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

κ-exponential function edit

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

${\displaystyle \exp _{\kappa }(x)={\begin{cases}{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x{\Big )}^{\frac {1}{\kappa }}&{\text{if }}0<\kappa <1.\\[6pt]\exp(x)&{\text{if }}\kappa =0,\\[8pt]\end{cases}}}$ 

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

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

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

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

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

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:

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

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

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

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

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

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

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

For a real number ${\displaystyle r}$ , the κ-exponential has the property:

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

κ-logarithm function edit

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

${\displaystyle \ln _{\kappa }(x)={\begin{cases}{\frac {x^{\kappa }-x^{-\kappa }}{2\kappa }}&{\text{if }}0<\kappa <1,\\[8pt]\ln(x)&{\text{if }}\kappa =0,\\[8pt]\end{cases}}}$ 

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

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

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

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

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

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

following the rule

${\displaystyle \ln _{\kappa }(1+x)=\sum _{n=1}^{\infty }b_{n}(\kappa )\,(-1)^{n-1}\,{\frac {x^{n}}{n}}}$ 

with ${\displaystyle b_{1}(\kappa )=1}$ , and

${\displaystyle b_{n}(\kappa )(x)={\begin{cases}1&{\text{if }}n=1,\\[8pt]{\frac {1}{2}}{\Big (}1-\kappa {\Big )}{\Big (}1-{\frac {\kappa }{2}}{\Big )}...{\Big (}1-{\frac {\kappa }{n-1}}{\Big )},\,+\,{\frac {1}{2}}{\Big (}1+\kappa {\Big )}{\Big (}1+{\frac {\kappa }{2}}{\Big )}...{\Big (}1+{\frac {\kappa }{n-1}}{\Big )}&{\text{for }}n>1,\\[8pt]\end{cases}}}$ 

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

Basic properties

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

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

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

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

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

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

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

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

For a real number ${\displaystyle r}$ , the κ-logarithm has the property:

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

κ-Algebra edit

κ-sum edit

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

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

that can also be written in form:

${\displaystyle x{\stackrel {\kappa }{\oplus }}y={1 \over \kappa }\,\sinh \left({\rm {arcsinh}}\,(\kappa x)\,+\,{\rm {arcsinh}}\,(\kappa y)\,\right)}$ ,

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

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

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

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

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

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

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

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

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

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

κ-product edit

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

${\displaystyle x{\stackrel {\kappa }{\otimes }}y={1 \over \kappa }\,\sinh \left(\,{1 \over \kappa }\,\,{\rm {arcsinh}}\,(\kappa x)\,\,{\rm {arcsinh}}\,(\kappa y)\,\right)}$ ,

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

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

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

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

${\displaystyle {\text{3. inverse element:}}\quad x{\stackrel {\kappa }{\otimes }}{\overline {x}}={\overline {x}}{\stackrel {\kappa }{\otimes }}x=I\quad {\text{for}}\quad {\overline {x}}=\kappa ^{-1}\sinh(\kappa ^{2}/{\rm {arcsinh}}\,(\kappa x))}$ 

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

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

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

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

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

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

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

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

κ-Calculus edit

κ-Differential edit

The Kaniadakis differential (or κ-differential) of ${\displaystyle x}$  is defined by:

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

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

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

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

κ-Integral edit

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

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

which recovers the ordinary integral in the classical limit ${\displaystyle \kappa \rightarrow 0}$ .

κ-Trigonometry edit

κ-Cyclic Trigonometry edit

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:

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

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

where the κ-generalized Euler formula is

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

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

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

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

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

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

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

The κ-cyclic trigonometric functions become the ordinary trigonometric function in the classical limit ${\displaystyle \kappa \rightarrow 0}$ .

κ-Inverse cyclic function

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

${\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {1-x^{2}}}+ix\right)}$ ,

${\displaystyle {\rm {arccos}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)}$ ,

${\displaystyle {\rm {arctan}}_{\kappa }(x)=i\ln _{\kappa }\left({\sqrt {\frac {1-ix}{1+ix}}}\right)}$ ,

${\displaystyle {\rm {arccot}}_{\kappa }(x)=i\ln _{\kappa }\left({\sqrt {\frac {ix+1}{ix-1}}}\right)}$ .

κ-Hyperbolic Trigonometry edit

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

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

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

where the κ-Euler formula is

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

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

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

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

The κ-hyperbolic trigonometric functions become the ordinary hyperbolic trigonometric functions in the classical limit ${\displaystyle \kappa \rightarrow 0}$ .

