Statistical physics approach
Kaniadakis statistics (also known as κ-statistics ) is a generalization of Boltzmann–Gibbs statistical mechanics ,[ 1] based on a relativistic [ 2] [ 3] [ 4] 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,[ 5] 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 ,[ 6] [ 7] 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 ,[ 8] [ 9] condensed matter , quantum physics ,[ 10] [ 11] seismology ,[ 12] [ 13] genomics ,[ 14] [ 15] economics ,[ 16] [ 17] epidemiology ,[ 18] and many others.
The mathematical formalism of κ-statistics is generated by κ-deformed functions, especially the κ-exponential function.
κ-exponential function[ edit ] Plot of the κ-exponential function exp κ ( x ) {\displaystyle \exp _{\kappa }(x)} for three different κ-values. The solid black curve corresponding to the ordinary exponential function exp ( x ) {\displaystyle \exp(x)} ( κ = 0 {\displaystyle \kappa =0} ). The Kaniadakis exponential (or κ-exponential) function is a one-parameter generalization of an exponential function, given by:
exp κ ( x ) = { ( 1 + κ 2 x 2 + κ x ) 1 κ if 0 < κ < 1. exp ( x ) if κ = 0 , {\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 exp − κ ( x ) = exp κ ( x ) {\displaystyle \exp _{-\kappa }(x)=\exp _{\kappa }(x)} .
The κ-exponential for 0 < κ < 1 {\displaystyle 0<\kappa <1} can also be written in the form:
exp κ ( x ) = exp ( 1 κ arcsinh ( κ x ) ) . {\displaystyle \exp _{\kappa }(x)=\exp {\Bigg (}{\frac {1}{\kappa }}{\text{arcsinh}}(\kappa x){\Bigg )}.} The first five terms of the Taylor expansion of exp κ ( x ) {\displaystyle \exp _{\kappa }(x)} are given by:
exp κ ( x ) = 1 + x + x 2 2 + ( 1 − κ 2 ) x 3 3 ! + ( 1 − 4 κ 2 ) x 4 4 ! + ⋯ {\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:
exp κ ( x ) ∈ C ∞ ( R ) {\displaystyle \exp _{\kappa }(x)\in \mathbb {C} ^{\infty }(\mathbb {R} )} d d x exp κ ( x ) > 0 {\displaystyle {\frac {d}{dx}}\exp _{\kappa }(x)>0} d 2 d x 2 exp κ ( x ) > 0 {\displaystyle {\frac {d^{2}}{dx^{2}}}\exp _{\kappa }(x)>0} exp κ ( − ∞ ) = 0 + {\displaystyle \exp _{\kappa }(-\infty )=0^{+}} exp κ ( 0 ) = 1 {\displaystyle \exp _{\kappa }(0)=1} exp κ ( + ∞ ) = + ∞ {\displaystyle \exp _{\kappa }(+\infty )=+\infty } exp κ ( x ) exp κ ( − x ) = − 1 {\displaystyle \exp _{\kappa }(x)\exp _{\kappa }(-x)=-1} For a real number r {\displaystyle r} , the κ-exponential has the property:
[ exp κ ( x ) ] r = exp κ / r ( r x ) {\displaystyle {\Big [}\exp _{\kappa }(x){\Big ]}^{r}=\exp _{\kappa /r}(rx)} . κ-logarithm function[ edit ] Plot of the κ-logarithmic function ln κ ( x ) {\displaystyle \ln _{\kappa }(x)} for three different κ-values. The solid black curve corresponding to the ordinary logarithmic function ln ( x ) {\displaystyle \ln(x)} ( κ = 0 {\displaystyle \kappa =0} ). The Kaniadakis logarithm (or κ-logarithm) is a relativistic one-parameter generalization of the ordinary logarithm function,
ln κ ( x ) = { x κ − x − κ 2 κ if 0 < κ < 1 , ln ( x ) if κ = 0 , {\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 ln − κ ( x ) = ln κ ( x ) {\displaystyle \ln _{-\kappa }(x)=\ln _{\kappa }(x)} , which is the inverse function of the κ-exponential:
ln κ ( exp κ ( x ) ) = exp κ ( ln κ ( x ) ) = x . {\displaystyle \ln _{\kappa }{\Big (}\exp _{\kappa }(x){\Big )}=\exp _{\kappa }{\Big (}\ln _{\kappa }(x){\Big )}=x.} The κ-logarithm for 0 < κ < 1 {\displaystyle 0<\kappa <1} can also be written in the form:
ln κ ( x ) = 1 κ sinh ( κ ln ( x ) ) {\displaystyle \ln _{\kappa }(x)={\frac {1}{\kappa }}\sinh {\Big (}\kappa \ln(x){\Big )}}
The first three terms of the Taylor expansion of ln κ ( x ) {\displaystyle \ln _{\kappa }(x)} are given by:
ln κ ( 1 + x ) = x − x 2 2 + ( 1 + κ 2 2 ) x 3 3 − ⋯ {\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
