Nonelementary integral
Integrals not expressible in closed-form from elementary functions
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Related on Lably
Through Mathematics, Antiderivative and Polynomial
Examples
Examples of functions with nonelementary antiderivatives include:
- (elliptic integral)
- (logarithmic integral)
- (error function, Gaussian integral)
- and (Fresnel integral)
- (sine integral, Dirichlet integral)
- (exponential integral)
- (in terms of the exponential integral)
- (in terms of the logarithmic integral)
- (incomplete gamma function); for the antiderivative can be written in terms of the exponential integral; for in terms of the error function; for any positive integer, the antiderivative 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.
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Properties
Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can always be integrated 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 though it is not 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
The closure under integration of the set of the elementary functions is the set of the Liouvillian functions.