Search results
Results from the WOW.Com Content Network
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 ...
These 2 latter inequalities follow from the convexity of the exponential function (or from an analysis of the function ). Letting u = x 2 {\displaystyle u=x^{2}} and making use of the basic properties of improper integrals (the convergence of the integrals is obvious), we obtain the inequalities:
Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac ...
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives .
Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean. [1] Define sequences a n and g n, where a 0 = 1, g 0 = √ 1 − k 2 = k ′ and the recurrence relations a n + 1 = a n + g n / 2 , g n + 1 = √ a n g n hold.
Args: f: The function to integrate. a: Lower limit of integration. b: Upper limit of integration. max_steps: Maximum number of steps. acc: Desired accuracy. Returns: The approximate value of the integral.
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
In theory of vibrations, Duhamel's integral is a way of calculating the response of linear systems and structures to arbitrary time-varying external perturbation. Introduction [ edit ]