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A periodic Bernoulli polynomial P n (x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
Bernoulli's principle is a key concept in fluid dynamics that relates pressure, density, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum. [1]:
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus .
Gregory coefficients G n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, ... and Schröder's integral formula [19] ...
Bernoulli equation may refer to: Bernoulli differential equation; Bernoulli's equation, in fluid dynamics; Euler–Bernoulli beam equation, in solid mechanics
The logistic equation is a special case of the Bernoulli differential equation and has the following solution: f ( x ) = e x e x + C . {\displaystyle f(x)={\frac {e^{x}}{e^{x}+C}}.} Choosing the constant of integration C = 1 {\displaystyle C=1} gives the other well known form of the definition of the logistic curve:
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...