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In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. The classic example involves the two roots of a quadratic equation, which can be expressed in closed form in terms of addition and subtraction, multiplication and division, and square root extraction. Mathematics is the study of quantity, structure, space and change. ...
In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...
In mathematics, equation solving is the problem of finding what values (numbers, functions, sets. ...
In mathematics, a quadratic equation is a polynomial equation of the second degree. ...
When no closed-form solutions exist – as is the case for fifth-order or higher polynomial equations, for example – such equations have to be solved numerically, typically by using some root-finding algorithm. In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ...
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
A root-finding algorithm is a numerical method or algorithm for finding a value x such that f(x) = 0, for a given function f. ...
The precise meaning of closed-form solution depends on what operations are considered to be well-known. For example, 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. 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. In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ...
In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...
The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ...
Traditionally, the well-known functions were limited to the elementary functions. Also excluded were infinite series, limits, continued fractions, etc. In mathematics, several functions are important enough to deserve their own name. ...
In mathematics, a series is the sum of a sequence of terms. ...
Limit of a sequence is one of the oldest concepts in mathematical analysis. ...
In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...
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