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1 - 3 of 3 for ArcCos

Exact iterates [in iterated maps]
For any integer a the n th iterate of x FractionalPart[a x] can be written as FractionalPart[a n x] , or equivalently 1/2 - ArcTan[Cot[a n π x]]/ π . In the specific case a = 2 the iterates of If[x < 1/2, a x, a (1 - x)] have the form ArcCos[Cos[2 n π x]]/ π .

In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms
Sin[2 t ArcSin[ √ x ]] 2
(1 - Cos[2 t ArcCos[1 - 2 x]])/2
(1 - ChebyshevT[2 t , 1 - 2x])/2
where these follow from functional relations such as
Sin[2x] 2 4 Sin[x] 2 (1 - Sin[x] 2 )
ChebyshevT[m n, x] ChebyshevT[m, ChebyshevT[n, x]]
For a = 2 it also turns out that there is a complete formula:
(1 - (1 - 2 x) 2 t )/2
And the same is true for a = -2 :
1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))]
In all these examples t enters essentially only in a t . … When a = 2 it is Exp[x] and when a = -2 it is 2 Cos[(1/3) ( π - √ 3 x)] .

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.