Laplace Limit

Author

Eric W. Weisstein
July 15, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/LaplaceLimit.nb.

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Convergence Region

With[{z=x+I y},RegionPlot[Abs[((z Exp[(Sqrt[1-(z^2)])])/(1+(Sqrt[1-(z^2)])))]≤1,{x,-1,1},{y,-.7,.7},PlotPoints->40,AspectRatio->Automatic]]

With[{z=x+I y},ContourPlot[Abs[((z Exp[(Sqrt[1-(z^2)])])/(1+(Sqrt[1-(z^2)])))]==1,{x,-1,1},{y,-.7,.7},PlotPoints->40,AspectRatio->Automatic]]

Cartesian

FullSimplify[ComplexExpand[Abs[((z Exp[(Sqrt[1-(z^2)])])/(1+(Sqrt[1-(z^2)])))]/.z->x+I y],(x|y)∈Reals]

ContourPlot[(E^(((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) Arg[1-(((x+I y))^2)]])) √(((x^2)+(y^2))/(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) Arg[1-(((x+I y))^2)]]))==1,{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

1/Divide@@ComplexExpand[Through[{Re,Im}[1-(((x+I y))^2)]]]

FullSimplify[Cos[(1/2)ArcTan[-((2 x y)/(1-(x^2)+(y^2)))]],(x|y)∈Reals]

ContourPlot[(E^(((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])) √(((x^2)+(y^2))/(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]))==1,{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

(E^(((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])) √(((x^2)+(y^2))/(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]))//TraditionalForm

ContourPlot[(E^(-((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])) ==√(((x^2)+(y^2))/(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

ContourPlot[(E^(-2((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])) ==(((x^2)+(y^2))/(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

ContourPlot[(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]) ==((x^2)+(y^2))(E^(2((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]])),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

TraditionalForm[(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]) ==((x^2)+(y^2))(E^(2((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]))]

FullSimplify[FullSimplify[(Sqrt[((1+Cos[ ArcTan[x]])/2)])]/.x->((2 x y)/(-(x^2)+(y^2)+1)),(x|y)∈Reals]

(1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+2 ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]) ==((x^2)+(y^2))(E^(2((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) Cos[(1/2) ArcTan[((2 x y)/(1-(x^2)+(y^2)))]]))/._Cos:>((Sqrt[1+(1/(Sqrt[1+((4 (x^2) (y^2))/(((1-(x^2)+(y^2)))^2))]))])/(Sqrt[2]))

1/Together[1+((4 (x^2) (y^2))/(((1-(x^2)+(y^2)))^2))]

FullSimplify[1+(Sqrt[((((-1+(x^2)-(y^2)))^2)/(1-2 (x^2)+(x^4)+2 (y^2)+2 (x^2) (y^2)+(y^4)))]),(x|y)∈Reals]

ContourPlot[1+(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])+(Sqrt[2]) ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) (Sqrt[1+((1-(x^2)+(y^2))/(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)]))])==(E^((Sqrt[2]) ((((x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)))^(1/4)) (Sqrt[1+((1-(x^2)+(y^2))/(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)]))]))) ((x^2)+(y^2)),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]

With[{z=(Sqrt[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)])},ContourPlot[1+z+(Sqrt[2z])  (Sqrt[1+((1-(x^2)+(y^2))/z)])==(E^((Sqrt[2z])  (Sqrt[1+((1-(x^2)+(y^2))/z)]))) ((x^2)+(y^2)),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]]

Factor[(x^4)+2 (x^2) (-1+(y^2))+(((1+(y^2)))^2)]

Factor[(1-2 x+(x^2)+(y^2)) (1+2 x+(x^2)+(y^2))+1-x^2+y^2]

With[{z=(Sqrt[(1-2 x+(x^2)+(y^2)) (1+2 x+(x^2)+(y^2))])},ContourPlot[1+z+(Sqrt[2(z+1-(x^2)+(y^2))])==Exp[(Sqrt[2(z+1-(x^2)+(y^2))])] ((x^2)+(y^2)),{x,-1,1},{y,-.7,.7},AspectRatio->Automatic]]

