Fixed Point

Author

Eric W. Weisstein
September 26, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FixedPoint.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FixedPoint.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Trigs

Plot[{#[x],x},{x,-5,5}]&/@{Cos,Cot,Csc,Sec,Sin,Tan}

Cos

FindRoot[Cos[x]==x,{x,1}]

Solve[Cos[x]==x,x]

Reduce[Cos[x]==x,x,Reals]

Reduce[Cos[x]==x&&0<x<100,x,Reals]

A003957

Cot

FindRoot[Cot[x]==x,{x,1}]

Solve[Cot[x]==x,x]

Reduce[Cot[x]==x,x,Reals]

Reduce[Cot[x]==x&&1/10<x<1,x]

A069855

Csc

FindRoot[Csc[x]==x,{x,1}]

Solve[Csc[x]==x,x]

Reduce[Csc[x]==x,x,Reals]

Reduce[Csc[x]==x&&1/10<x<2,x]

XXXXX

Sec

FindRoot[Sec[x]==x,{x,5}]

Solve[Sec[x]==x,x]

Reduce[Sec[x]==x,x,Reals]

Reduce[Sec[x]==x&&4<x<5,x]

XXXXX

Sin

FindRoot[Sin[x]==x,{x,0}]

Solve[Sin[x]==x,x]

Reduce[Sin[x]==x,x,Reals]

Reduce[Sin[x]==x&&-1<x<1,x]

Tan

FindRoot[Tan[x]==x,{x,4}]

Solve[Tan[x]==x,x]

Reduce[Tan[x]==x,x,Reals]

Reduce[Tan[x]==x&&4<x<5,x]

A115365

ArcTrig

Plot[{#[x],x},{x,-5,5}]&/@{ArcCos,ArcCot,ArcCsc,ArcSec,ArcSin,ArcTan}

ArcCos

FindRoot[ArcCos[x]==x,{x,0}]

Solve[ArcCos[x]==x,x]

Reduce[ArcCos[x]==x,x]

Reduce[ArcCos[x]==x,x,Reals]

A003957

ArcCot

FindRoot[ArcCot[x]==x,{x,0}]

Solve[ArcCot[x]==x,x]

Reduce[ArcCot[x]==x,x,Reals]

Reduce[ArcCot[x]==x&&1/10<x<1,x]

A069855

ArcCsc

FindRoot[ArcCsc[x]==x,{x,1.1}]

Solve[ArcCsc[x]==x,x]

Reduce[ArcCsc[x]==x,x,Reals]

Reduce[ArcCsc[x]==x&&1/10<x<12/10,x]

XXXXX

Hyperbolic Trig

Plot[{#[x],x},{x,-5,5}]&/@{Cosh,Coth,Csch,Sech,Sinh,Tanh}

Cosh

Plot[{x,Cosh[x]},{x,-2,2}]

FindRoot[Cosh[x]==x,{x,1},WorkingPrecision->30]

Solve[Cosh[x]==x,x]

Reduce[Cosh[x]==x,x,Reals]

Reduce[Cosh[x]==x&&-1<x<1,x]

Coth

FindRoot[Coth[x]==x,{x,1}]

Solve[Coth[x]==x,x]

Reduce[Coth[x]==x,x,Reals]

Reduce[Coth[x]==x&&0<x<2,x]

A085984

Csch

FindRoot[Csch[x]==x,{x,1}]

Solve[Csch[x]==x,x]

Reduce[Csch[x]==x,x,Reals]

Reduce[Csch[x]==x&&0<x<2,x]

XXX

Sech

FindRoot[Sech[x]==x,{x,1}]

Solve[Sech[x]==x,x]

Reduce[Sech[x]==x,x,Reals]

Reduce[Sech[x]==x&&0<x<2,x]

A069814

Hyperbolic ArcTrig

Plot[{#[x],x},{x,-5,5}]&/@{ArcCosh,ArcCoth,ArcCsch,ArcSech,ArcSinh,ArcTanh}

ArcCoth

FindRoot[ArcCoth[x]==x,{x,2}]

Solve[ArcCoth[x]==x,x]

Reduce[ArcCoth[x]==x,x,Reals]

Reduce[ArcCoth[x]==x&&0<x<2,x]

A085984

ArcCsch

FindRoot[ArcCsch[x]==x,{x,2}]

Solve[ArcCsch[x]==x,x]

Reduce[ArcCsch[x]==x,x,Reals]

Reduce[ArcCsch[x]==x&&0<x<1,x]

XXXXX

ArcSech

FindRoot[ArcSech[x]==x,{x,2}]

Solve[ArcSech[x]==x,x]

Reduce[ArcSech[x]==x,x,Reals]

A069814

Complex Plane

FixedPointList[Cos,-10.-10. I,3]

data=With[{dx=.02,n=5,nmax=50},Table[Quiet@FixedPointList[Cos,x+I y,nmax],{x,-n,n,dx},{y,-n,n,dx}]];//Timing

GraphicsGrid[{ArrayPlot[Abs[Map[#,data,{2}]],ColorFunction->(Hue[100#]&)]&/@{Last,Length}},ImageSize->500]

data=With[{dx=.02,n=5,nmax=50},Table[Quiet@FixedPointList[Sin,x+I y,nmax],{x,-n,n,dx},{y,-n,n,dx}]];//Timing

GraphicsGrid[{ArrayPlot[Abs[Map[#,data,{2}]],ColorFunction->(Hue[100#]&)]&/@{Last,Length}},ImageSize->500]

data=With[{dx=.01,n=1,nmax=50},Table[Quiet@FixedPointList[Tan,x+I y,nmax],{x,-n,n,dx},{y,-n,n,dx}]];//Timing

ArrayPlot[Abs[Map[#,data,{2}]],ColorFunction->(Hue[500#]&)]&/@{Last,Length}