ln κ ( 1 + x ) = ∑ n = 1 ∞ b n ( κ ) ( − 1 ) n − 1 x n n {\displaystyle \ln _{\kappa }(1+x)=\sum _{n=1}^{\infty }b_{n}(\kappa )\,(-1)^{n-1}\,{\frac {x^{n}}{n}}} with b 1 ( κ ) = 1 {\displaystyle b_{1}(\kappa )=1} , and
b n ( κ ) ( x ) = { 1 if n = 1 , 1 2 ( 1 − κ ) ( 1 − κ 2 ) . . . ( 1 − κ n − 1 ) , + 1 2 ( 1 + κ ) ( 1 + κ 2 ) . . . ( 1 + κ n − 1 ) for n > 1 , {\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 b n ( 0 ) = 1 {\displaystyle b_{n}(0)=1} and b n ( − κ ) = b n ( κ ) {\displaystyle b_{n}(-\kappa )=b_{n}(\kappa )} . The two first terms of the Taylor expansion of ln κ ( x ) {\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:
ln κ ( x ) ∈ C ∞ ( R + ) {\displaystyle \ln _{\kappa }(x)\in \mathbb {C} ^{\infty }(\mathbb {R} ^{+})} d d x ln κ ( x ) > 0 {\displaystyle {\frac {d}{dx}}\ln _{\kappa }(x)>0} d 2 d x 2 ln κ ( x ) < 0 {\displaystyle {\frac {d^{2}}{dx^{2}}}\ln _{\kappa }(x)<0} ln κ ( 0 + ) = − ∞ {\displaystyle \ln _{\kappa }(0^{+})=-\infty } ln κ ( 1 ) = 0 {\displaystyle \ln _{\kappa }(1)=0} ln κ ( + ∞ ) = + ∞ {\displaystyle \ln _{\kappa }(+\infty )=+\infty } ln κ ( 1 / x ) = − ln κ ( x ) {\displaystyle \ln _{\kappa }(1/x)=-\ln _{\kappa }(x)} For a real number r {\displaystyle r} , the κ-logarithm has the property:
ln κ ( x r ) = r ln r κ ( x ) {\displaystyle \ln _{\kappa }(x^{r})=r\ln _{r\kappa }(x)} For any x , y ∈ R {\displaystyle x,y\in \mathbb {R} } and | κ | < 1 {\displaystyle |\kappa |<1} , the Kaniadakis sum (or κ-sum) is defined by the following composition law:
x ⊕ κ y = x 1 + κ 2 y 2 + y 1 + κ 2 x 2 {\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:
x ⊕ κ y = 1 κ sinh ( a r c s i n h ( κ x ) + a r c s i n h ( κ y ) ) {\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 κ → 0 {\displaystyle \kappa \rightarrow 0} : x ⊕ 0 y = x + y {\displaystyle x{\stackrel {0}{\oplus }}y=x+y} .
The κ-sum, like the ordinary sum, has the following properties:
1. associativity: ( x ⊕ κ y ) ⊕ κ z = x ⊕ κ ( y ⊕ κ z ) {\displaystyle {\text{1. associativity:}}\quad (x{\stackrel {\kappa }{\oplus }}y){\stackrel {\kappa }{\oplus }}z=x{\stackrel {\kappa }{\oplus }}(y{\stackrel {\kappa }{\oplus }}z)} 2. neutral element: x ⊕ κ 0 = 0 ⊕ κ x = x {\displaystyle {\text{2. neutral element:}}\quad x{\stackrel {\kappa }{\oplus }}0=0{\stackrel {\kappa }{\oplus }}x=x} 3. opposite element: x ⊕ κ ( − x ) = ( − x ) ⊕ κ x = 0 {\displaystyle {\text{3. opposite element:}}\quad x{\stackrel {\kappa }{\oplus }}(-x)=(-x){\stackrel {\kappa }{\oplus }}x=0} 4. commutativity: x ⊕ κ y = y ⊕ κ x {\displaystyle {\text{4. commutativity:}}\quad x{\stackrel {\kappa }{\oplus }}y=y{\stackrel {\kappa }{\oplus }}x} The κ-difference ⊖ κ {\displaystyle {\stackrel {\kappa }{\ominus }}} is given by x ⊖ κ y = x ⊕ κ ( − y ) {\displaystyle x{\stackrel {\kappa }{\ominus }}y=x{\stackrel {\kappa }{\oplus }}(-y)} .
The fundamental property exp κ ( − x ) exp κ ( x ) = 1 {\displaystyle \exp _{\kappa }(-x)\exp _{\kappa }(x)=1} arises as a special case of the more general expression below: exp κ ( x ) exp κ ( y ) = e x p κ ( x ⊕ κ y ) {\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:
ln κ ( x y ) = ln κ ( x ) ⊕ κ ln κ ( y ) {\displaystyle \ln _{\kappa }(x\,y)=\ln _{\kappa }(x){\stackrel {\kappa }{\oplus }}\ln _{\kappa }(y)} For any x , y ∈ R {\displaystyle x,y\in \mathbb {R} } and | κ | < 1 {\displaystyle |\kappa |<1} , the Kaniadakis product (or κ-product) is defined by the following composition law:
x ⊗ κ y = 1 κ sinh ( 1 κ a r c s i n h ( κ x ) a r c s i n h ( κ y ) ) {\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 κ → 0 {\displaystyle \kappa \rightarrow 0} : x ⊗ 0 y = x × y = x y {\displaystyle x{\stackrel {0}{\otimes }}y=x\times y=xy} .
