Nonelementary integral

{{Short description|Integrals not expressible in closed-form from elementary functions}}

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function.Weisstein, Eric W. "Elementary Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ElementaryFunction.html From MathWorld Accessed 24 Apr 2017. A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist.{{cite book| title=The Calculus Gallery| first=William| last=Dunham |isbn=978-0-691-13626-4 |publisher=Princeton |year=2005| page=119}} This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.

Examples

Examples of functions with nonelementary antiderivatives include:

$\sqrt{1 - x^4}$ (elliptic integral)
$\frac{1}{\ln x}$ [http://www2.maths.ox.ac.uk/cmi/library/academy/LectureNotes05/Conrad.pdf Impossibility theorems for elementary integration]; Brian Conrad. Clay Mathematics Institute: 2005 Academy Colloquium Series. Accessed 14 Jul 2014. (logarithmic integral)
$e^{-x^2}$ (error function, Gaussian integral)
$\sin(x^2)$ and $\cos(x^2)$ (Fresnel integral)
$\frac{\sin(x)}{x} = \operatorname{sinc}(x)$ (sine integral, Dirichlet integral)
$\frac{e^{-x}}{x}$ (exponential integral)
$e^{e^x} \,$ (in terms of the exponential integral)
$\ln(\ln x) \,$ (in terms of the logarithmic integral)
${x^{c-1}}e^{-x}$ (incomplete gamma function); for $c = 0,$ the antiderivative can be written in terms of the exponential integral; for $c = \tfrac{1}{2},$ in terms of the error function; for $c =$ any positive integer, the antiderivative {{em|is}} elementary.

Some common non-elementary antiderivative functions are given names, defining so-called special functions, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.

Properties

Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can Analytic function term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral Taylor series.

Even if it isn't always possible to evaluate the antiderivative in elementary terms, one can approximate a corresponding definite integral by numerical integration. There are also cases where there is no elementary antiderivative, but specific definite integrals (often improper integrals over unbounded intervals) can be evaluated in elementary terms: most famously the Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt \pi.$ {{Cite web |last=Weisstein |first=Eric W. |title=Gaussian Integral |url=https://mathworld.wolfram.com/GaussianIntegral.html |access-date=2025-05-06 |website=mathworld.wolfram.com |language=en}}

The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.

References

[http://www.sosmath.com/calculus/integration/fant/fant.html Integration of Nonelementary Functions], S.O.S MATHematics.com; accessed 7 Dec 2012.

Nonelementary integral

Examples

Properties

See also

References

Further reading