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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.
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 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 ...
The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan, [1] [2] but their history may also be traced back to the much earlier works. [ 3 ] Integral representations
where li(z) is the integral logarithm and () is the binomial coefficient. It is also known that the zeta function , the gamma function , the polygamma functions , the Stieltjes constants and many other special functions and constants may be expressed in terms of infinite series containing these numbers.
The generating function of the Bernoulli polynomials is given by: = = ()! These polynomials are given in terms of the Hurwitz zeta function: (,) = = (+)by (,) = for .Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation: [6]
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696). [2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733.
Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the Bernoulli differential equation, ′ = + ().