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However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation log ( x y ) = log ( x ) + log ( y ) . {\displaystyle \log(xy)=\log(x)+\log(y).}
In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).
Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} holds with the same definition of Δ y , but instead of ε being infinitesimal, we have lim Δ x → 0 ε = 0 {\displaystyle ...
A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), ..., f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
Download as PDF; Printable version; ... In algebra, a septic equation is an equation of the form ... The equation may be obtained from the function by setting f(x) ...
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation = where the function f is given and g is unknown. Here, L stands for a linear differential operator.
While some real life problems, such as population dynamics, can be modeled by algebraic difference equations, difference algebra also has applications in pure mathematics. For example, there is a proof of the Manin–Mumford conjecture using methods of difference algebra. [4] The model theory of difference fields has been studied.
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