If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis. For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

The specific form of the continuous generalization of the modulo 2 function used is λ [x_] := Exp[-10 (x - 1) 2 ] + Exp[-10 (x - 3) 2 ] Each cell in the system is then updated according to λ [a + c] for rule 90, and λ [a + b + c + b c] for rule 30.

On this page and the ones that follow [ 165 , 166 ] the initial conditions used are u=Exp[-x 2 ] , D[u,t]=0 .

The solution to this equation with an impulse initial condition is Exp[-x 2 /t] , and with a block from -a to a it is (Erf[(a - x)/ √ t ] + Erf[(a + x)/ √ t ])/a .

Fourier transforms In a typical Fourier transform, one uses basic forms such as Exp[  π r x/n] with r running from 1 to n . … Fourier[data] can be thought of as multiplication by the n × n matrix Array[Exp[2 π  #1 #2/n] &, {n, n}, 0] .

Repetition in continuous systems A standard approach to partial differential equations (PDEs) used for more than a century is so-called linear stability analysis, in which one assumes that small fluctuations around some kind of basic solution can be treated as a superposition of waves of the form Exp[  k x] Exp[  ω t] .

With k colors each giving a string of the same length s the recurrence relation is Thread[Map[ ϕ [#][t + 1, ω ] &, Range[k] - 1]  Apply[Plus, MapIndexed[Exp[  ω (Last[#2] - 1) s t ] ϕ [#1][t, ω ] &, Range[k] - 1 /. rules, {-1}], {1}]/ √ s ] Some specific properties of the examples shown include: (a) (Thue–Morse sequence) The spectrum is essentially Nest[Range[2 Length[#]] Join[#, Reverse[#]] &, {1}, t] . … After t steps a continuous approximation to the spectrum is Product[1 - Exp[2 s  ω ], {s, t}] , which is an example of a type of product studied by Frigyes Riesz in 1918 in connection with questions about the convergence of trigonometric series. … After t steps a continuous approximation to the spectrum is Product[1 + Exp[3 s 2  ω ], {s, t}] .

Mathematical impossibilities It is sometimes said that in the 1800s problems such as trisecting angles, squaring the circle, solving quintics, and integrating functions like Exp[x 2 ] were proved mathematically impossible.

: ExpIntegralEi[1] - EulerGamma .

[Examples of] short computations Some properties include: (a) The regions are bounded by the hyperbolas x y  Exp[n/2] for successive integers n .