The κ-product, like the ordinary product, has the following properties:
1. associativity: ( x ⊗ κ y ) ⊗ κ z = x ⊗ κ ( y ⊗ κ z ) {\displaystyle {\text{1. associativity:}}\quad (x{\stackrel {\kappa }{\otimes }}y){\stackrel {\kappa }{\otimes }}z=x{\stackrel {\kappa }{\otimes }}(y{\stackrel {\kappa }{\otimes }}z)} 2. neutral element: x ⊗ κ I = I ⊗ κ x = x for I = κ − 1 sinh κ ⊕ κ x = x {\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} 3. inverse element: x ⊗ κ x ¯ = x ¯ ⊗ κ x = I for x ¯ = κ − 1 sinh ( κ 2 / a r c s i n h ( κ 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))} 4. commutativity: x ⊗ κ y = y ⊗ κ x {\displaystyle {\text{4. commutativity:}}\quad x{\stackrel {\kappa }{\otimes }}y=y{\stackrel {\kappa }{\otimes }}x} The κ-division ⊘ κ {\displaystyle {\stackrel {\kappa }{\oslash }}} is given by x ⊘ κ y = x ⊗ κ y ¯ {\displaystyle x{\stackrel {\kappa }{\oslash }}y=x{\stackrel {\kappa }{\otimes }}{\overline {y}}} .
The κ-sum ⊕ κ {\displaystyle {\stackrel {\kappa }{\oplus }}} and the κ-product ⊗ κ {\displaystyle {\stackrel {\kappa }{\otimes }}} obey the distributive law: z ⊗ κ ( x ⊕ κ y ) = ( z ⊗ κ x ) ⊕ κ ( z ⊗ κ y ) {\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 ln κ ( 1 / x ) = − ln κ ( x ) {\displaystyle \ln _{\kappa }(1/x)=-\ln _{\kappa }(x)} arises as a special case of the more general expression below:
ln κ ( x y ) = ln κ ( x ) ⊕ κ ln κ ( y ) {\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: exp κ ( x ) ⊗ κ exp κ ( y ) = exp κ ( x + y ) {\displaystyle \exp _{\kappa }(x){\stackrel {\kappa }{\otimes }}\exp _{\kappa }(y)=\exp _{\kappa }(x\,+\,y)} ln κ ( x ⊗ κ y ) = ln κ ( x ) + ln κ ( y ) {\displaystyle \ln _{\kappa }(x\,{\stackrel {\kappa }{\otimes }}\,y)=\ln _{\kappa }(x)+\ln _{\kappa }(y)} The Kaniadakis differential (or κ-differential) of x {\displaystyle x} is defined by:
d κ x = d x 1 + κ 2 x 2 {\displaystyle \mathrm {d} _{\kappa }x={\frac {\mathrm {d} \,x}{\displaystyle {\sqrt {1+\kappa ^{2}\,x^{2}}}}}} . So, the κ-derivative of a function f ( x ) {\displaystyle f(x)} is related to the Leibniz derivative through:
d f ( x ) d κ x = γ κ ( x ) d f ( x ) d x {\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} _{\kappa }x}}=\gamma _{\kappa }(x){\frac {\mathrm {d} f(x)}{\mathrm {d} x}}} , where γ κ ( x ) = 1 + κ 2 x 2 {\displaystyle \gamma _{\kappa }(x)={\sqrt {1+\kappa ^{2}x^{2}}}} is the Lorentz factor. The ordinary derivative d f ( x ) d x {\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} x}}} is a particular case of κ-derivative d f ( x ) d κ x {\displaystyle {\frac {\mathrm {d} f(x)}{\mathrm {d} _{\kappa }x}}} in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
The Kaniadakis integral (or κ-integral) is the inverse operator of the κ-derivative defined through
∫ d κ x f ( x ) = ∫ d x 1 + κ 2 x 2 f ( x ) {\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .
κ-Cyclic Trigonometry[ edit ] [click on the figure] Plot of the κ-sine and κ-cosine functions for κ = 0 {\displaystyle \kappa =0} (black curve) and κ = 0.1 {\displaystyle \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 κ ( x ) = exp κ ( i x ) − exp κ ( − i x ) 2 i {\displaystyle \sin _{\kappa }(x)={\frac {\exp _{\kappa }(ix)-\exp _{\kappa }(-ix)}{2i}}} , cos κ ( x ) = exp κ ( i x ) + exp κ ( − i x ) 2 {\displaystyle \cos _{\kappa }(x)={\frac {\exp _{\kappa }(ix)+\exp _{\kappa }(-ix)}{2}}} , where the κ-generalized Euler formula is
exp κ ( ± i x ) = cos κ ( x ) ± i sin κ ( x ) {\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:
cos κ 2 ( x ) + sin κ 2 ( x ) = 1 {\displaystyle \cos _{\kappa }^{2}(x)+\sin _{\kappa }^{2}(x)=1} sin κ ( x ⊕ κ y ) = sin κ ( x ) cos κ ( y ) + cos κ ( x ) sin κ ( y ) {\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:
tan κ ( x ) = sin κ ( x ) cos κ ( x ) {\displaystyle \tan _{\kappa }(x)={\frac {\sin _{\kappa }(x)}{\cos _{\kappa }(x)}}} cot κ ( x ) = cos κ ( x ) sin κ ( 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 κ → 0 {\displaystyle \kappa \rightarrow 0} .
κ-Inverse cyclic function
The Kaniadakis inverse cyclic functions (or κ-inverse cyclic functions) are associated to the κ-logarithm:
a r c s i n κ ( x ) = − i ln κ ( 1 − x 2 + i x ) {\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {1-x^{2}}}+ix\right)} , a r c c o s κ ( x ) = − i ln κ ( x 2 − 1 + x ) {\displaystyle {\rm {arccos}}_{\kappa }(x)=-i\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)} , a r c t a n κ ( x ) = i ln κ ( 1 − i x 1 + i x ) {\displaystyle {\rm {arctan}}_{\kappa }(x)=i\ln _{\kappa }\left({\sqrt {\frac {1-ix}{1+ix}}}\right)} , a r c c o t κ ( x ) = i ln κ ( i x + 1 i x − 1 ) {\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:
sinh κ ( x ) = exp κ ( x ) − exp κ ( − x ) 2 {\displaystyle \sinh _{\kappa }(x)={\frac {\exp _{\kappa }(x)-\exp _{\kappa }(-x)}{2}}} , cosh κ ( x ) = exp κ ( x ) + exp κ ( − x ) 2 {\displaystyle \cosh _{\kappa }(x)={\frac {\exp _{\kappa }(x)+\exp _{\kappa }(-x)}{2}}} , where the κ-Euler formula is
exp κ ( ± x ) = cosh κ ( x ) ± sinh κ ( x ) {\displaystyle \exp _{\kappa }(\pm x)=\cosh _{\kappa }(x)\pm \sinh _{\kappa }(x)} . The κ-hyperbolic tangent and κ-hyperbolic cotangent functions are given by:
tanh κ ( x ) = sinh κ ( x ) cosh κ ( x ) {\displaystyle \tanh _{\kappa }(x)={\frac {\sinh _{\kappa }(x)}{\cosh _{\kappa }(x)}}} coth κ ( x ) = cosh κ ( x ) sinh κ ( 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 κ → 0 {\displaystyle \kappa \rightarrow 0} .
From the κ-Euler formula and the property exp κ ( − x ) exp κ ( x ) = 1 {\displaystyle \exp _{\kappa }(-x)\exp _{\kappa }(x)=1} the fundamental expression of κ-hyperbolic trigonometry is given as follows:
cosh κ 2 ( x ) − sinh κ 2 ( x ) = 1 {\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:
a r c s i n h κ ( x ) = ln κ ( 1 + x 2 + x ) {\displaystyle {\rm {arcsinh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {1+x^{2}}}+x\right)} , a r c c o s h κ ( x ) = ln κ ( x 2 − 1 + x ) {\displaystyle {\rm {arccosh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {x^{2}-1}}+x\right)} , a r c t a n h κ ( x ) = ln κ ( 1 + x 1 − x ) {\displaystyle {\rm {arctanh}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1+x}{1-x}}}\right)} , a r c c o t h κ ( x ) = ln κ ( 1 − x 1 + x ) {\displaystyle {\rm {arccoth}}_{\kappa }(x)=\ln _{\kappa }\left({\sqrt {\frac {1-x}{1+x}}}\right)} , in which are valid the following relations:
a r c s i n h κ ( x ) = s i g n ( x ) a r c c o s h κ ( 1 + x 2 ) {\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {sign}}(x){\rm {arccosh}}_{\kappa }\left({\sqrt {1+x^{2}}}\right)} , a r c s i n h κ ( x ) = a r c t a n h κ ( x 1 + x 2 ) {\displaystyle {\rm {arcsinh}}_{\kappa }(x)={\rm {arctanh}}_{\kappa }\left({\frac {x}{\sqrt {1+x^{2}}}}\right)} , a r c s i n h κ ( x ) = a r c c o t h κ ( 1 + x 2 x ) {\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:
s i n κ ( x ) = − i s i n h κ ( i x ) {\displaystyle {\rm {sin}}_{\kappa }(x)=-i{\rm {sinh}}_{\kappa }(ix)} , c o s κ ( x ) = c o s h κ ( i x ) {\displaystyle {\rm {cos}}_{\kappa }(x)={\rm {cosh}}_{\kappa }(ix)} , t a n κ ( x ) = − i t a n h κ ( i x ) {\displaystyle {\rm {tan}}_{\kappa }(x)=-i{\rm {tanh}}_{\kappa }(ix)} , c o t κ ( x ) = i c o t h κ ( i x ) {\displaystyle {\rm {cot}}_{\kappa }(x)=i{\rm {coth}}_{\kappa }(ix)} , a r c s i n κ ( x ) = − i a r c s i n h κ ( i x ) {\displaystyle {\rm {arcsin}}_{\kappa }(x)=-i\,{\rm {arcsinh}}_{\kappa }(ix)} , a r c c o s κ ( x ) ≠ − i a r c c o s h κ ( i x ) {\displaystyle {\rm {arccos}}_{\kappa }(x)\neq -i\,{\rm {arccosh}}_{\kappa }(ix)} , a r c t a n κ ( x ) = − i a r c t a n h κ ( i x ) {\displaystyle {\rm {arctan}}_{\kappa }(x)=-i\,{\rm {arctanh}}_{\kappa }(ix)} , a r c c o t κ ( x ) = i a r c c o t h κ ( i x ) {\displaystyle {\rm {arccot}}_{\kappa }(x)=i\,{\rm {arccoth}}_{\kappa }(ix)} . The Kaniadakis statistics is based on the Kaniadakis κ-entropy, which is defined through:
S κ ( p ) = − ∑ i p i ln κ ( p i ) = ∑ i p i ln κ ( 1 p i ) {\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 p = { p i = p ( x i ) ; x ∈ R ; i = 1 , 2 , . . . , N ; ∑ i p i = 1 } {\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 X {\displaystyle X} , and 0 ≤ | κ | < 1 {\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.
