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Nonelementary integral. In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) 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 ...
Elementary function. In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x1/n). [1]
Liouville's theorem (differential algebra) In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, [1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be ...
Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.
Also subharmonic function and superharmonic function. Elementary function: composition of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations. Special functions: non-elementary functions that have established names and notations due to their importance.
Fubini's theorem. In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle , then one can evaluate the double integral as an ...
Transcendental function. In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using the basic operations of addition, subtraction, multiplication, and division.
As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey [19] and Fekih-Ahmed [20] obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n. [19] If v is the unique solution of