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The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in = | | + there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the ...
More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO( n ) , one example of which is ...
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
The Rankine scale is used in engineering systems where heat computations are done using degrees Fahrenheit. [3] The symbol for degrees Rankine is °R [2] (or °Ra if necessary to distinguish it from the Rømer and Réaumur scales). By analogy with the SI unit kelvin, some authors term the unit Rankine, omitting the degree symbol. [4] [5]
Spaces within a formula must be directly managed (for example by including explicit hair or thin spaces). Variable names must be italicized explicitly, and superscripts and subscripts must use an explicit tag or template. Except for short formulas, the source of a formula typically has more markup overhead and can be difficult to read.
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The h-calculus is the calculus of finite differences, which was studied by George Boole and others, and has proven useful in combinatorics and fluid mechanics. In a sense, q -calculus dates back to Leonhard Euler and Carl Gustav Jacobi , but has only recently begun to find usefulness in quantum mechanics , given its intimate connection with ...
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