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In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Γ ( n ) = ( n − 1 ) ! . {\displaystyle ...
Viète. de Moivre. Euler. Fourier. v. t. e. In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.
- Properties
- Two-Dimensional Gaussian Function
- Multi-Dimensional Gaussian Function
- Estimation of Parameters
- Discrete Gaussian
- Applications
- See Also
- External Links
Gaussian functions arise by composing the exponential function with a concave quadratic function: 1. α = − 1 / 2 c 2 , {\displaystyle \alpha =-1/2c^{2},} 2. β = b / c 2 , {\displaystyle \beta =b/c^{2},} 3. γ = ln a − ( b 2 / 2 c 2 ) . {\displaystyle \gamma =\ln a-(b^{2}/2c^{2}).} (Note: in ln a , a = 1 / ( σ 2 π ) {\displaystyle \ln a,a=1/(\sig...
Base form: In two dimensions, the power to which e is raised in the Gaussian function is any negative-definite quadratic form. Consequently, the level setsof the Gaussian will always be ellipses. A particular example of a two-dimensional Gaussian function is Here the coefficient A is the amplitude, x0, y0 is the center, and σx, σy are the x and y s...
In an n {\displaystyle n} -dimensional space a Gaussian function can be defined as The integral of this Gaussian function over the whole n {\displaystyle n} -dimensional space is given as It can be easily calculated by diagonalizing the matrix C {\displaystyle C} and changing the integration variables to the eigenvectors of C {\displaystyle C} . Mo...
A number of fields such as stellar photometry, Gaussian beam characterization, and emission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. There are three unknown parameters for a 1D Gaussian function (a, b, c) and five for a 2D Gaussian f...
One may ask for a discrete analog to the Gaussian;this is necessary in discrete applications, particularly digital signal processing. A simple answer is to sample the continuous Gaussian, yielding the sampled Gaussian kernel. However, this discrete function does not have the discrete analogs of the properties of the continuous function, and can lea...
Gaussian functions appear in many contexts in the natural sciences, the social sciences, mathematics, and engineering. Some examples include: 1. In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the centra...
Bensimhoun Michael, N-Dimensional Cumulative Function, And Other Useful Facts About Gaussians and Normal Densities (2009)Binomial distribution for = with n and k as in Pascal's triangleThe probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is /. In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no ...
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although ...
The Planck constant, or Planck's constant, denoted by , [1] is a fundamental physical constant [1] of foundational importance in quantum mechanics: a photon 's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a matter wave equals the Planck constant divided by the associated particle momentum.
