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
Canonical name	ExpressibleInClosedForm
Date of creation	2013-03-22 18:29:09
Last modified on	2013-03-22 18:29:09
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Definition
Classification	msc 30A99
Classification	msc 26E99
Related topic	ClosedForm
Related topic	IrreducibilityOfBinomialsWithUnityCoefficients
Related topic	ReductionOfEllipticIntegralsToStandardForm
Defines	closed form