Nonelementary integral

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations).[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.

Some examples of such functions are:

• ${\displaystyle {\sqrt {1-x^{4}}}}$[1] (see Elliptic integral)
• ${\displaystyle {\frac {1}{\ln x}}}$[3] (see Logarithmic integral).
• ${\displaystyle e^{-{\frac {x^{2}}{2}}}\,}$[1] (see Error function and Gaussian integral)
• ${\displaystyle \sin(x^{2})}$ and ${\displaystyle \cos(x^{2})}$ (see Fresnel integral)
• ${\displaystyle {\frac {\sin(x)}{x}}=\operatorname {sinc} (x)}$ (see Sine integral and Dirichlet integral)
• ${\displaystyle {\frac {e^{x}}{x}}}$ (see Exponential integral).
• The function ${\displaystyle e^{e^{x}}\,}$can also be shown to have a nonelementary antiderivative that can be written in terms of the exponential integral. Likewise, ${\displaystyle \ln(\ln x)\,}$has a nonelementary antiderivative in terms of the logarithmic integral.
• ${\displaystyle {x^{z-1}}e^{-x}}$, for some values of z, such as 0 (which has an antiderivative in terms of the exponential integral) and 0.5 (which has an antiderivative in terms of the error function). However for values of z including 1 and 2, the antiderivative is elementary. See Incomplete gamma function.

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.

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