In order for the surface to stay flat its growth rate Log[h[x, y]] must therefore solve Laplace's equation, and hence must be a harmonic function Re[f[x +  y]] . … The pictures below show results for several growth rate functions; in the last case, the function is not harmonic, and the surface cannot be drawn in the plane without tearing. … Harmonic growth rate functions can potentially be obtained from the large-time effects of a chemical subject to diffusion.

Ackermann functions A convenient example is f[1, n_] := n; f[m_, 1] := f[m - 1, 2] f[m_, n_] := f[m - 1, f[m, n - 1] + 1] The original function constructed by Wilhelm Ackermann around 1926 is essentially f[1, x_, y_] := x + y; f[m_, x_, y_] := Nest[f[m - 1, x, #] &, x, y - 1] or f[m_, x_, y_]:= Nest[Function[z, Nest[#, x, z - 1]] &, x + # &, m - 1][y] For successive m (following the so-called Grzegorczyk hierarchy) this is x + y , x y , x y , Nest[x # &, 1, y] , .... f[4, x, y] can also be written Array[x &, y, 1, Power] and is sometimes called tetration and denoted x ↑ ↑ y .

[Computation of] mathematical functions The number of bit operations needed to add two n -digit numbers is of order n . … Many standard continuous mathematical functions just increase or decrease smoothly at large x (see page 917 ). … Unlike for continuous mathematical functions, known algorithms for number theoretical functions such as FactorInteger[x] or MoebiusMu[x] typically seem to require a number of operations that grows faster with the number of digits n in x than any power of n (see page 1090 ).

Following the introduction of so-called primitive recursive functions (see page 907 ) in the 1880s, there had by the 1920s emerged the idea that perhaps any reasonable function could be computed using the small set of operations on which primitive recursive functions are based. … But the discovery of the Ackermann function in the late 1920s (see page 906 ) showed that there are reasonable functions that are not primitive recursive. The proof of Gödel's Theorem in 1931 made use of so-called general recursive functions (see page 1121 ) as a way to represent possible functions in arithmetic.

Recognizing repetition [in sounds] The curve of the function Sin[x] + Sin[ √ 2 x] shown on page 146 looks complicated to the eye. … However, if one uses the function to generate a score—say playing a note at the position of each peak—then no such simplicity can be recognized.

Binary decision diagrams One can specify a Boolean function of n variables by giving a finite automaton (and thus a network) in which paths exist only for those lists of values for which the function yields True . … Out of all possible Boolean functions the number that require BDDs of sizes 1, 2, ... is for n = 2 : {1, 0, 6, 9} and for n = 3 : {1, 0, 0, 27, 36, 132, 60} ; the absolute maximum grows roughly like 2 n . … In practical system design BDDs have become fairly popular in the past ten years, and by maintaining minimality when logical combinations of functions are formed, cases with millions of nodes have been studied.

Gegenbauer functions Introduced by Leopold Gegenbauer in 1893 GegenbauerC[n, m, z] is a polynomial in z with integer coefficients for all integer n and m . … The GegenbauerC[n, d/2 - 1, z] form a set of orthogonal functions on a d -dimensional sphere.

Circuit complexity Any function with a fixed size of input can be computed by a circuit of the kind shown on page 619 . … Note that much as on page 662 one can construct universal circuits that can be arranged by appropriate choice of parts of their input to compute any function of a given input size.

Proofs in Mathematica Most of the individual built-in functions of Mathematica I designed to be as predictable as possible—applying transformations in definite ways and using algorithms that are never of fundamentally unknown difficulty. … And in many cases these functions end up trying to prove theorems; so for example FullSimplify[(a + b)/2 ≥ Sqrt[a b], a > 0 && b > 0] must in effect prove a theorem to get the result True .

Digit sequence encryption One can consider using as encrypting sequences the digit sequences of numbers obtained from standard mathematical functions. … But in many cases one can immediately tell how a sequence was made just by globally applying appropriate mathematical functions. … If no linear equation is satisfied by any combination of known functions of x , however, the method fails, and it seems quite likely that in such cases secure encrypting sequences can be generated, albeit less efficiently than with systems like cellular automata.