Closed-form expression

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an <a href="/facts/Mathematical_expression/MPrMlYbE">expression</a> or <a href="/facts/Equation/pxFzLAVR">equation</a> is in closed form if it is formed with <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a>, <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>, and a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> considered as basic and connected by arithmetic operations (+, −, ×, /, and <a href="/facts/Exponentiation/pac6QY2g">integer powers</a>) and <a href="/facts/Function_composition/r5Pvnmth">function composition</a>. Commonly, the basic functions that are allowed in closed forms are <a href="/facts/Nth_root/XrRjtJGe">nth root</a>, <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>, <a href="/facts/Logarithm/QeXaV97q">logarithm</a>, and <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>. However, the set of basic functions depends on the context. For example, if one adds <a href="/facts/Polynomial_root/mlK3dhoT">polynomial roots</a> to the basic functions, the functions that have a closed form are called <a href="/facts/Elementary_function/wVFyDwu8">elementary functions</a>.
The closed-form problem arises when new ways are introduced for specifying <a href="/facts/Mathematical_objects/Juhyow9M">mathematical objects</a>, such as <a href="/facts/Limit_(mathematics)/j8JBhj4e">limits</a>, <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>, and <a href="/facts/Integral/V7lyV997">integrals</a>: 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.

Closed-form expression open-in-new

Closed-form expression