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

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

κ-Inverse hyperbolic function

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

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {1+x^{2}}}+x\right)}$ ,

${\displaystyle {\rm {arccosh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)}$ ,

${\displaystyle {\rm {arctanh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1+x}{1-x}}}\right)}$ ,

${\displaystyle {\rm {arccoth}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1-x}{1+x}}}\right)}$ ,

in which are valid the following relations:

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {sign}}(x){\rm {arccosh}}_{\kappa }\left({\sqrt {1+x^{2}}}\right)}$ ,

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {arctanh}}_{\kappa }\left({\frac {x}{\sqrt {1+x^{2}}}}\right)}$ ,

${\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {arccoth}}_{\kappa }\left({\frac {\sqrt {1+x^{2}}}{x}}\right)}$ .

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

${\displaystyle {\rm {sin}}_{\kappa }(x)=-i{\rm {sinh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {cos}}_{\kappa }(x)={\rm {cosh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {tan}}_{\kappa }(x)=-i{\rm {tanh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {cot}}_{\kappa }(x)=i{\rm {coth}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\,{\rm {arcsinh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {arccos}}_{\kappa }(x)\neq -i\,{\rm {arccosh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {arctan}}_{\kappa }(x)=-i\,{\rm {arctanh}}_{\kappa }(ix)}$ ,

${\displaystyle {\rm {arccot}}_{\kappa }(x)=i\,{\rm {arccoth}}_{\kappa }(ix)}$ .

Kaniadakis entropy edit

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

${\displaystyle S_{\kappa }{\big (}p{\big )}=-\sum _{i}p_{i}\ln _{\kappa }{\big (}p_{i}{\big )}=\sum _{i}p_{i}\ln _{\kappa }{\bigg (}{\frac {1}{p_{i}}}{\bigg )}}$ 

where ${\displaystyle p=\{p_{i}=p(x_{i});x\in \mathbb {R} ;i=1,2,...,N;\sum _{i}p_{i}=1\}}$  is a probability distribution function defined for a random variable ${\displaystyle X}$ , and ${\displaystyle 0\leq |\kappa |<1}$  is the entropic index.

The Kaniadakis κ-entropy is thermodynamically and Lesche stable^[19][20] and obeys the Shannon-Khinchin axioms of continuity, maximality, generalized additivity and expandability.

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.

κ-Exponential distribution edit

κ-Gaussian distribution edit

κ-Gamma distribution edit

κ-Weibull distribution edit

κ-Logistic distribution edit

κ-Laplace Transform edit

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 ${\displaystyle f}$  of a real variable ${\displaystyle t}$  to a new function ${\displaystyle F_{\kappa }(s)}$  in the complex frequency domain, represented by the complex variable ${\displaystyle s}$ . This κ-integral transform is defined as:^[21]

${\displaystyle F_{\kappa }(s)={\cal {L}}_{\kappa }\{f(t)\}(s)=\int _{\,0}^{\infty }\!f(t)\,[\exp _{\kappa }(-t)]^{s}\,dt}$ 

The inverse κ-Laplace transform is given by:

${\displaystyle f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)={{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\!F_{\kappa }(s)\,{\frac {[\exp _{\kappa }(t)]^{s}}{\sqrt {1+\kappa ^{2}t^{2}}}}\,ds}}$ 

The ordinary Laplace transform and its inverse transform are recovered as ${\displaystyle \kappa \rightarrow 0}$ .

Properties

Let two functions ${\displaystyle f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)}$  and ${\displaystyle g(t)={\cal {L}}_{\kappa }^{-1}\{G_{\kappa }(s)\}(t)}$ , and their respective κ-Laplace transforms ${\displaystyle F_{\kappa }(s)}$  and ${\displaystyle G_{\kappa }(s)}$ , the following table presents the main properties of κ-Laplace transform:^[21]

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

κ-Fourier Transform edit

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:^[22]

${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }f(x)\,\exp _{\kappa }(-x\otimes _{\kappa }\omega )^{i}\,d_{\kappa }x}$ 

which can be rewritten as

${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }f(x)\,{\exp(-i\,x_{\{\kappa \}}\,\omega _{\{\kappa \}}) \over {\sqrt {1+\kappa ^{2}\,x^{2}}}}\,dx}$ 

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

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

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

The inverse κ-Fourier transform is defined as:^[22]

${\displaystyle {\cal {F}}_{\kappa }[{\hat {f}}(\omega )](x)={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }{\hat {f}}(\omega )\,\exp _{\kappa }(\omega \otimes _{\kappa }x)^{i}\,d_{\kappa }\omega }$ 

Let ${\displaystyle u_{\kappa }(x)={\frac {1}{\kappa }}\cosh {\Big (}\kappa \ln(x){\Big )}}$ , the following table shows the κ-Fourier transforms of several notable functions:^[22]

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