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In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion = + + +.
The Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of 1 / 1 − x is the geometric series + + + +. So, by substituting x for 1 − x, the Taylor series of 1 / x at a = 1 is
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.
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.
The Maclaurin series of an even function includes only even powers. The Maclaurin series of an odd function includes only odd powers. The Fourier series of a periodic even function includes only cosine terms. The Fourier series of a periodic odd function includes only sine terms. The Fourier transform of a purely real-valued even function is ...
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In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k {\displaystyle y=ax^{k}} – appear as straight lines in a log–log graph, with the exponent corresponding to ...
Maclaurin attributed the series to Brook Taylor, though the series was known before to Newton and Gregory, and in special cases to Madhava of Sangamagrama in fourteenth century India. [6] Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series. [7]