Parametric

ParametricPlot[(Sqrt[((2u)/(Sinh[2u]))]){Sign[u](Sqrt[1-Sinh[u](u Cosh[u]-Sinh[u])]),(Sqrt[Sinh[u](u Cosh[u]-Sinh[u])])},{u,-1.199,1.199}]

FullSimplify[(Sqrt[((2u)/(Sinh[2u]))]){(Sqrt[1-Sinh[u](u Cosh[u]-Sinh[u])]),(Sqrt[Sinh[u](u Cosh[u]-Sinh[u])])},0<u<1.19]

Plot[{(Sqrt[u (-u+Coth[u])]),(Sqrt[u (u-Tanh[u])])},{u,0,1.199}]

Solve[u (-u+Coth[u])==0,u]

Solve[(-u+Coth[u])==0,u]

Reduce[ (-u+Coth[u])==0&&0<u<3/2,u,Reals]

ParametricPlot[{(Sqrt[u (-u+Coth[u])]),(Sqrt[u (u-Tanh[u])])},{u,0,1.199}]

Solve[Simplify[Abs[((z Exp[(Sqrt[1-(z^2)])])/(1+(Sqrt[1-(z^2)])))]/.z->I y,0<y<1]==1,y]

FullSimplify[(E^(Sqrt[1+(y^2)])) y-1+(Sqrt[1+(y^2)])]

Reduce[(((E^(Sqrt[1+(y^2)])) y)/(1+(Sqrt[1+(y^2)])))==1,y,Reals]

Reduce[(((E^(Sqrt[1+(y^2)])) y)/(1+(Sqrt[1+(y^2)])))==1,y,Reals]

umax=Root[{-Coth[#1]+#1&,1.1996786402577338339163698486411419442614587884186072089166`30.102999566398122}];

FullSimplify[(Sqrt[((2u)/(Sinh[2u]))])/.u->umax]

N[%,20]

umax=N[u /.FindRoot[Sinh[u](u Cosh[u]-Sinh[u])==1,{u,1.}],20]

N[Sqrt[2u/Sinh[2u]] /. u->umax,20]

umax=1.19967874;

f[u_]:={Sign[u]Sqrt[1-Sinh[u](u Cosh[u]-Sinh[u])],Sqrt[Sinh[u](u Cosh[u]-Sinh[u])]}

g[u_]:={{1,0},{0,-1}}.f[u]

Show[{
	ParametricPlot[Sqrt[2u/Sinh[2u]]f[u],{u,-umax,umax},
			AspectRatio->Automatic,DisplayFunction->Identity],
	ParametricPlot[Sqrt[2u/Sinh[2u]]g[u],{u,-umax,umax},
			AspectRatio->Automatic,DisplayFunction->Identity]},
	DisplayFunction->$DisplayFunction
]

Properties

Area

With[{z=(Sqrt[((((1-x))^2)+(y^2)) ((((1+x))^2)+(y^2))])},RegionPlot[1+z+(Sqrt[2(z+1-(x^2)+(y^2))])>Exp[(Sqrt[2(z+1-(x^2)+(y^2))])] ((x^2)+(y^2)),{x,-1,1},{y,-.7,.7}]]

FullSimplify[Abs[((z Exp[(Sqrt[1-(z^2)])])/(1+(Sqrt[1-(z^2)])))]≤1/.z->x +I y,(x|y)∈Reals]

NIntegrate[Boole[(E^(Re[(Sqrt[1-(((x+I y))^2)])])) Abs[((x+I y)/(1+(Sqrt[1-(((x+I y))^2)])))]≤1],{x,-2,2},{y,-2,2}]//Timing

