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In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier ...
The Laplace–Stieltjes transform of a real-valued function g is given by a Lebesgue–Stieltjes integral of the form ()for s a complex number.As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that g be of bounded variation on the region of integration.
In mathematics, the Stieltjes transformation S ρ (z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside I by the formula S ρ ( z ) = ∫ I ρ ( t ) d t t − z , z ∈ C ∖ I . {\displaystyle S_{\rho }(z)=\int _{I}{\frac {\rho (t)\,dt}{t-z}},\qquad z\in \mathbb {C} \setminus I.}
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics. [1] Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in
The distribution of the product of correlated non-central normal samples was derived by Cui et al. [11] and takes the form of an infinite series of modified Bessel functions of the first kind. Moments of product of correlated central normal samples. For a central normal distribution N(0,1) the moments are
Plot of the hypergeometric function 2F1(a,b; c; z) with a=2 and b=3 and c=4 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , the Gaussian or ordinary hypergeometric function 2 F 1 ( a , b ; c ; z ) is a special function represented by the hypergeometric series , that ...
The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1]. The logit-normal distribution on (0,1). The Dirac delta function , although not strictly a probability distribution, is a limiting form of many continuous probability functions.
Non-logarithmized series that are growing exponentially often appear to have increasing variability as the series rises over time. The variability in percentage terms may, however, be rather stable. Use a different specification for the model (different X variables, or perhaps non-linear transformations of the X variables).