Einstein Functions

Author

Eric W. Weisstein
April 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/EinsteinFunctions.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Code

Einstein[1,x_]:=x^2 Exp[x]/(Exp[x]-1)^2
Einstein[2,x_]:=x/(Exp[x]-1)
Einstein[3,x_]:=Log[1-Exp[-x]]
Einstein[4,x_]:=x/(Exp[x]-1)-Log[1-Exp[-x]]

Plot

MakeBoxes[label[n_],TraditionalForm]:=RowBox[{SubscriptBox["E",MakeBoxes[n,TraditionalForm]],"(","x",")"}]

Internal`DeactivateMessages[
Show[GraphicsArray[Partition[Block[{$DisplayFunction=Identity},Plot[Einstein[#,x],{x,-5,5},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,label[#]}]&/@Range[4]
],2]],ImageSize->500]
]

Inflection Points

(Subscript[E, 1])(x)

d2e1=FullSimplify[D[Einstein[1,x],{x,2}]]

Plot[d2e1,{x,-3,3},PlotStyle->Red]

Solve[d2e1==0,x]

Solve[((2+(x^2)) Cosh[x]+2 (-1+(x^2)-2 x Sinh[x]))==0,x]

Solve[TrigToExp[((2+(x^2)) Cosh[x]+2 (-1+(x^2)-2 x Sinh[x]))]==0,x]

FindRoot[((2+(x^2)) Cosh[x]+2 (-1+(x^2)-2 x Sinh[x])),{x,2},WorkingPrecision->200]

Transcendental Root

V6.1

Reduce[((2+(x^2)) Cosh[x]+2 (-1+(x^2)-2 x Sinh[x]))==0&&0<x<10,x,Reals]

Reduce[((2+(x^2)) Cosh[x]+2 (-1+(x^2)-2 x Sinh[x]))==0&&-10<x<10,x,Reals]

(Subscript[E, 2])(x)

d2e2=FullSimplify[D[Einstein[2,x],{x,2}]]

Plot[d2e2,{x,-10,10},PlotStyle->Red,PlotRange->{0,.2}]

(Subscript[E, 3])(x)

d2e3=FullSimplify[D[Einstein[3,x],{x,2}]]

Plot[d2e3,{x,-3,3},PlotStyle->Red,PlotRange->{-10,1}]

(Subscript[E, 4])(x)

d2e4=FullSimplify[D[Einstein[4,x],{x,2}]]

Plot[d2e4,{x,-3,3},PlotStyle->Red,PlotRange->{0,10}]