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The Mandelbrot set ( / ˈmændəlbroʊt, - brɒt /) [ 1][ 2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers for which the function does not ...
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
This is a polar plot of the first 20 real values r n of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. The values of r n in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1 / 2 and, according to the Riemann hypothesis, they all lie on the line σ = 1 / 2 . This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical ...
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. This quadratic polynomial has two roots, and. The golden ratio is also closely related to the polynomial. which has roots and As the root of a quadratic polynomial, the golden ratio is a constructible number.
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The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, [2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend ...