In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff.
Motivation Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function, f ( x ) = e x {\displaystyle f(x)=e^{x}} . If we differentiate this function, we get e x {\displaystyle e^{x}} again; that is,
f ′ ( x ) = e x = f ( x ) . {\displaystyle f'(x)=e^{x}=f(x).}
Another example is the reciprocal function, g ( x ) = 1 / x {\displaystyle g(x)=1/x} . Differentiating this function, we see that
g ′ ( x ) = − 1 x 2 = − g ( x ) 2 . {\displaystyle g^{\prime }(x)={\frac {-1}{x^{2}}}=-g(x)^{2}.}
Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example, if we take the function h ( x ) = e x log x {\displaystyle h(x)=e^{x}\log x} , we have that
h ′ ( x ) = e x log x + x − 1 e x = h ( x ) + f ( x ) g ( x ) . {\displaystyle h^{\prime }(x)=e^{x}\log x+x^{-1}e^{x}=h(x)+f(x)g(x).}
Functions like these form the links in a so-called Pfaffian chain. Such a chain is a sequence of functions f 1 , f 2 , … {\displaystyle f_{1},f_{2},\dots } with the property that if we differentiate any of the functions in this chain then the result can be written in terms of the function itself and all the functions preceding it in the chain (specifically as a polynomial in those functions and the variables involved). Thus, with the functions above, we have that f , g , h {\displaystyle f,g,h} is a Pfaffian chain. A Pfaffian function is then just a polynomial in the functions appearing in a Pfaffian chain and the function argument. So with the Pfaffian chain just mentioned, functions such as F ( x ) = x 3 f ( x ) 2 − 2 g ( x ) h ( x ) {\displaystyle F(x)=x^{3}f(x)^{2}-2g(x)h(x)} are Pfaffian.
Formal definition Let U {\displaystyle U} be an open domain in R n {\displaystyle \mathbb {R} ^{n}} . A Pfaffian chain of order r ≥ 0 {\displaystyle r\geq 0} and degree α ≥ 1 {\displaystyle \alpha \geq 1} in U {\displaystyle U} is a sequence of real analytic functions f 1 , … , f r {\displaystyle f_{1},\dots ,f_{r}} in U {\displaystyle U} satisfying differential equations
∂ f i ∂ x j = P i , j ( x , f 1 ( x ) , … , f i ( x ) ) {\displaystyle {\frac {\partial f_{i}}{\partial x_{j}}}=P_{i,j}({\boldsymbol {x}},f_{1}({\boldsymbol {x}}),\ldots ,f_{i}({\boldsymbol {x}}))}
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