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Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer". While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using ...
where K ν (z) is the ν th order modified Bessel function of the second kind. These functions are named after William Thomson, 1st Baron Kelvin. While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments xe iφ, 0 ≤ φ ...
The Möbius function ... The sum of the Möbius function over all positive divisors of n (including n itself and 1) is zero except when n = 1:
If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4.The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1.
where ρ runs over the non-trivial zeros of the Riemann zeta function and t > 0. The sum over non-trivial zeta zeros in the formula for π 0 (x) describes the fluctuations of π 0 (x) while the remaining terms give the "smooth" part of prime-counting function, [28] so one can use
If n is an integer, then n < 0 must imply that n ≤ −1, while n > 0 must imply that the function attains unity at n = 1. Therefore the "step function" exhibits ramp-like behavior over the domain of [−1, 1], and cannot authentically be a step function, using the half-maximum convention. Unlike the continuous case, the definition of H[0] is ...
For instance, betainc (incomplete beta function) in MATLAB and GNU Octave, pbeta (probability of beta distribution) in R, or special.betainc in SciPy compute the regularized incomplete beta function—which is, in fact, the cumulative beta distribution—and so, to get the actual incomplete beta function, one must multiply the result of betainc ...