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Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
and the problem is, given the continuous kernel function and the function , to find the function .. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely (,) = (), and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions and and therefore ...
Earlier systems such as Macsyma had a few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the Maple [ 4 ] system ...
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li s (z) of order s and argument z.Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function.
Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: [3] () + (,) = where g(t) vanishes at least once in the interval [a,b] [4] [5] or where g(t) vanishes at a finite number of points in (a,b).
Bayesian quadrature is a statistical approach to the numerical problem of computing integrals and falls under the field of probabilistic numerics. It can provide a full handling of the uncertainty over the solution of the integral expressed as a Gaussian process posterior variance.
Contour plot of the beta function. In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients.
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy -plane bounded by the graph of f , the x -axis, and the lines x = a and x = b , such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total.