NIntegrate[Boole[(E^(Re[(Sqrt[1-(((x+I y))^2)])])) Abs[((x+I y)/(1+(Sqrt[1-(((x+I y))^2)])))]≤1],{x,-2,2},{y,-2,2},PrecisionGoal->4,WorkingPrecision->20]//Timing

With[{z=(Sqrt[((((1-x))^2)+(y^2)) ((((1+x))^2)+(y^2))])},NIntegrate[Boole[1+z+(Sqrt[2(z+1-(x^2)+(y^2))])>Exp[(Sqrt[2(z+1-(x^2)+(y^2))])] ((x^2)+(y^2))],{x,-1,1},{y,-.7,.7}]]//Timing

With[{z=(Sqrt[((((1-x))^2)+(y^2)) ((((1+x))^2)+(y^2))])},NIntegrate[Boole[Log[1+z+(Sqrt[2(z+1-(x^2)+(y^2))])]>(Sqrt[2(z+1-(x^2)+(y^2))])+Log[((x^2)+(y^2))]],{x,-1,1},{y,-.7,.7}]]//Timing

Integrate[Boole[(E^(Re[(Sqrt[1-(((x+I y))^2)])])) Abs[((x+I y)/(1+(Sqrt[1-(((x+I y))^2)])))]≤1],{x,-2,2},{y,-2,2}]//Timing

...hangs...

With[{z=(Sqrt[((((1-x))^2)+(y^2)) ((((1+x))^2)+(y^2))])},Integrate[Boole[1+z+(Sqrt[2(z+1-(x^2)+(y^2))])>Exp[(Sqrt[2(z+1-(x^2)+(y^2))])] ((x^2)+(y^2))],{x,-1,1},{y,-.7,.7}]]//Timing

Curvature

Curvature[With[{z=Sqrt[(1-2 x+x^2+y^2) (1+2 x+x^2+y^2)]},1+z+Sqrt[2 (z+1-x^2+y^2)]==Exp[Sqrt[2 (z+1-x^2+y^2)]] (x^2+y^2)],{x,y}]//FullSimplify

Slope

Slope[With[{z=Sqrt[(1-2 x+x^2+y^2) (1+2 x+x^2+y^2)]},1+z+Sqrt[2 (z+1-x^2+y^2)]==Exp[Sqrt[2 (z+1-x^2+y^2)]] (x^2+y^2)],{x,y}]//FullSimplify//Timing

ByteCount/@%69[[2]]

Function

f[x_]:=((x Exp[(Sqrt[1+(x^2)])])/(1+(Sqrt[1+(x^2)])))

Plot[f[x],{x,0,1},PlotStyle->Red,Epilog->{Blue,Dashing[{.02}],Line[{{#,0},{#,1},{0,1}}]&[0.66274341934]},AxesLabel->TraditionalForm/@{x,HoldForm[f[x]]}]

Value

FindRoot

FindRoot[f[x]-1,{x,1},WorkingPrecision->50]

FindRoot[f[x]-1,{x,1},WorkingPrecision->1000]

Transcendental Root

V7

laplace=Reduce[((x Exp[(Sqrt[1+(x^2)])])/(1+(Sqrt[1+(x^2)])))==1,x,Reals][[2]]

N[Root[{1-E^Sqrt[1+#1^2] #1+Sqrt[1+#1^2]&,0.6627434193491815810}],100]

Element[laplace,Algebraics]

Sqrt[x^2-1]/. FindRoot[x==Coth[x],{x,1}

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

Sqrt[(NumericalApproximation[Root, (RowDefault["1.20", FEExpressionsNumericalApproximationElider]), Root[{-1+#1-(E^(-2 #1)) (1+#1)&,1.19967864025773383391605572375129828977`20.3060665829672}], ])^2-1]

N[%,20]

Sqrt[-1 + Root[{#-(1+#)/E^(2#)-1 &, 1.1996786}]^2]

N[%,20]

RealDigits[Sqrt[ Root[{#-(1+#)/E^(2#)-1 &, 1.1996786}]^2-1],10,100][[1]]