Central Limit Theorem Averages of large collections of random numbers tend to follow a Gaussian or normal distribution in which the probability of getting value x is Exp[-(x - μ ) 2 /(2 σ 2 )] / (Sqrt[2 π ] σ ) The mean μ and standard deviation σ are determined by properties of the random numbers, but the form of the distribution is always the same.

The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] .

The continued fractions for Exp[2/k] and Tan[1/2k] have simple forms (as discussed by Leonhard Euler in the mid-1700s); other rational powers of E and tangents do not appear to. … An example discovered by Srinivasa Ramanujan around 1913 is Exp[ π √ 163 ] , which is an integer to one part in 10 30 , and has second continued fraction term 1,333,462,407,511. … Other less spectacular examples include Exp[ π ]- π and 163/Log[163] .

Different current methods asymptotically require slightly different numbers of steps—but all typically at least Exp[Sqrt[Log[n]]] .

(This can be done by repeatedly making use of functional relations such as Exp[2x]  Exp[x] 2 which express f[2x] as a polynomial in f[x] ; such an approach is known to work for elementary, elliptic, modular and other functions associated with ArithmeticGeometricMean and for example DedekindEta .)

The way the path integral for a quantum field theory works, each possible configuration of the field is in effect taken to make a contribution Exp[  s/ ℏ ] , where s is the so-called action for the field configuration (given by the integral of the Lagrangian density—essentially a modified energy density), and ℏ is a basic scale factor for quantum effects (Planck's constant divided by 2 π ). … And indeed standard perturbation theory is based on starting from these and then looking at the expansion of Exp[  s/ ℏ ] in powers of the coupling constant. … Formally continuing to Euclidean space makes path integrals easier to define with traditional mathematics, and gives them weights of the form Exp[- β s] —analogous to constant temperature systems in statistical mechanics.

With new tip positions as on page 400 given by {p Exp[  θ ], p Exp[-  θ ], q} , rough {p, q, θ } for at least some versions of some common plants include: wild carrot (Queen Anne's lace) {0.4, 0.7, 30 ° } , cypress {0.4, 0.7, 45 ° } , coralbells {0.5, 0.4, 0 ° } , ivy {0.5, 0.6, 0 ° } , grape {0.5, 0.6, 15 ° } , sycamore {0.5, 0.6, 15 ° } , mallow {0.5, 0.6, 30 ° } , goosefoot {0.55, 0.8, 30 ° } , willow {0.55, 0.8, 80 ° } , morning glory {0.7, 0.8, 0 ° } , cucumber {0.7, 0.8, 15 ° } , ginger {0.65, 0.6, 15 ° } .

With odd n the same turns out to be true for sequences Exp[2 π  Mod[Range[n] 2 , n]/n] —a fact used in the design of acoustic diffusers (see page 1183 ).

For equations of the form ∂ tt u[t, x]  ∂ xx u[t, x] + f[u[t, x]] one can set up a simple finite difference method by taking f in the form of pure function and creating from it a kernel with space step dx and time step dt : PDEKernel[f_, {dx_, dt_}] := Compile[{a,b,c,d}, Evaluate[(2 b - d) + ((a + c - 2 b)/dx 2 + f[b]) dt 2 ]] Iteration for n steps is then performed by PDEEvolveList[ker_, {u0_, u1_}, n_] := Map[First, NestList[PDEStep[ker, #]&, {u0, u1}, n]] PDEStep[ker_, {u1_, u2_}] := {u2, Apply[ker, Transpose[ {RotateLeft[u2], u2, RotateRight[u2], u1}], {1}]} With this approach an approximation to the top example on page 165 can be obtained from PDEEvolveList[PDEKernel[ (1 - # 2 )(1 + #)&, {.1, .05}], Transpose[ Table[{1, 1} N[Exp[-x 2 ]], {x, -20, 20, .1}]], 400] For both this example and the middle one the results converge rapidly as dx decreases.

It is known that Exp[n] and Log[n] for whole numbers n (except 0 and 1 respectively) are transcendental.