Kaniadakis distributions [ edit ] 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 ] 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 {\displaystyle f} of a real variable t {\displaystyle t} to a new function F κ ( s ) {\displaystyle F_{\kappa }(s)} in the complex frequency domain, represented by the complex variable s {\displaystyle s} . This κ-integral transform is defined as:[ 21]
F κ ( s ) = L κ { f ( t ) } ( s ) = ∫ 0 ∞ f ( t ) [ exp κ ( − t ) ] s d t {\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:
f ( t ) = L κ − 1 { F κ ( s ) } ( t ) = 1 2 π i ∫ c − i ∞ c + i ∞ F κ ( s ) [ exp κ ( t ) ] s 1 + κ 2 t 2 d s {\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .
Properties
Let two functions f ( t ) = L κ − 1 { F κ ( s ) } ( t ) {\displaystyle f(t)={\cal {L}}_{\kappa }^{-1}\{F_{\kappa }(s)\}(t)} and g ( t ) = L κ − 1 { G κ ( s ) } ( t ) {\displaystyle g(t)={\cal {L}}_{\kappa }^{-1}\{G_{\kappa }(s)\}(t)} , and their respective κ-Laplace transforms F κ ( s ) {\displaystyle F_{\kappa }(s)} and G κ ( s ) {\displaystyle G_{\kappa }(s)} , the following table presents the main properties of κ-Laplace transform:[ 21]
Properties of the κ-Laplace transform Property f ( t ) {\displaystyle f(t)} F κ ( s ) {\displaystyle F_{\kappa }(s)} Linearity a f ( t ) + b g ( t ) {\displaystyle a\,f(t)+b\,g(t)} a F κ ( s ) + b G κ ( s ) {\displaystyle a\,F_{\kappa }(s)+b\,G_{\kappa }(s)} Time scaling f ( a t ) {\displaystyle f(at)} 1 a F κ / a ( s a ) {\displaystyle {\frac {1}{a}}\,F_{\kappa /a}({\frac {s}{a}})} Frequency shifting f ( t ) [ exp κ ( − t ) ] a {\displaystyle f(t)\,[\exp _{\kappa }(-t)]^{a}} F κ ( s − a ) {\displaystyle F_{\kappa }(s-a)} Derivative d f ( t ) d t {\displaystyle {\frac {d\,f(t)}{dt}}} s L κ { f ( t ) 1 + κ 2 t 2 } ( s ) − f ( 0 ) {\displaystyle s\,{\cal {L}}_{\kappa }\left\{{\frac {f(t)}{\sqrt {1+\kappa ^{2}t^{2}}}}\right\}(s)-f(0)} Derivative d d t 1 + κ 2 t 2 f ( t ) {\displaystyle {\frac {d}{dt}}\,{\sqrt {1+\kappa ^{2}t^{2}}}\,f(t)} s F κ ( s ) − f ( 0 ) {\displaystyle s\,F_{\kappa }(s)-f(0)} Time-domain integration 1 1 + κ 2 t 2 ∫ 0 t f ( w ) d w {\displaystyle {\frac {1}{\sqrt {1+\kappa ^{2}t^{2}}}}\,\int _{0}^{t}f(w)dw} 1 s F κ ( s ) {\displaystyle {\frac {1}{s}}\,F_{\kappa }(s)} f ( t ) [ ln ( exp κ ( t ) ) ] n {\displaystyle f(t)\,[\ln(\exp _{\kappa }(t))]^{n}} ( − 1 ) n d n F κ ( s ) d s n {\displaystyle (-1)^{n}{\frac {d^{n}F_{\kappa }(s)}{ds^{n}}}} f ( t ) [ ln ( exp κ ( t ) ) ] − n {\displaystyle f(t)\,[\ln(\exp _{\kappa }(t))]^{-n}} ∫ s + ∞ d w n ∫ w n + ∞ d w n − 1 . . . ∫ w 3 + ∞ d w 2 ∫ w 2 + ∞ d w 1 F κ ( w 1 ) {\displaystyle \int _{s}^{+\infty }dw_{n}\int _{w_{n}}^{+\infty }dw_{n-1}...