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Roots of polynomials can thus almost always be found with NSolve in about Log[n] m[n] operations. If one evaluates NIntegrate or NDSolve by effectively fitting functions to order s polynomials the difficulty of getting results with n -digit precision typically increases like 2 n/s . An adaptive algorithm such as Romberg integration reduces this to about 2^ √ n .

Results about primes
Prime[n] is given approximately by n Log[n] + n Log[Log[n]] . ( Prime[10 9 ] is 22,801,763,489 while the approximation gives 2.38 × 10 10. ) A first approximation to PrimePi[n] is n/Log[n] . A somewhat better approximation is LogIntegral[n] , equal to Integrate[1/Log[t], {t, 2, n}] . … According to the Riemann Hypothesis, the difference between PrimePi[n] and LogIntegral[n] is of order √ n Log[n] .

For rational functions f[x] , Integrate[f[x], {x, 0, 1}] must always be a linear function of Log and ArcTan applied to algebraic numbers ( f[x] = 1/(1 + x 2 ) for example yields π /4 ). Multiple integrals of rational functions can be more complicated, as in
Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}] HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan
and presumably often cannot be expressed at all in terms of standard mathematical functions. … If f[n] is a rational function, Sum[f[n], {n, ∞ }] must just be a linear combination of PolyGamma functions, but again the multivariate case can be much more complicated.

Here a typical orthogonality property is Integrate[f[r, x] f[s, x], {x, 0, 1}] KroneckerDelta[r, s] . As discovered by Joseph Fourier around 1810, this is satisfied for basis functions such as Sin[2 n π x]/ √ 2 .

Intrinsically defined curves
With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 )
NDSolve[{x'[s] Cos[ θ [s]], y'[s] Sin[ θ [s]], θ '[s] f[s], x[0] y[0] θ [0] 0}, {x, y, θ }, {s, 0, s max }]
For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve :
f[s] = 1: {Sin[ θ ], Cos[ θ ]}
f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]}
f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]}
f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]}
f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]}
f[s] = s n : result involves Gamma[1/n, ± θ n/n ]
f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

The perceived color of light with a given wavelength distribution is basically determined by the three numbers obtained by integrating these responses. … Mixing n specific colors in different proportions allows one to reach any point in an n -cornered polytope. For n = 3 this polytope comes close to filling the region of all possible colors, but for no n can it completely fill it—which is why practical displays and printing processes can produce only limited ranges of colors.

In the 1970s they also became widely used for high-dimensional numerical integration, notably for Feynman diagram evaluation in quantum electrodynamics. … In the late 1970s it was noted that by evaluating PowerMod[a, n - 1, n] 1 for several random integers a one can with high probability quickly deduce PrimeQ[n] .