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Concept
Closed-form expression
Summary
In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, ÷, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions . However, the set of basic functions depends on the context.
The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed-form expression of this object, that is, an expression of this object in terms of previous ways of specifying it.
The quadratic formula
is a closed form of the solutions
of the general quadratic equation
More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed basic functions are reduced to only nth-roots. In fact, field theory allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.
There are expressions in radicals for all solutions of cubic equations (degree 3) and quartic equations (degree 4). However, they are rarely written explicitly because they are too complicated for being useful.
In higher degrees, Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. The simplest example is the equation Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.
Symbolic integration consists essentially of the search of closed forms for antiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly logarithms, exponential function and polynomial roots.
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