30A99-ExpressibleInClosedForm.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ExpressibleInClosedForm}
\pmcreated{2013-03-22 18:29:09}
\pmmodified{2013-03-22 18:29:09}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{expressible in closed form}
\pmrecord{8}{41162}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30A99}
\pmclassification{msc}{26E99}
\pmrelated{ClosedForm}
\pmrelated{IrreducibilityOfBinomialsWithUnityCoefficients}
\pmrelated{ReductionOfEllipticIntegralsToStandardForm}
\pmdefines{closed form}

\endmetadata

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\begin{document}
An expression is \PMlinkescapetext{{\em 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 \PMlinkescapetext{contain} e.g. limit signs, integral signs, sum signs and ``\ldots''.

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 \PMlinkescapetext{scope} of the ``elementary functions'' may be enlarged by allowing in it some other functions, e.g. the error function.

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