# expressible in closed form

An expression is , 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 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 of the “elementary functions” may be enlarged by allowing in it some other functions, e.g. the error function.

Title expressible in closed form ExpressibleInClosedForm 2013-03-22 18:29:09 2013-03-22 18:29:09 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 30A99 msc 26E99 ClosedForm IrreducibilityOfBinomialsWithUnityCoefficients ReductionOfEllipticIntegralsToStandardForm closed form