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Exponential function: raises a fixed number to a variable power. Hyperbolic functions: formally similar to the trigonometric functions. Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions. Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
A ray through the unit hyperbola = in the point (,), where is twice the area between the ray, the hyperbola, and the -axis. The earliest and most widely adopted symbols use the prefix arc-(that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).
The matrix exponential of another matrix (matrix-matrix exponential), [24] is defined as = = for any normal and non-singular n×n matrix X, and any complex n×n matrix Y. For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y , because the multiplication operator for matrix ...
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters .
The ordinary exponential function of mathematical analysis is a special case of the exponential map when is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however ...
The operations are pointwise addition and multiplication of functions. Let X be a set, and let R be a ring. Then the set of all functions from X to R forms a ring, which is commutative if R is commutative. The ring of quadratic integers, the integral closure of in a quadratic extension of .
while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2). A more abstract version of formula for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of for an analytic (at a and −a) function is provided by application of Sylvester's formula, [3]
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
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