A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives". Finite difference approximations are finite difference quotients in the terminology employed above. Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms () and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals. Basic types Three basic types are commonly considered: forward, backward, and central finite differences. A forward difference, denoted of a function is a function defined as Depending on the application, the spacing may be variable or constant. When omitted, is taken to be 1; that is, A backward difference uses the function values at and , instead of the values at and : Finally, the central difference is given by Relation with derivatives Finite difference is often used as an approximation of the derivative, typically in numerical differentiation. The derivative of a function at a point is defined by the limit. If has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written Hence, the forward difference divided by approximates the derivative when is small. The error in this approximation can be derived from Taylor's theorem. Assuming that is twice differentiable, we have The same formula holds for the backward difference: However, the central (also called centered) difference yields a more accurate