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In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a stencil made up of the point itself together with its eight "neighbors".
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.
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
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 [ 11 ] as follows [ 10 ] D − 1 f ( x ) {\displaystyle D^{-1}f(x)} for a first antiderivative,
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
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