega {\otimes }_{\kappa }x{)}^i{d}_{\kappa }\omega$

Let $u_{\kappa }(x)=\frac{1}{\kappa }\cosh (\kappa \ln (x))$, the following table shows the κ-Fourier transforms of several notable functions:

κ-Fourier transform of several functions

	$f(x)$	${ℱ}_{\kappa }[f(x)](\omega )$
Step function	$\theta (x)$	$\sqrt{2\pi }\delta (\omega )+\frac{1}{\sqrt{2\pi }i{\omega }_{\{\kappa \}}}$
Modulation	${\cos }_{\kappa }(a\stackrel{\kappa }{\oplus }x)$	$\sqrt{\frac{\pi }{2}}{u}_{\kappa }({\exp }_{\kappa }(a))(\delta (\omega +a)+\delta (\omega -a))$
Causal $\kappa$-exponential	$\theta (x){\exp }_{\kappa }(-a\stackrel{\kappa }{\otimes }x)$	$\frac{1}{\sqrt{2\pi }}\frac{1}{a_{\{\kappa \}}+i{\omega }_{\{\kappa \}}}$
Symmetric $\kappa$-exponential	${\exp }_{\kappa }(-a\stackrel{\kappa }{\otimes }|x|)$	$\sqrt{\frac{2}{\pi }}\frac{a_{\{\kappa \}}}{a_{\{\kappa \}}^2+{\omega }_{\{\kappa \}}^2}$
Constant	$1$	$\sqrt{2\pi }\delta (\omega )$
$\kappa$-Phasor	${\exp }_{\kappa }(a\stackrel{\kappa }{\otimes }x{)}^i$	$\sqrt{2\pi }{u}_{\kappa }({\exp }_{\kappa }(a))\delta (\omega -a)$
Impuslse	$\delta (x-a)$	$\frac{1}{\sqrt{2\pi }}\frac{{\exp }_{\kappa }(\omega \stackrel{\kappa }{\otimes }a{)}^i}{u_{\kappa }({\exp }_{\kappa }(a))}$
Signum	Sgn$(x)$	$\sqrt{\frac{2}{\pi }}\frac{1}{i{\omega }_{\{\kappa \}}}$
Rectangular	$\mathrm{\Pi }\left(\frac{x}{a}\right)$	$\sqrt{\frac{2}{\pi }}{a}_{\{\kappa \}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}}_{\kappa }(\omega \stackrel{\kappa }{\otimes }a)$

The κ-deformed version of the Fourier transform preserves the main properties of the ordinary Fourier transform, as summarized in the following table.

κ-Fourier properties

$f(x)$	${ℱ}_{\kappa }[f(x)](\omega )$
Linearity	${ℱ}_{\kappa }[\alpha f(x)+\beta g(x)](\omega )=\alpha {ℱ}_{\kappa }[f(x)](\omega )+\beta {ℱ}_{\kappa }[g(x)](\omega )$
Scaling	${ℱ}_{\kappa }[f(\alpha x)](\omega )=\frac{1}{\alpha }{ℱ}_{{\kappa }^{\prime }}[f(x)]({\omega }^{\prime })$ where ${\kappa }^{\prime }=\kappa /\alpha$ and ${\omega }^{\prime }=(a/\kappa )\sinh (\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\kappa \omega )/a^2)$
$\kappa$-Scaling	${ℱ}_{\kappa }[f(\alpha \stackrel{\kappa }{\otimes }x)](\omega )=\frac{1}{{\alpha }_{\{\kappa \}}}{ℱ}_{\kappa }[f(x)]\left(\frac{1}{\alpha }\stackrel{\kappa }{\otimes }\omega \right)$
Complex conjugation	${ℱ}_{\kappa }[f(x){]}^{\ast }(\omega )={ℱ}_{\kappa }[f(x)](-\omega )$
Duality	${ℱ}_{\kappa }\left[{ℱ}_{\kappa }\right[f(x)](\nu )](\omega )=f(-\omega )$
Reverse	${ℱ}_{\kappa }[f(-x)](\omega )={ℱ}_{\kappa }[f(x)](-\omega )$
$\kappa$-Frequency shift	${ℱ}_{\kappa }[{\exp }_{\kappa }({\omega }_0\stackrel{\kappa }{\otimes }x{)}^{i}f(x)](\omega )={ℱ}_{\kappa }[f(x)](\omega \stackrel{\kappa }{\ominus }{\omega }_0)$
$\kappa$-Time shift	${ℱ}_{\kappa }[f(x\stackrel{\kappa }{\oplus }{x}_0)](\omega )={\exp }_{\kappa }(\omega \stackrel{\kappa }{\otimes }{x}_0{)}^i{ℱ}_{\kappa }[f(x)](\omega )$
Transform of $\kappa$-derivative	${ℱ}_{\kappa }\left[\frac{df(x)}{d_{\kappa }x}\right](\omega )=i{\omega }_{\{\kappa \}}{ℱ}_{\kappa }[f(x)](\omega )$
$\kappa$-Derivative of transform	$\frac{d}{d_{\kappa }\omega }{ℱ}_{\kappa }[f(x)](\omega )=-i{\omega }_{\{\kappa \}}{ℱ}_{\kappa }[x_{\{\kappa \}}f(x)](\omega )$
Transform of integral	${ℱ}_{\kappa }[\underset{-\mathrm{\infty }}{\int }{\mathrm{\backslash limits}}^{x}f(y)dy](\omega )=\frac{1}{i{\omega }_{\{\kappa \}}}{ℱ}_{\kappa }[f(x)](\omega )+2\pi {ℱ}_{\kappa }[f(x)](0)\delta (\omega )$
$\kappa$-Convolution	${ℱ}_{\kappa }[(f\stackrel{\kappa }{⊛}g)(x)](\omega )=\sqrt{2\pi }{ℱ}_{\kappa }[f(x)](\omega ){ℱ}_{\kappa }[g(x)](\omega )$ where $(f\stackrel{\kappa }{⊛}g)(x)=\underset{-\mathrm{\infty }}{\int }{\mathrm{\backslash limits}}^{+\mathrm{\infty }}f(y)g(x\stackrel{\kappa }{\ominus }y)d_{\kappa }y$
Modulation	${ℱ}_{\kappa }[f(x)g(x)](\omega )=\frac{1}{\sqrt{2\pi }}\left({ℱ}_{\kappa }[f(x)]\stackrel{\kappa }{⊛}{ℱ}_{\kappa }[g(x)]\right)(\omega )$

The properties of the κ-Fourier transform presented in the latter table reduce to the corresponding ordinary Fourier transforms in the classical limit $\kappa \to 0$.

See also

• Giorgio Kaniadakis
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

References

• Template:Creative Commons text attribution notice

External links

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org