Analytic solution
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.
Problems are said to be tractable if they can be solved in terms of a closed-form expression.
The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation:
is tractable since its solutions can be expressed as closed-form expression, i.e. in terms of elementary functions:
Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closed-form solutions using elementary functions, such as x5 − x + 1 = 0.
An area of study in mathematics referred to broadly as Galois theory involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.
Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well-known, since numerical implementations are widely available.
An analytic expression (or expression in analytic form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the nth root), logarithms, and trigonometric functions.
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