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It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor, [8] after whom the series are now named. The Maclaurin series was named after Colin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.
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. [1] [17] [18] [28] [29]
A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution , and X i , i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log( p ) distribution, then
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
This expansion is a Maclaurin series, so the n th cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero: [1] = (). If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Furthermore, every polynomial is its own Maclaurin series. The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x 0 (as in the definition) but for all values of x (real or complex). The trigonometric functions, logarithm, and the power functions are analytic on any open set of their ...
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines , each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines ...