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n = 1 that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(a n,b n)} N n = 1 for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset. [16]
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.
The x antiderivative of y and the second antiderivative of f, Euler notation. D-notation can be used for antiderivatives in the same way that Lagrange's notation is [ 10 ] as follows [ 9 ] D − 1 f ( x ) {\displaystyle D^{-1}f(x)} for a first antiderivative,
In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function F(x) such that
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [ 2 ]
5.4 Derivation for the integral form of the remainder. ... for x near the point a, ... and a similar estimate if x < a. This is a simple consequence of the Lagrange ...
If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ( quadrature or squaring ...
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g.More precisely, given an open set in the complex plane and a function :, the antiderivative of is a function : that satisfies =.