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The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.
Generalization. The exponential integral may also be generalized to. which can be written as a special case of the upper incomplete gamma function: [8] The generalized form is sometimes called the Misra function [9] , defined as. Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
Probability theory. In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take ...
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 expressed as elementary functions.
Laplace's method. In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form. where is a twice- differentiable function, is a large number, and the endpoints and could be infinite. This technique was originally presented in the book by Laplace (1774).
It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1770). [1]