\int _{w_{3}}^{+\infty }dw_{2}\int _{w_{2}}^{+\infty }dw_{1}\,F_{\kappa }(w_{1})} Dirac delta-function δ ( t − τ ) {\displaystyle \delta (t-\tau )} [ exp κ ( − τ ) ] s {\displaystyle [\exp _{\kappa }(-\tau )]^{s}} Heaviside unit function u ( t − τ ) {\displaystyle u(t-\tau )} s 1 + κ 2 τ 2 + κ 2 τ s 2 − κ 2 [ exp κ ( − τ ) ] s {\displaystyle {\frac {s{\sqrt {1+\kappa ^{2}\tau ^{2}}}+\kappa ^{2}\tau }{s^{2}-\kappa ^{2}}}\,[\exp _{\kappa }(-\tau )]^{s}} Power function t ν − 1 {\displaystyle t^{\nu -1}} s 2 s 2 − κ 2 ν 2 Γ κ s ( ν + 1 ) ν s ν = s s + | κ | ν Γ ( ν ) | 2 κ | ν Γ ( s | 2 κ | − ν 2 ) Γ ( s | 2 κ | + ν 2 ) {\displaystyle {\frac {s^{2}}{s^{2}-\kappa ^{2}\nu ^{2}}}\,{\frac {\Gamma _{\frac {\kappa }{s}}(\nu +1)}{\nu \,s^{\nu }}}={\frac {s}{s+|\kappa |\nu }}\,{\frac {\Gamma (\nu )}{|2\kappa |^{\nu }}}\,{\frac {\Gamma \left({\frac {s}{|2\kappa |}}-{\frac {\nu }{2}}\right)}{\Gamma \left({\frac {s}{|2\kappa |}}+{\frac {\nu }{2}}\right)}}} Power function t 2 m − 1 , m ∈ Z + {\displaystyle t^{2m-1},\ \ m\in Z^{+}} ( 2 m − 1 ) ! ∏ j = 1 m [ s 2 − ( 2 j ) 2 κ 2 ] {\displaystyle {\frac {(2m-1)!}{\prod _{j=1}^{m}\left[s^{2}-(2j)^{2}\kappa ^{2}\right]}}} Power function t 2 m , m ∈ Z + {\displaystyle t^{2m},\ \ m\in Z^{+}} ( 2 m ) ! s ∏ j = 1 m + 1 [ s 2 − ( 2 j − 1 ) 2 κ 2 ] {\displaystyle {\frac {(2m)!\,s}{\prod _{j=1}^{m+1}\left[s^{2}-(2j-1)^{2}\kappa ^{2}\right]}}}
The κ-Laplace transforms presented in the latter table reduce to the corresponding ordinary Laplace transforms in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
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]
F κ [ f ( x ) ] ( ω ) = 1 2 π ∫ − ∞ + ∞ f ( x ) exp κ ( − x ⊗ κ ω ) i d κ x {\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
F κ [ f ( x ) ] ( ω ) = 1 2 π ∫ − ∞ + ∞ f ( x ) exp ( − i x { κ } ω { κ } ) 1 + κ 2 x 2 d x {\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 x { κ } = 1 κ a r c s i n h ( κ x ) {\displaystyle x_{\{\kappa \}}={\frac {1}{\kappa }}\,{\rm {arcsinh}}\,(\kappa \,x)} and ω { κ } = 1 κ a r c s i n h ( κ ω ) {\displaystyle \omega _{\{\kappa \}}={\frac {1}{\kappa }}\,{\rm {arcsinh}}\,(\kappa \,\omega )} . The κ-Fourier transform imposes an asymptotically log-periodic behavior by deforming the parameters x {\displaystyle x} and ω {\displaystyle \omega } in addition to a damping factor, namely 1 + κ 2 x 2 {\displaystyle {\sqrt {1+\kappa ^{2}\,x^{2}}}} .
