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# expressible in closed form

An expression is expressible in a closed form, if it can be converted (simplified) into an expression containing only elementary functions, combined by a finite amount of rational operations and compositions. Thus, such a closed form must not contain e.g. limit signs, integral signs, sum signs and “…”.

For example,

$\int\!\!\frac{dx}{x^{4}\!+\!1},$ |

may be expressed in the closed form

$\frac{1}{4\sqrt{2}}\ln\frac{x^{2}\!+\!x\sqrt{2}\!+\!1}{x^{2}\!-\!x\sqrt{2}\!+% \!1}+\frac{1}{2\sqrt{2}}\arctan\frac{x\sqrt{2}}{1\!-\!x^{2}}+C$ |

but for

$\int\!\!\frac{dx}{\sqrt{x^{4}\!+\!1}}\,dx,$ |

there exists no closed form.

In certain contexts, the scope of the “elementary functions” may be enlarged by allowing in it some other functions, e.g. the error function.

Defines:

closed form

Related:

ClosedForm, IrreducibilityOfBinomialsWithUnityCoefficients, ReductionOfEllipticIntegralsToStandardForm

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

30A99*no label found*26E99

*no label found*

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