Nonelementary integral

In mathematics, a nonelementary antiderivative is an antiderivative for which it can be shown that there exists no formula in terms of elementary functions (i.e. involving polynomials, and the standard functions sin, cos, exp, and so on).^[1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist.^[2] This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives. It can be shown that, if one is given a function of any complexity, the probability that it will have an elementary antiderivative is very low.

Some examples of such functions are:

${\sqrt {1-x^{4}}}$ ^[1]
$\ln(\ln x)\,$
${\frac {1}{\ln x}}$ ^[3] (see Logarithmic integral)
${\frac {e^{x}}{x}}$ (see Exponential integral)
$e^{e^{x}}\,$
$e^{-{\frac {x^{2}}{2}}}\,$ ^[1] (see Normal distribution)
$\sin(x^{2})$ and $\cos(x^{2})$ (see Fresnel integral)

The evaluation of nonelementary antiderivatives can often be done using Taylor series. This is because Taylor series can always be integrated as one would an ordinary polynomial (using the fact that any Taylor series is uniformly convergent within its radius of convergence), even if there is no elementary antiderivative of the function that generated the Taylor series.

However, in some cases it is not possible to rely on Taylor series. For example, if the function is not infinitely differentiable, one cannot generate a Taylor series. Even if a Taylor series can be generated, there is a good possibility that it will diverge and not represent the function one is attempting to antidifferentiate; there even exist non-analytic but infinitely differentiable real-valued functions (see bump function). Many functions which are infinitely differentiable have higher order derivatives that are unmanageable by hand. In these cases, it is not possible to evaluate indefinite integrals, but definite integrals can be evaluated numerically, for instance by Simpson's rule. There are yet other cases (such as the Gaussian integral) where definite integrals can be evaluated exactly without numerical methods, but indefinite integrals cannot, for lack of an elementary antiderivative.

The integrals for many of these functions can be written down if one allows so-called “special” (nonelementary) functions. For example, the first example’s integral is expressible using incomplete elliptic integrals of the first kind, the second and third use the logarithmic integral, the fourth the exponential integral, and the sixth the error function. Still, there exist functions, such as $x^{x}$ and $\sin(\sin(x))$ for which no notation currently exists to describe their integrals (other than the use of the integrals themselves).

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

References

1 2 3 Weisstein, Eric W. "Elementary Function." From MathWorld--A Wolfram Web Resource. Accessed 7 Dec 2012.
↑ Dunham, William (2005). The Calculus Gallery. Princeton. p. 119. ISBN 978-0-691-13626-4.
↑ Impossibility theorems for elementary integration; Brian Conrad. Clay Mathematics Institute: 2005 Academy Colloquium Series. Accessed 14 Jul 2014.

Integration of Nonelementary Functions, S.O.S MATHematics.com; accessed 7 Dec 2012.

Nonelementary integral

See also

References

Further reading