Real (top panel) and imaginary (bottom panel) part of the kernel h κ ( x , ω ) {\displaystyle h_{\kappa }(x,\omega )} for typical κ {\displaystyle \kappa } -values and ω = 1 {\displaystyle \omega =1} . The kernel of the κ-Fourier transform is given by:
h κ ( x , ω ) = exp ( − i x { κ } ω { κ } ) 1 + κ 2 x 2 {\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]
F κ [ f ^ ( ω ) ] ( x ) = 1 2 π ∫ − ∞ + ∞ f ^ ( ω ) exp κ ( ω ⊗ κ x ) i d κ ω {\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 u κ ( x ) = 1 κ cosh ( κ ln ( x ) ) {\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]
κ-Fourier transform of several functions f ( x ) {\displaystyle f(x)} F κ [ f ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )} Step function θ ( x ) {\displaystyle \theta (x)} 2 π δ ( ω ) + 1 2 π i ω { κ } {\displaystyle {\sqrt {2\,\pi }}\,\delta (\omega )+{1 \over {\sqrt {2\,\pi }}\,i\,\omega _{\{\kappa \}}}} Modulation cos κ ( a ⊕ κ x ) {\displaystyle \cos _{\kappa }(a{\stackrel {\kappa }{\oplus }}x)} π 2 u κ ( exp κ ( a ) ) ( δ ( ω + a ) + δ ( ω − a ) ) {\displaystyle {\sqrt {\pi \over 2}}\,u_{\kappa }(\exp _{\kappa }(a))\,\left(\delta (\omega +a)+\delta (\omega -a)\right)} Causal κ {\displaystyle \kappa } -exponential θ ( x ) exp κ ( − a ⊗ κ x ) {\displaystyle \theta (x)\,\exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}x)} 1 2 π 1 a { κ } + i ω { κ } {\displaystyle {1 \over {\sqrt {2\,\pi }}}{1 \over a_{\{\kappa \}}+i\,\omega _{\{\kappa \}}}} Symmetric κ {\displaystyle \kappa } -exponential exp κ ( − a ⊗ κ | x | ) {\displaystyle \exp _{\kappa }(-a{\stackrel {\kappa }{\otimes }}|x|)} 2 π a { κ } a { κ } 2 + ω { κ } 2 {\displaystyle {\sqrt {2 \over \pi }}\,{a_{\{\kappa \}} \over a_{\{\kappa \}}^{2}+\omega _{\{\kappa \}}^{2}}} Constant 1 {\displaystyle 1} 2 π δ ( ω ) {\displaystyle {\sqrt {2\,\pi }}\,\delta (\omega )} κ {\displaystyle \kappa } -Phasor exp κ ( a ⊗ κ x ) i {\displaystyle \exp _{\kappa }\,(a{\stackrel {\kappa }{\otimes }}x)^{i}} 2 π u κ ( exp κ ( a ) ) δ ( ω − a ) {\displaystyle {\sqrt {2\,\pi }}\,u_{\kappa }(\exp _{\kappa }(a))\,\delta (\omega -a)} Impuslse δ ( x − a ) {\displaystyle \delta (x-a)} 1 2 π exp κ ( ω ⊗ κ a ) i u κ ( exp κ ( a ) ) {\displaystyle {1 \over {\sqrt {2\,\pi }}}{\exp _{\kappa }\,(\omega {\stackrel {\kappa }{\otimes }}a)^{i} \over u_{\kappa }\left(\exp _{\kappa }\,(a)\right)}} Signum Sgn ( x ) {\displaystyle (x)} 2 π 1 i ω { κ } {\displaystyle {\sqrt {2 \over \pi }}\,\,{1 \over i\,\omega _{\{\kappa \}}}} Rectangular Π ( x a ) {\displaystyle \Pi \left({x \over a}\right)} 2 π a { κ } s i n c κ ( ω ⊗ κ a ) {\displaystyle {\sqrt {2 \over \pi }}\,\,a_{\{\kappa \}}\,{\rm {sinc}}_{\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 ) {\displaystyle f(x)} F κ [ f ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )} Linearity F κ [ α f ( x ) + β g ( x ) ] ( ω ) = α F κ [ f ( x ) ] ( ω ) + β F κ [ g ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }[\alpha \,f(x)+\beta \,g(x)](\omega )=\alpha \,{\cal {F}}_{\kappa }[f(x)](\omega )+\beta \,{\cal {F}}_{\kappa }[g(x)](\omega )} Scaling F κ [ f ( α x ) ] ( ω ) = 1 α F κ ′ [ f ( x ) ] ( ω ′ ) {\displaystyle {\cal {F}}_{\kappa }\left[f(\alpha \,x)\right](\omega )={1 \over \alpha }\,{\cal {F}}_{\kappa ^{\prime }}\left[f(x)\right](\omega ^{\prime })} where κ ′ = κ / α {\displaystyle \kappa ^{\prime }=\kappa /\alpha } and ω ′ = ( a / κ ) sinh ( a r c s i n h ( κ ω ) / a 2 ) {\displaystyle \omega ^{\prime }=(a/\kappa )\,\sinh \left({\rm {arcsinh}}(\kappa \,\omega )/a^{2}\right)} κ {\displaystyle \kappa } -Scaling F κ [ f ( α ⊗ κ x ) ] ( ω ) = 1 α { κ } F κ [ f ( x ) ] ( 1 α ⊗ κ ω ) {\displaystyle {\cal {F}}_{\kappa }\left[f(\alpha {\stackrel {\kappa }{\otimes }}x)\right](\omega )={1 \over \alpha _{\{\kappa \}}}\,{\cal {F}}_{\kappa }[f(x)]\left({\frac {1}{\alpha }}{\stackrel {\kappa }{\otimes }}\omega \right)} Complex conjugation F κ [ f ( x ) ] ∗ ( ω ) = F κ [ f ( x ) ] ( − ω ) {\displaystyle {\cal {F}}_{\kappa }{\big [}f(x){\big ]}^{\ast }(\omega )={\cal {F}}_{\kappa }{\big [}f(x){\big ]}(-\omega )} Duality F κ [ F κ [ f ( x ) ] ( ν ) ] ( ω ) = f ( − ω ) {\displaystyle {\cal {F}}_{\kappa }{\Big [}{\cal {F}}_{\kappa }{\big [}f(x){\big ]}(\nu ){\Big ]}(\omega )=f(-\omega )} Reverse F κ [ f ( − x ) ] ( ω ) = F κ [ f ( x ) ] ( − ω ) {\displaystyle {\cal {F}}_{\kappa }\left[f(-x)\right](\omega )={\cal {F}}_{\kappa }[f(x)](-\omega )} κ {\displaystyle \kappa } -Frequency shift F κ [ exp κ ( ω 0 ⊗ κ x ) i f ( x ) ] ( ω ) = F κ [ f ( x ) ] ( ω ⊖ κ ω 0 ) {\displaystyle {\cal {F}}_{\kappa }\left[\exp _{\kappa }(\omega _{0}{\stackrel {\kappa }{\otimes }}x)^{i}f(x)\right](\omega )={\cal {F}}_{\kappa }[f(x)](\omega {\stackrel {\kappa }{\ominus }}\omega _{0})} κ {\displaystyle \kappa } -Time shift F κ [ f ( x ⊕ κ x 0 ) ] ( ω ) = exp κ ( ω ⊗ κ x 0 ) i F κ [ f ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }\left[f(x\,{\stackrel {\kappa }{\oplus }}\,x_{0})\right](\omega )=\exp _{\kappa }(\omega \,{\stackrel {\kappa }{\otimes }}\,x_{0})^{i}\,{\cal {F}}_{\kappa }[f(x)](\omega )} Transform of κ {\displaystyle \kappa } -derivative F κ [ d f ( x ) d κ x ] ( ω ) = i ω { κ } F κ [ f ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }\left[{\frac {d\,f(x)}{d_{\kappa }x}}\right](\omega )=i\,\omega _{\{\kappa \}}\,{\cal {F}}_{\kappa }[f(x)](\omega )} κ {\displaystyle \kappa } -Derivative of transform d d κ ω F κ [ f ( x ) ] ( ω ) = − i ω { κ } F κ [ x { κ } f ( x ) ] ( ω ) {\displaystyle {\frac {d}{d_{\kappa }\omega }}\,{\cal {F}}_{\kappa }[f(x)](\omega )=-i\,\omega _{\{\kappa \}}\,{\cal {F}}_{\kappa }\left[x_{\{\kappa \}}\,f(x)\right](\omega )} Transform of integral F κ [ ∫ − ∞ x f ( y ) d y ] ( ω ) = 1 i ω { κ } F κ [ f ( x ) ] ( ω ) + 2 π F κ [ f ( x ) ] ( 0 ) δ ( ω ) {\displaystyle {\cal {F}}_{\kappa }\left[\int \limits _{-\infty }\limits ^{x}f(y)\,dy\right](\omega )={1 \over i\,\omega _{\{\kappa \}}}{\cal {F}}_{\kappa }[f(x)](\omega )+2\,\pi \,{\cal {F}}_{\kappa }[f(x)](0)\,\delta (\omega )} κ {\displaystyle \kappa } -Convolution F κ [ ( f ⊛ κ g ) ( x ) ] ( ω ) = 2 π F κ [ f ( x ) ] ( ω ) F κ [ g ( x ) ] ( ω ) {\displaystyle {\cal {F}}_{\kappa }\left[(f\,{\stackrel {\kappa }{\circledast }}\,g)(x)\right](\omega )={\sqrt {2\,\pi }}\,{\cal {F}}_{\kappa }[f(x)](\omega )\,{\cal {F}}_{\kappa }[g(x)](\omega )} where ( f ⊛ κ g ) ( x ) = ∫ − ∞ + ∞ f ( y ) g ( x ⊖ κ y ) d κ y {\displaystyle (f\,{\stackrel {\kappa }{\circledast }}\,g)(x)=\int \limits _{-\infty }\limits ^{+\infty }f(y)\,g(x\,{\stackrel {\kappa }{\ominus }}\,y)\,d_{\kappa }y} Modulation F κ [ f ( x ) g ( x ) ] ( ω ) = 1 2 π ( F κ [ f ( x ) ] ⊛ κ F κ [ g ( x ) ] ) ( ω ) {\displaystyle {\cal {F}}_{\kappa }\left[f(x)\,g(x)\right](\omega )={1 \over {\sqrt {2\,\pi }}}\left({\cal {F}}_{\kappa }\left[f(x)\right]\,{\stackrel {\kappa }{\circledast }}\,{\cal {F}}_{\kappa }\left[g(x)\right]\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .
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